stability of neuronal pulses composed of concatenated unstable kinks we demonstrate that a traveling pulse solution emerging from the concatenation of two unstable kinks can be stable by means of stability analysis and numerical simulations we show the stability of neuronal pulses action potentials with increasing refractory periods which decompose into two radiationally unstable kinks in the limit these action potentials are solutions of an ultrarefractory version of the fitzhugh nagumo system

nonlinear finite dimensional control of a class of nonlinear distributed parameter systems using residual mode filters a proof of local exponential stability abstract in any attempt to control an infinite dimensional distributed parameter system dps with a finite dimensional controller some compensation for deleterious controller interaction with unmodeled modes is needed we introduce the concepts of both a finite dimensional nonlinear controller based on any reasonable nonlinear reduced order model of the dps and stability compensation by a finite dimensional nonlinear residual mode filter under appropriate lipschitz continuity conditions on the nonlinearities this new pair of interconnected finite dimensional systems will produce closed loop exponentially stable control of the nonlinear dps

pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction diffusion model we address the stability of solitary pulses as well as some other traveling structures near the onset of spatiotemporal chaos in a two species reaction diffusion model describing the oxidation of c

feedback mediated control of spiral waves abstract a theoretical approach is developed which allows one to describe the drift of a spiral wave mediated by a feedback signal taken as an integral of a system variable over a certain domain of an excitable medium the underlying reaction diffusion equations are reduced to an autonomous system of ordinary differential equations for the coordinates of the core center of the spiral wave a new method to determine the drift velocity field of the spiral core is proposed based on a superposition principle this approach is applied to analyze the drift velocity field for a feedback signal generated from a one point detector and from integration domains of different size and shape including global feedback it is shown that a variation of the shape of the integration domain can induce bifurcations that change the attractor structure of the drift velocity field these theoretical predictions are in good quantitative agreement with numerical simulations performed for the oregonator model and the complex ginzburg landau equation the results are also confirmed by experiments with the light sensitive belousov zhabotinsky reaction

stabilization of a coupled transport diffusion system with boundary input we consider the problem of stabilizing a coupled transport diffusion system with boundary input the system is described by two linear transport diffusion equations and is not asymptotically stable in order to stabilize the system with boundary input sensor influence functions are assumed to be located at interior of the domain first we formulate the system as an evolution equation with unbounded output operators in a hilbert space using variable transformation next we derive a reduced order model with a finite dimensional state variable for the infinite dimensional system then a stabilizing controller is constructed for the reduced order model under an additional assumption it is shown that the finite dimensional controller together with a residual mode filter plays a role of a finite dimensional stabilizing controller for the original infinite dimensional system if the order of the residual mode filter is chosen sufficiently large finally the validity of the design method is demonstrated through a numerical simulation

global aspects of turbulence induced by heteroclinic cycles in competitive diffusion lotka volterra equation abstract this paper presents detailed numerical results of the competitive diffusion lotka volterra equation may leonard type first we derive the global phase diagrams of attractors in the parameter space including the system size where transition lines between simple attractors are clearly obtained in accordance with the results of linear stability analysis but the transition borders become complex when multi basin structures appear the complex aspects of the transition borders are studied in the case when the system size decreases next we show the statistical aspects of the turbulence with special attention to the onset of the supercritical hopf bifurcation several characteristic quantities such as correlation length correlation time lyapunov spectra and lyapunov dimension are investigated in detail near the onset of turbulence our data show the critical scaling law near the onset only in the restricted parameter domain however even when the critical indices are not determined accurately it is shown that the empirical scaling relations are obtained in a wide parameter domain far from the onset point and those scaling indices satisfy several relations these scaling relations are discussed in comparison with the result derived by the phase reduction method lastly we make a conjecture about the stability of an ecosystem based on the bifurcation diagram the ecosystem obeying the lotka volterra equation in the case of may leonard type is stabilized more as the system size increases

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

quasi equilibrium grid algorithm geometric construction for model reduction the method of invariant grid mig is an iterative procedure for model reduction in chemical kinetics which is based on the notion of slow invariant manifold sim a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 a n gorban i v karlin a y zinovyev invariant grids for reaction kinetics physica a 333 2004 106 154 a n gorban i v karlin invariant manifolds for physical and chemical kinetics lecture notes physics 660 springer berlin heidelberg 2005 doi 10 1007 b98103 important role in that method is played by the initial grid which once refined gives a description of the invariant manifold the invariant grid a convenient way to get a first approximation of the sim is given by the spectral quasi equilibrium manifold sqem a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 in the present paper a flexible numerical method to construct the discrete analog of a quasi equilibrium manifold in any dimension is presented that object is named quasi equilibrium grid qeg while the procedure quasi equilibrium grid algorithm qega extensions of the qem notion are also suggested the qega is a numerical tool which can be used to find a grid based approximation for the locus of minima of a convex function under some linear constraints the method is validated by construction of one and two dimensional grids for a model of hydrogen oxidation reaction

targeting characteristic wave properties in reaction diffusion systems by optimization of external forcing we consider the targeted manipulation of reaction diffusion waves by optimization of an external forcing parameter as an example we present numerical results for the fitzhugh nagumo system exploiting model based optimization capable of targeting characteristic wave properties such as wavelength shape and propagation speed by spatiotemporally controlling electric current the conceptual basis of our approach is optimal control of periodic orbits in a wave variable coordinate system the results are transferred back to the partial differential equation context and validated in numerical simulations the whole procedure is applicable to any reaction diffusion model

t points a codimension two heteroclinic bifurcation the local bifurcation structure of a heteroclinic bifurcation which has been observed in the lorenz equations is analyzed the existence of a particular heteroclinic loop at one point in a two dimensional parameter space a t point implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits as well as countably many other topologically more complicatedt points in a small neighborhood in parameter space

robust feed back control of travelling waves in a class of reaction diffusion distributed biological systems abstract reaction diffusion rd mechanisms can describe many biological phenomena such as neuron firing in the brain the heartbeat cellular organization activities or even biological disorders such as fibrillation the fitzhugh nagumo fhn model is a particular case of rd systems it is able to capture the key features of many biological processes and since it is relatively simple it has been widely employed during recent years some examples of its predictive capabilities include the representation of the normal behavior of some physiological phenomena related to a travelling plane wave as well as biological disorders associated with spiral or irregular fronts the objective of this work is to design a control law that is able to stabilize complex behaviors travelling plane wave in biological systems using the fhn model as a case study since in biological systems there usually exists a lack of detailed information on the system structure our control law will be designed to be robust i e it must be able to reach the predefined reference regardless the presence of structural uncertainties to this purpose we will extend some classical results on the finite dimensional robust control theory to rd systems by means of order reduction techniques in particular the proper orthogonal decomposition method

feedback control of the kuramoto sivashinsky equation abstract this work focuses on linear finite dimensional output feedback control of the kuramoto sivashinsky equation kse with periodic boundary conditions under the assumption that the linearization of the kse around the zero solution is controllable and observable linear finite dimensional output feedback controllers are synthesized that achieve stabilization of the zero solution for any value of the instability parameter the controllers are synthesized on the basis of finite dimensional approximations of the kse which are obtained through galerkin s method the performance of the controllers is successfully tested through computer simulations

a galerkin neural network based design of guaranteed cost control for nonlinear distributed parameter systems this paper presents a galerkin neural network based guaranteed cost control gcc design for a class of parabolic partial differential equation pde systems with unknown nonlinearities a parabolic pde system typically involves a spatial differential operator with eigenspectrum that can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement motivated by this in the proposed control scheme galerkin method is initially applied to the pde system to derive an ordinary differential equation ode system with unknown nonlinearities which accurately describes the dynamics of the dominant slow modes of the pde system the resulting nonlinear ode system is subsequently parameterized by a multilayer neural network mnn with one hidden layer and zero bias terms then based on the neural model and a lure type lyapunov function a linear modal feedback controller is developed to stabilize the closed loop pde system and provide an upper bound for the quadratic cost function associated with the finite dimensional slow system for all admissible approximation errors of the network the outcome of the gcc problem is formulated as a linear matrix inequality lmi problem moreover by using the existing lmi optimization technique a suboptimal guaranteed cost controller in the sense of minimizing the cost bound is obtained finally the proposed design method is applied to the control of the temperature profile of a catalytic rod

h _ infty fuzzy observer based control for a class of nonlinear distributed parameter systems with control constraints an hinfin fuzzy observer based control design is proposed for a class of nonlinear parabolic partial differential equation pde systems with control constraints for which the eigenspectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement in the proposed control scheme galerkin s method is initially applied to the pde system to derive a nonlinear ordinary differential equation ode system that accurately describes the dynamics of the dominant slow modes of the pde system the resulting nonlinear ode system is subsequently represented by the takagi sugeno t s fuzzy model then based on the t s fuzzy model a fuzzy observer based controller is developed to stabilize the nonlinear pde system and achieve an optimized hinfin disturbance attenuation performance for the finite dimensional slow system while control constraints are respected the outcome of the hinfin fuzzy observer based control problem is formulated as a bilinear matrix inequality bmi optimization problem a local optimization algorithm that treats the bmi as a double linear matrix inequality is presented to solve this bmi optimization problem finally the proposed design method is applied to the control of the temperature profile of a catalytic rod to illustrate its effectiveness

robust stabilization of the distributed parameter system with time delay via fuzzy control in this paper stabilization of the distributed parameter system dps with time delay is studied using galerkin s method and fuzzy control with the help of galerkin s method the dynamics of dps with time delay can be first converted into a group of low order functional ordinary differential equations which will be used for design of the robust fuzzy controller the fuzzy controller designed can guarantee exponential stability of the closed loop dps some sufficient conditions are derived for the stabilization together with the linear matrix inequality design approach the effectiveness of the proposed control design methodology is demonstrated in numerical simulations

second order and stability analysis for optimal sparse control of the fitzhugh nagumo equation optimal sparse control problems are considered for the fitzhugh nagumo system including the so called schlogl model the nondifferentiable objective functional of tracking type includes a quadratic tikhonov regularization term and the l 1 norm of the control that accounts for the sparsity though the objective functional is not differentiable a theory of second order sufficient optimality conditions is established for tikhonov regularization parameter nu 0 and also for the case nu 0 in this context also local minima are discussed that are strong in the sense of the calculus of variations the second order conditions are used as the main assumption for proving the stability of locally optimal solutions with respect to nu to 0 and with respect to perturbations of the desired state functions the theory is confirmed by numerical examples that are resolved with high precision to confirm that the optimal solution obeys the system of necessary optimality conditions

accumulations of t points in a model for solitary pulses in an excitable reaction diffusion medium we consider a family of differential equations that describes traveling waves in a reaction diffusion equation modeling oxidation of carbon oxide on a platinum surface near the onset of spatio temporal chaos the organizing bifurcation for the bifurcation structure with small carbon oxide pressures turns out to be a codimension 3 bifurcation involving a homoclinic orbit to an equilibrium undergoing a transcritical bifurcation we show how infinitely many t point bifurcations of multi loop heteroclinic cycles occur in the unfolding

the accumulation of eigenvalues in a stability problem abstract when two waves propagating in a one dimensional medium are locked together as a composite wave a natural question arises as to whether the new wave is stable an interesting and novel instability mechanism is exposed here in which a cascade of eigenvalues accumulates at a distinguished point in the unstable half plane the underlying assumption is that the transition between the two waves occurs at an unstable homogeneous steady state of the partial differential equations this causes the individual waves to have an unstable continuous spectrum but the instability of the full wave cannot be predicted from the configuration of these spectra alone

lq optimal boundary control of infinite dimensional systems with yosida type approximate boundary observation a class of boundary control systems with boundary observation is considered for which the unbounded operators often lead to technical difficulties an extended model for this class of systems is described and analyzed which involves no unbounded operator except for the dynamics generator a method for the resolution of the lq optimal control problem for this model is described and the solution provides a stabilizing feedback for the nominal system with unbounded operators in the sense that in closed loop the state trajectories converge to zero exponentially fast the model consists of an extended abstract differential equation whose state components are the boundary input the state up to an affine transformation and a yosida type approximation of the output of the nominal system it is shown that under suitable conditions the model is well posed and in particular that the dynamics operator is the generator of a c 0 semigroup moreover the model is shown to be observable and to carry controllability stabilizability and detectability properties from the nominal system a general method of resolution based on the problem of spectral factorization of a multi dimensional operator valued spectral density is described in order to solve a lq optimal control problem for this model it is expected that this approach will lead hopefully to a good trade off between the cost of modeling and the efficiency of methods of resolution of control problems for such systems

non linear feedback control of parabolic partial differential difference equation systems this paper proposes a general method for the synthesis of non linear output feedback controllers for single input singleoutput quasi linear parabolic partial differential difference equation pdde systems for which the eigenspectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement initially a non linear model reduction scheme which is based on combination of galerkin s method with the concept of approximate inertial manifold is employed for the derivation of differential difference equation dde systems that describe the dominant dynamics of the pdde system then these dde systems are used as the basis for the explicit construction of non linear output feedback controllers through combination of geometric and lyapunov techniques the controllers guarantee stability and enforce output tracking in the closed loop parabolic pdde system independently of the size of the state delay provided that the separation o

transient spatiotemporal chaos is extensive in three reaction diffusion networks extensive asymptotic spatiotemporal chaos is comprised of statistically similar subsystems that interact only weakly a systematic study of transient spatiotemporal chaos reveals extensive system behavior in all three reaction diffusion networks for various boundary conditions the lyapunov dimension the sum of positive lyapunov exponents and the logarithm of the transient lifetime grow linearly with the system size the unstable manifold of the chaotic saddle has nearly the same dimension as the saddle itself and the stable manifold is nearly space filling

on locating connecting orbits abstract an extension of a method for computing connecting orbits that locates such orbits in a systematic manner is presented as an application we compute homoclinic orbits in a singular perturbation problem

traveling waves of calcium in a pancreatic acinar cells model construction and bifurcation analysis abstract we construct and study a model for intracellular calcium wave propagation with particular attention to pancreatic acinar cells the model is based on a model of the inositol trisphosphate ip3 receptor which assumes that calcium modulates the binding affinity of ip3 to the receptor two versions of the model one simpler than the other are studied numerically in both versions solitary waves in the excitable regime arise via homoclinic bifurcations in the traveling wave equations as the background concentration of ip3 is increased the wave speed increases and for some values of the ip3 concentration the initial pulse gives rise to secondary pulses that travel in both directions this can give rise to irregular spatio temporal behavior or to trains of pulses in the simpler model these secondary waves are related to the presence of a t point a heteroclinic cycle and an associated spiral of homoclinic orbits which terminate the branch of homoclinic orbits

the set of positive solutions of semilinear equations in large balls the exact number of positive solutions of u f u 0 on finite balls in n is determined the assumptions about f u are similar to those imposed by serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of n see 7 8 for finite balls of sufficiently large radius it is shown here that there are exactly two positive and hence radial solutions to this end we first prove the linear nondegeneracy of the positive solution of n this is obtained by applying the technique of monotone separation of graphs 7 to the linearised equations somewhat sharper estimates are required here see part i section 2

numerical bifurcation and stability analysis of solitary pulses in an excitable reaction diffusion medium abstract we present a systematic computer assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space the reaction diffusion model used is of the activator inhibitor type only the activator diffuses in this medium the control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold this study focuses on travelling pulses their domain of existence and the bifurcations that render them unstable these pulses are approximated as a homoclinic orbits in a travelling wave ode frame and b as solutions of the full partial differential equation pde with periodic boundary conditions in large domains a variety of bifurcations in the travelling wave ode frame are observed including heteroclinic loops so called t points a r champneys and y a kuznetsov numerical detection and continuation of codimension 2 homoclinic bifurcations int j bif chaos 4 1994 785 h kokobu homoclinic and heteroclinic bifurcations of vectorfields japan j appl math 5 1988 455 instabilities in the full pde frame include both hopf bifurcations to modulated travelling waves involving the discrete pulse spectrum as well as transitions involving the continuous spectrum such as the so called backfiring transition m bar m hildebrand m eiswirth m falcke h engel and m neufeld chemical turbulence and standing waves in a surface reaction model the influence of global coupling and wave instabilities chaos 4 1994 499 the stability of modulated pulses is computed through numerical floquet analysis and a cascade of period doubling bifurcations is observed as well as certain global bifurcations these results corroborated by observations from direct numerical integration provide a skeleton around which many features of the overall complex spatiotemporal dynamics of the pde are organized

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

metrics with conical singularities on the sphere and sharp extensions of the theorems of landau and schottky an explicit formula for the generalized hyperbolic metric on the thrice punctured sphere mathbb p backslash z 1 z 2 z 3 with singularities of order j 1 at zj is obtained in all possible cases 1 2 3 2 the existence and uniqueness of such a metric was proved long time ago by picard j reine angew math 130 243 258 1905 and heins nagoya math j 21 1 60 1962 while explicit formulas for the cases 1 2 1 were given earlier by agard ann acad sci fenn ser a i 413 1968 and recently by anderson et al math z to appear doi 10 1007 s00209 009 0560 5 we also establish precise and explicit lower bounds for the generalized hyperbolic metric this extends work of hempel j lond math soc ii ser 20 435 445 1979 and minda complex var 8 129 144 1987 as applications sharp versions of landau and schottky type theorems for meromorphic functions are obtained

nonlinear feedback control of parabolic partial differential equation systems with time dependent spatial domains partial differential equations pdes with time dependent spatial domains whose dynamics can be partitioned into slow and fast ones initially a nonlinear model reduction scheme similar to the one introduced in christofides and daoutidis j math anal appl 216 1997 398 420 which is based on combinations of galerkin s method with the concept of approximate inertial manifold is employed for the derivation of low order ordinary differential equation ode systems that yield solutions which are close up to a desired accuracy to the ones of the pde system for almost all times then these ode systems are used as the basis for the explicit construction of nonlinear time varying output feedback controllers via geometric control methods the controllers guarantee stability and enforce the output of the closed loop parabolic pde system to follow up to a desired accuracy a prespecified response for almost all times provided that the separation of the slow and fast dynamics is sufficiently large differences in the nature of the model reduction and control problems between parabolic pde systems with fixed and moving spatial domains are identified and discussed the proposed control method is used to stabilize an unstable steady state of a diffusion reaction process whose spatial domain changes with time it is shown to lead to a significant reduction on the order of the stabilizing nonlinear output feedback controller and outperform a nonlinear controller synthesis method that does not account for the variation of the spatial domain q 1999 academic press

brief analysis and control of parabolic pde systems with input constraints this paper develops a general framework for the analysis and control of parabolic partial differential equations pde systems with input constraints initially galerkin s method is used for the derivation of ordinary differential equation ode system that capture the dominant dynamics of the pde system this ode systems are then used as the basis for the synthesis via lyapunov techniques of stabilizing bounded nonlinear state and output feedback control laws that provide an explicit characterization of the sets of admissible initial conditions and admissible control actuator locations that can be used to guarantee closed loop stability in the presence of constraints precise conditions that guarantee stability of the constrained closed loop parabolic pde system are provided in terms of the separation between the fast and slow eigenmodes of the spatial differential operator the theoretical results are used to stabilize an unstable steady state of a diffusion reaction process using constrained control action

on the monotonicity of the time map the equation au y u y 0 on the unit ball with homogeneous linear boundary conditions has been much studied cf 1 2 and the references therein the change of variable x l i transforms this equation to mx a 0 1x1 r 1 1 where r fi for radial solutions of this equation lying in say the kth nodal class r depends only on p u o r t p this function which describes how the size of the balls on which radial solutions exist vary with the quantity u o turns out to be an important function determining much of the behavior of the solutions thus if we consider 1 1 together with the boundary conditions cu x i du x dn 0 1x1 r

stability of distributed parameter systems with finite dimensional controller compensators using singular perturbations abstract using a singular perturbation formulation of the linear time invariant distributed parameter system we develop a method to design finite dimensional feedback compensators of any fixed order which will stabilize the infinite dimensional distributed parameter system the synthesis conditions are given entirely in terms of a finite dimensional reduced order model the stability results depend on an infinite dimensional version of the klimushchev krasovskii lemma presented here

from quasi 2d to 3d turing patterns in ramped systems abstract we elaborate on the transition from quasi two dimensional to three dimensional turing patterns in a chemical reaction diffusion system confined in gradients of chemicals between two feed boundaries this transition is observed in open spatial reactors specially designed to make possible the unfolding of a pattern sequence in one direction of the plane of observation in this direction the confinement of the structure is progressively relaxed complementary observations from two reactor geometries allow the dimensionality of the structure to be elucidated quasi two dimensional and three dimensional patterns respectively correspond to patterns developing in monolayers and in bilayers beyond the now classical hexagonal and stripe patterns various new stable planforms are shown to result from the coupling of these two classical pattern modes which develop in two adjacent layers with well defined phase relations between the two pattern modes

reduced order optimal control of the convective fitzhugh nagumo equations abstract in this paper we compare three model order reduction methods the proper orthogonal decomposition pod discrete empirical interpolation method deim and dynamic mode decomposition dmd for the optimal control of the convective fitzhugh nagumo fhn equations the convective fhn equations consist of the semi linear activator and the linear inhibitor equations modeling blood coagulation in moving excitable media the semilinear activator equation leads to a non convex optimal control problem ocp the most commonly used method in reduced optimal control is pod we use deim and dmd to approximate efficiently the nonlinear terms in reduced order models we compare the accuracy and computational times of three reduced order optimal control solutions with the full order discontinuous galerkin finite element solution of the convection dominated fhn equations with terminal controls numerical results show that pod is the most accurate whereas pod dmd is the fastest

a direct method for the computation of hopf bifurcation points we present a method for the numerical computation of a hopf bifurcation point of an evolution equation an augmented time independent system of equations has to be solved for which the hopf bifurcation point is an isolated solution in the case of a system of ordinary differential equations we propose an efficient implementation of newton s method for the solution of the resulting system of algebraic equations if the method is applied directly to a partial differential system with one space variable a two point boundary value problem is obtained we also compare this technique with a related method and results are given for two systems of partial differential equations describing chemical reacting systems

local stability theory of solitary pulses in an active medium abstract if the spatio temporal chaos of an extended active medium is driven by repulsive or attractive interaction among localized coherent structures such as kinks and pulses its dynamics can often be described by a low dimensional dynamical system this welcomed reduction in the degrees of freedom only occurs when the localized structures remain stable during their mutual interaction it is shown that the structures can be destroyed by two types of local disturbance an expanding radiation packet and a localized disturbance which travels with the structure these two instabilities and the bifurcations they induce select the proper coherent structure and specify the dynamics of the generalized kuramoto sivashinsky gks equation in an extended domain the possibility of describing pulse radiation interaction by resonance poles also suggests that a more general coherent structure theory can be developed

layer oscillations in reaction diffusion systems this paper considers a two component system of reaction diffusion equations involving two parameters varepsilon and tau varepsilon tau u 1 varepsilon 2 u xx f u v qquad v

on stability of two degenerate reaction diffusion systems abstract in this paper we construct a partially degenerate reaction diffusion equation subject to the neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable in this way we solve an open problem proposed by casten and holland 1977 4 moreover we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross diffusion effects which cannot be determined by a simple spectral analysis based on the well developed semigroup theory

an application of inertial manifold theory to boundary stabilization of semilinear diffusion systems this paper deals with an application of inertial manifold theory to boundary stabilization of semilinear diffusion systems on a one dimensional domain it is shown that a sakawa type of controller plays an important role in the construction of an inertial manifold for the closed loop system with a given semilinear diffusion system this means that the use of a sakawa type of controller reduces the stabilization problem for the closed loop system to the one on the inertial manifold

finite dimensional control of parabolic pde systems using approximate inertial manifolds abstract this paper introduces a methodology for the synthesis of nonlinear finite dimensional output feedback controllers for systems of quasi linear parabolic partial differential equations pdes for which the eigenspectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement combination of galerkin s method with a novel procedure for the construction of approximate inertial manifolds for the pde system is employed for the derivation of ordinary differential equation ode systems whose dimension is equal to the number of slow modes that yield solutions which are close up to a desired accuracy to the ones of the pde system for almost all times these ode systems are used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed loop system to follow up to a desired accuracy a prespecified response for almost all times

turbulence in diffusion replicator equation dynamical behaviors in the diffusion replicator equation of three species are numerically studied we point out the significant role of the heteroclinic cycle in the equation and analyze the details of the turbulent solution that appeared in this system firstly the bifurcation diagram for a certain parameter setting is drawn then it is shown that the turbulence appears with the supercritical hopf bifurcation of a stationary uniform solution and it disappears under a subcritical type bifurcation secondly the statistical property of the turbulence near the supercritical hopf onset point is analyzed precisely further the correlation lengths and correlation times obey some characteristic scaling laws

large amplitude stationary waves in an excitable lateral inhibitory medium pattern formation in a system of reaction diffusion equations which exhibit both lateral inhibition and excitability is investigated the system studied is derived from the fitzhugh nagumo nerve conduction model the existence of solitary and periodic stationary wave solutions is proved numerical computations show that some of these waves are stable as solutions of the partial differential equations further arguments are given which indicate that these patterns arise as a result of excitability of the system and not as a result of a turing mechanism

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

model reduction of parabolic pdes using multivariate splines abstracta new methodology is presented for model reduction of linear parabolic partial differential equations pdes on general geometries using multivariate splines on triangulations state space descriptions are derived that can be used for control design this method uses galerkin projection with b splines to derive a finite set of ordinary differential equations odes any desired smoothness conditions between elements as well as the boundary conditions are flexibly imposed as a system of side constraints on the set of odes projection of the set of odes on the null space of the system of side constraints naturally produces a reduced order model that satisfies these constraints this method can be applied for both in domain control and boundary control of parabolic pdes with spatially varying coefficients on general geometries the reduction method is applied to design and implement feedback controllers for stabilisation of a 1 d unstable heat equation and a more challenging 2 d reaction convection d

stationary wave solutions of a system of reaction diffusion equations derived from the fitzhugh nagumo equations we consider an extension of the fitzhugh nagumo model namely the system u t d 1 u xx f u w qquad w t d 2 w xx varepsilon u gamma w where varepsilon 0 gamma 0 d 1 0 d 2 0 and f u is cubic we allow gamma to be large which implies that there are three constant solutions we show that over an appropriate range of parameters the system has time independent pulse solutions and an infinite number of periodic solutions depending on the particular choice of parameters we show that the pulse solution leads to either the first constant solution or the third constant solution

finite dimensional approximation of nonlinear problems part ii limit points note univ paris 6 f 75230 paris 05 france ecole polytech ctr math appl cnrs equipe rech 747 f 91128 palaiseau france univ pavia cnr ist anal numer i 27100 pavia italy brezzi f univ pavia cnr ist math appl i 27100 pavia italy isi document delivery no lr761times cited 54cited reference count 13 reference asn article 1981 002doi 10 1007 bf01396184 record created on 2006 08 24 modified on 2017 05 12

slow switching in globally coupled oscillators robustness and occurrence through delayed coupling the phenomenon of slow switching in populations of globally coupled oscillators is discussed this characteristic collective dynamics which was first discovered in a particular class of the phase oscillator model is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population we argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom we also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators although a heteroclinic loop might seem to arise only exceptionally we find that it appears rather easily by introducing time delay into a population which would otherwise exhibit perfect phase synchrony we argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle node bifurcations slow switching arises when a system with a heteroclinic loop is weakly perturbed this will be demonstrated with a vector model by applying weak noises other types of weak symmetry breaking perturbations can also cause slow switching

on the bifurcation curve for an elliptic system of fitzhugh nagumo type abstract we study a system of partial differential equations derived from the fitzhugh nagumo model in one dimension solutions are required to satisfy zero dirichlet boundary conditions on the interval 1 1 estimates are given to describe bounds on the range of parameters over which solutions exist numerical computations provide the global bifurcation diagram for families of symmetric and asymmetric solutions in the two dimensional case we use numerical methods for zero dirichlet boundary conditions on the square domain 1 1 1 1 numerical computations are given both for symmetric and asymmetric and for stable and unstable solutions

dynamics and control of the 2 d navier stokes equations abstract this paper deals with the dynamics and control of the two dimensional 2 d navier stokes n s equations with a spatially periodic and temporally steady forcing term first we construct a dynamical system of nine nonlinear differential equations by fourier expansion and truncation of the 2 d n s equations then we study the dynamics of the obtained reduced order system by analyzing the system s attractors for different values of the reynolds number r e by applying the symmetry of the equations on one of the system s attractors a symmetric limit trajectory that is part of the dynamics is obtained moreover a lyapunov based control strategy to control the dynamics of the system for a given r e is designed finally numerical simulations are undertaken to validate the theoretical developments

a model for fast computer simulation of waves in excitable media abstract starting from a two variable system of reaction diffusion equations an algorithm is devised for efficient simulation of waves in excitable media the spatio temporal resolution of the simulation can be varied continuously for fine resolutions the algorithm provides accurate solution of the underlying reaction diffusion equations for coarse resolutions the algorithm provides qualitative simulations at small computational cost

a sampled data singularly perturbed boundary control for a heat conduction system with noncollocated observation this note presents a sampled data strategy for a boundary control problem of a heat conduction system modeled by a parabolic partial differential equation pde using the zero order hold the control law becomes a piecewise constant signal in which a step change of value occurs at each sampling instant through the dasialiftingpsila technique the pde is converted into a sequence of constant input problems to be solved individually for a sampled data formulation the eigenspectrum of the parabolic system can be partitioned into two groups a finite number of slow modes and an infinite number of fast modes which is studied via the theory of singular perturbations controllability and observability of the sampled data system are preserved irrelevant to the sampling period a noncollocated output feedback design based upon the state observer is employed for set point regulation the state observer serves as an output feedback compensator with no static feedback directly from the output satisfying the so called dasialow pass propertypsila the feedback controller is thus robust against the observation error due to the neglected fast modes

homoclinic bifurcations in reversible hamiltonian systems abstract we study the existence of homoclic solutions for reversible hamiltonian systems taking the family of differential equations u iv au u f u b 0 as a model where f is an analytic function and a b real parameters these equations are important in several physical situations such as solitons and in the existence of finite energy stationary states of partial differential equations but no assumptions of any kind of discrete symmetry is made and the analysis here developed can be extended to others hamiltonian systems and successfully employed in situations where standard methods fail we reduce the problem of computing these orbits to that of finding the intersection of the unstable manifold with a suitable set and then apply it to concrete situations we also plot the homoclinic values configuration in parameters space giving a picture of the structural distribution and a geometrical view of homoclinic bifurcations

shaping wave patterns in reaction diffusion systems we present a method to control the two dimensional shape of traveling wave solutions to reaction diffusion systems such as interfaces and excitation pulses control signals that realize a pregiven wave shape are determined analytically from nonlinear evolution equation for isoconcentration lines as the perturbed nonlinear phase diffusion equation or the perturbed linear eikonal equation while the control enforces a desired wave shape perpendicular to the local propagation direction the wave profile along the propagation direction itself remains almost unaffected provided that the one dimensional wave profile of all state variables and its propagation velocity can be measured experimentally and the diffusion coefficients of the reacting species are given the new approach can be applied even if the underlying nonlinear reaction kinetics are unknown

heteroclinic contours and self replicated solitary waves in a reaction diffusion lattice with complex threshold excitation abstract the space time dynamics of the network system modeling collective behavior of electrically coupled nonlinear cells is investigated the dynamics of a local cell is described by the fitzhugh nagumo system with complex threshold excitation heteroclinic orbits defining traveling wave front solutions are investigated in a moving frame system a heteroclinic contour formed by separatrix manifolds of two saddle foci is found in the phase space the existence of such structure indicates the appearance of complex wave patterns in the network such solutions have been confirmed and analyzed numerically complex homoclinic orbits found in the neighborhood of the heteroclinic contour define the propagation of composite pulse excitations that can be self replicated in collisions leading to the appearance of complex wave patterns

turning points and traveling waves in fitzhugh nagumo type equations abstract consider the following fitzhugh nagumo type equation u t u x x f u w w t g u w where f u w u u a w 1 u for some smooth function a w and g u w u w by allowing a w to cross zero and one the corresponding traveling wave equation possesses special turning points which result in very rich dynamics in this work we examine the existence of fronts backs and pulses solutions in particular the co existence of different fronts will be discussed

an algorithm for the computation of hopf bifurcation points in comparison with other methods we present an algorithm for the computation of a hopf bifurcation point based on a direct method i e an augmented time independent system is solved the algorithm is designed to deal efficiently with problems representing a finite difference discretization of a partial differential system in this case the bandstructure of the jacobian matrix is exploited we also compare this technique with some other methods in regard to its efficiency and reliability

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

the bifurcations of separatrix contours and chaos abstract bifurcations of systems close to systems having contours composed of separatrices of a pair of saddle points are studied and cases of the appearance of nontrivial hyperbolic sets are pointed out

distributed fuzzy control design of nonlinear hyperbolic pde systems with application to nonisothermal plug flow reactor this paper considers the problem of fuzzy control design for a class of nonlinear distributed parameter systems that is described by first order hyperbolic partial differential equations pdes where the control actuators are continuously distributed in space the goal of this paper is to develop a fuzzy state feedback control design methodology for these systems by employing a combination of pde theory and concepts from takagi sugeno t s fuzzy control first the t s fuzzy hyperbolic pde model is proposed to accurately represent the nonlinear first order hyperbolic pde system subsequently based on the t s fuzzy pde model a lyapunov technique is used to analyze the closed loop exponential stability with a given decay rate then a fuzzy state feedback control design procedure is developed in terms of a set of spatial differential linear matrix inequalities sdlmis from the resulting stability conditions furthermore utilizing the finite difference approximation method with a backward difference for the spatial derivative a recursive algorithm is presented to solve the sdlmis via the existing lmi optimization techniques finally the developed design methodology is successfully applied to the control of a nonisothermal plug flow reactor

numerical approximation of a periodic linear parabolic problem we approximate a periodic linear parabolic problem using a galerkin method for the space variable and a spectral method for the time variable some optimal error estimates are derived

neumann boundary stabilization of one dimensional linear parabolic systems with input delay this paper addresses the boundary stabilization problem of one dimensional linear parabolic systems with input delay especially the case of neumann boundary control is studied the purpose is to derive the stabilizing controller of predictor type in a hilbert space by using a backstepping method combined with the semigroup theory the use of the semigroup theory makes the proof of continuity of the inverse transformation easy also it is shown that the implementation of the abstract controller is feasible by using a finite number of eigenvalues and eigenfunctions of the system operator

global stabilization of the kuramoto sivashinsky equation via distributed output feedback control abstract this work addresses the problem of global exponential stabilization of the kuramoto sivashinsky equation kse subject to periodic boundary conditions via distributed static output feedback control under the assumption that the number of measurements is equal to the total number of unstable and critically stable eigenvalues of the kse and a necessary and sufficient stability condition is satisfied linear static output feedback controllers are designed that globally exponentially stabilize the zero solution of the kse the controllers are designed on the basis of finite dimensional approximations of the kse which are obtained through galerkin s method the theoretical results are confirmed by computer simulations of the closed loop system

approximation of hopf bifurcation we make several assumptions on a nonlinear evolution problem ensuring the existence of a hopf bifurcation under a fairly general approximation condition we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions

computing absolute and essential spectra using continuation a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented the advantage of this approach compared with direct eigenvalue computations for the discretized operator are the ecient and accurate computation of selected parts of the spectrum typically those near the imaginary axis and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability we also discuss the implementation and usage of this approach with the software package auto and provide example computations for the fitzhugh nagumo and the complex ginzburg landau equation

influence of boundary conditions on the dynamics of oscillatory media abstract we study the pattern dynamics in a reaction diffusion model of the activator inhibitor type in the oscillatory regime we consider finite systems with partially absorptive boundary conditions analysing examples in different geometries in one and two dimensions we observe that the boundary conditions have important effects in the pattern forming properties of the systems in all the studied cases the arising complex behaviour is found to be dependent on the absorption parameter by changing this parameter we can control the asymptotic behaviour to be stationary periodic quasiperiodic or chaotic

boundary effects in extended dynamical systems abstract in the framework of spatially extended dynamical systems we present three examples in which the presence of walls leads to dynamic behavior qualitatively different from the one obtained in an infinite domain or under periodic boundary conditions for a nonlinear reaction diffusion model we obtain boundary induced spatially chaotic configurations nontrivial average patterns arising from boundaries are shown to appear in spatiotemporally chaotic states of the kuramoto sivashinsky model finally walls organize novel states in simulations of the complex ginzburg landau equation

isolated singularities of affine special kahler metrics in two dimensions we prove that there are just two types of isolated singularities of special kahler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities we also construct examples of such metrics

predictive control with guaranteed stability for water hammer equations we study the application of the receding horizon optimal control rhoc for hydraulic pipeline systems described by the so called water hammer equations sufficient conditions to guarantee an asymptotic stability to an equilibrium state are first introduced and then integrated in the rhoc scheme for the implementation calculus of variations is employed to characterize the optimal solution in terms of the adjoint state and the recently proposed lattice boltzmann method is used to solve both direct and adjoint partial differential equations this approach is finally validated in simulation

nonlinear control of incompressible fluid flow application to burgers equation and 2d channel flow abstract this paper proposes a methodology for the synthesis of nonlinear finite dimensional feedback controllers for incompressible newtonian fluid flows described by two dimensional navier stokes equations combination of galerkin s method with approximate inertial manifolds is employed for the derivation of low order ordinary differential equation ode systems that accurately describe the dominant dynamics of the flow these ode systems are subsequently used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed loop system to follow the reference input asymptotically the method is successfully used to synthesize nonlinear finite dimensional output feedback controllers for the burgers equation and the two dimensional channel flow that enhance the convergence rate to the spatially uniform steady state and the parabolic velocity profile respectively the performance of the proposed controllers is successfully tested through simulations and is shown to be superior to the one of linear controllers

pulse bifurcations and instabilities in an excitable medium computations in finite ring domains we investigate the instabilities and bifurcations of traveling pulses in a model excitable medium in particular we discuss three different scenarios involving either the loss of stability or disappearance of stable pulses in numerical simulations beyond the instabilities we observe replication of pulses backfiring resulting in complex periodic or spatiotemporally chaotic dynamics as well as modulated traveling pulses we approximate the linear stability of traveling pulses through computations in a finite albeit large domain with periodic boundary conditions the critical eigenmodes at the onset of the instabilities are related to the resulting spatiotemporal dynamics and act upon the back of the pulses the first scenario has been analyzed earlier m g zimmermann et al physica d 110 92 1997 for high excitability low excitation threshold it involves the collision of a stable pulse branch with an unstable pulse branch in a so called t point in the framework of traveling wave ordinary differential equations pulses correspond to homoclinic orbits and the t point to a double heteroclinic loop we investigate this transition for a pulse in a domain with finite length and periodic boundary conditions numerical evidence of the proximity of the infinite domain t point in this setup appears in the form of two saddle node bifurcations alternatively for intermediate excitation threshold an entire cascade of saddle nodes causing a spiraling of the pulse branch appears near the parameter values corresponding to the infinite domain t point backfiring appears at the first saddle node bifurcation which limits the existence region of stable pulses the third case found in the model for large excitation threshold is an oscillatory instability giving rise to breathing traveling pulses that periodically vary in width and speed

distributed parameter thermal controllability a numerical method for solving the inverse heat conduction problem this paper addresses the inverse heat conduction problem encountered in thermal manufacturing processes a numerical control algorithm is developed for distributed parameter conduction systems based on galerkin optimization of an energy index employing green s functions various temperature profiles of variable complexity are studied using the proposed technique in order to determine the surface heat input distribution necessary to generate the desired temperature field inside a solid body furthermore the effect of altering the iterative time step and duration of processing time on the convergence of the solution generated by the aforementioned method is investigated it is proved that despite the variations in numerical processing the iterative technique is able to solve the problem of inverse heat conduction in the thermal processing of materials copyright 2004 john wiley sons ltd

special kahler manifolds we give an intrinsic definition of the special geometry which arises in global n 2 supersymmetry in four dimensions the base of an algebraic integrable system exhibits this geometry and with an integrality hypothesis any special kahler manifold is so related to an integrable system the cotangent bundle of a special kahler manifold carries a hyperkahler metric we also define special geometry in supergravity in terms of the special geometry in global supersymmetry

oscillatory instability of traveling waves for a kdv burgers equation the stability of traveling wave solutions of a generalization of the kdv burgers equation o u u p o u 03u a o u is studied as the parameters p and a are varied the eigenvalue problem for the linearized evolution of perturbations is analyzed by numerically computing evans function d a an analytic function whose zeros correspond to discrete eigenvalues in particular the number of unstable eigenvalues in the complex plane is evaluated by computing the winding number of d a analytical and numerical evidence suggests that a hopf bifurcation occurs for oscillatory traveling wave profiles in certain parameter ranges dynamic simulations suggest that the bifurcation is subcritical no stable time periodic solution is found

finite dimensional approximation and control of non linear parabolic pde systems this article proposes a rigorous and practical methodology for the derivation of accurate finite dimensional approximations and the synthesis of non linear output feedback controllers for non linear parabolic pde systems for which the manipulated inputs the controlled and measured outputs are distributed in space the method consists of three steps first the karhunen loeve expansion is used to derive empirical eigenfunctions of the non linear parabolic pde system then the empirical eigenfunctions are used as basis functions within a galerkin s and approximate inertial manifold model reduction framework to derive low order ode systems that accurately describe the dominant dynamics of the pde system and finally these ode systems are used for the synthesis of non linear output feedback controllers that guarantee stability and enforce output tracking in the closed loop system the proposed method is used to perform model reduction and synthesize a non linear dynamic output feedback controller for a rapi

robust output feedback control of quasi linear parabolic pde systems abstract this paper proposes a methodology for the synthesis of nonlinear robust output feedback controllers for systems of quasi linear parabolic partial differential equations with time varying uncertain variables the method is successfully applied to a typical diffusion reaction process with uncertainty

twist geometry of the c map we discuss the geometry of the c map from projective special kahler to quaternionic kahler manifolds using the twist construction to provide a global approach to hitchin s description as found by alexandrov et al and alekseevsky et al this is related to the quaternionic flip of haydys we prove uniqueness statements for several steps of the construction in particular we show that given a hyperkahler manifold with a rotating symmetry there is essentially only a one parameter degree of freedom in constructing a quaternionic kahler manifold of the same dimension we demonstrate how examples on group manifolds arise from this picture