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

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

infinite period bifurcation and global bifurcation branches branches of periodic solutions which exhibit the alternatives of the global hopf bifurcation theorem are calculated for two general systems of differential equations in the first a branch of solu

invariant subspaces dilation theory and the structure of the predual of a dual algebra i on etudie la structure des preduaux d algebres d operateurs on obtient un nouveau theoreme de sous espace invariant on trouve des proprietes concernant le treillis sous espace invariant de l operateur decalage de bergman

dual subspaces of operators in this note we introduce some new constructions of dual spaces of operators which are of course at the same time operator spaces in the sense of pisier 2003 12 we exemplify the utility of these constructs by establishing in this more general setting a curious and little known result from the theory of dual algebras namely from chevreau and pearcy 1991 11

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

bifurcation phenomena near homoclinic systems a two parameter analysis the bifurcations of periodic orbits in a class of autonomous three variable nonlinear differential equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues i where 1 sil nikov s condition are studied in a two parameter space the perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period doubling bifurcations leading to periodic orbits which may be attractors if 1 2 the accumulation rate of the critical parameter values at the homoclinic system is exp 2 a global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity a countable set of these cusp bifurcations with scaling properties related to the eigenvalues i of the stationary state are shown to occur in infinitely contractive flows in the two parameter space the periodic orbit attractor domain exhibits a spiral structure globally around the set of homoclinic systems in which all the different periodic orbits are continuously connected

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

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

complex oscillations and waves of calcium in pancreatic acinar cells abstract we perform a bifurcation analysis of a model of ca2 wave propagation in the basal region of pancreatic acinar cells the model we consider was first presented in sneyd et al j sneyd k tsaneva atanasova j i e bruce s v straub d r giovannucci d i yule a model of calcium waves in pancreatic and parotid acinar cells biophys j 85 2003 1392 1405 where a partial bifurcation analysis was given of the model in the absence of diffusion we obtain more complete information about bifurcations of the diffusionless model via numerical studies then analyse the spatially extended model by numerical investigation of the travelling wave equations and direct numerical solution of the model equations we find solitary waves in the model equations arising from homoclinic bifurcations in the travelling wave equations the solitary waves exist and appear to be stable for a significant interval of the primary bifurcation parameter i e the concentration of inositol trisphosphate but are eventually replaced by irregular spatio temporal behaviour the homoclinic bifurcations are related to a number of complicated mathematical structures in the travelling wave equations including an anomalous homoclinic hopf bifurcation heteroclinic bifurcations between an equilibrium and a periodic orbit and homoclinic bifurcations of periodic orbits

chaotic pulse trains the authors study a third order nonlinear ordinary differential equation whose solutions under certain specific conditions are individual pulses these correspond to homoclinic orbits in the phase space of the equation and the possible pulse types are studied in some detail sufficiently close to the conditions under which a homoclinic orbit exists the solutions take the form of trains of well separated pulses a measure of closeness to homoclinic conditions provides a small parameter for the development of an asymptotic solution consisting of superposed isolated pulses the solvability condition in the resulting singular perturbation theory is a timing map relating successive pulse spacings this map of the real line onto itself together with the known form of the homoclinic orbit provides a concise and accurate solution of the equation

traveling waves for a model of gravity driven film flows in cylindrical domains abstract traveling wave solutions are studied for a recently derived model of a falling viscous film on the interior of a vertical rigid tube by identifying a hopf bifurcation and using numerical continuation software families of non trivial traveling wave solutions may be traced out in parameter space these families all contain a single solution at a turnaround point with larger film thickness than all others in the family in an earlier paper it was conjectured that this turnaround point may represent a critical thickness separating two distinct flow regimes observed in physical experiments as well as two distinct types of behavior in transient solutions to the model here these hypotheses are verified over a range of parameter values using a combination of numerical and analytical techniques the linear stability of these solutions is also discussed both large and small amplitude solutions are shown to be unstable though the instability mechanisms are different for each wave type specifically for small amplitude waves the region of relatively flat film away from the localized wave crest is subject to the same instability that makes the trivial flat film solution unstable for large amplitude waves this mechanism is present but dwarfed by a much stronger tendency to relax to a regime close to that followed by small amplitude waves

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

finite aspect ratio taylor couette flow shil nikov dynamics of 2 tori abstract the nonlinear dynamics of the flow in a short annular container driven by the rotation of the inner cylinder is studied using direct numerical simulations of the three dimensional navier stokes equations the basic state is s o 2 z 2 symmetric for aspect ratios between 3 6 and 4 4 we have located three codimension two bifurcations a cusp a double hopf and a fold hopf bifurcation all these local bifurcations are z 2 invariant the breaking of z 2 symmetry involves very complex shil nikov type dynamics not directly connected to any of the three codimension two bifurcations but associated with five unstable limit cycles and a wealth of heteroclinic connections between them period adding cascades both direct and reverse of 2 tori have been found narrow regions of chaotic dynamics are interspersed within these quasiperiodic solutions

hyper reflexivity and the factorization of linear functionals abstract it is shown that hyper reflexivity of a space of linear operators on a hilbert space follows from a factorization property of linear functionals continuous in the weak operator topology this provides new examples of hyper reflexive algebras and new proofs for the hyper reflexivity of the noncommutative disk algebras

a reflexivity theorem for weakly closed subspaces of operators it was proved in 4 that the ultraweakly closed algebras generated by certain contractions on hilbert space have a remarkable property this property in conjunction with the fact that these algebras are isomorphic to h was used in 3 to show that such ultraweakly closed algebras are reflexive in the present paper we prove an analogous result that does not require isomorphism with h and applies even to linear spaces of operators our result contains the reflexivity theorems of 3 2 and 9 as particular cases let g denote the algebra of linear bounded operators acting on the hilbert spale x and let xf denote a linear subspace of then ff is endowed with the weak and ultraweak topologies that it inherits from y 3 cf 6 15 for two arbitrary vectors x y e h we can define the ultra weakly continuous functional x y on kby x c y a ax y a e x where k stands for the scalar product in j definition 1 let n be a natural number n 1 the subspace 4 has property bn respectively an if for every positive number e there exists a positive number 8 6 n such that for every system f i 1 i j n of weakly respectively ultraweakly continuous functionals on and every system xi yj 1 i j n of vectors in x satisfying the inequalities irk1j xi yj 11 8 there exist vectors x yj 1 i n in3 such that oi x yj i i va n and x x li e iiy y i j n since every weakly continuous functional on xf is also ultraweakly continuous property bn is weaker than an added in proof it was pointed out by c apostol that b7 and an are in fact equivalent this fact is not used below we recall now from 8 that a linear subspacea1 of y i is said to be reflexive if it contains every operator t e y i with the property that tx e x for every received by the editors january 31 1984 1980 mathemnatics sub ect classification primary 47c05 the author was partially supported by a grant from the national science foundation 1985 american mathematical society 0002 9947 85 1 00 25 per page 139 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 39 22 utc all use subject to http about jstor org terms

experimental unfolding of the nonlinear dynamics of a cable mass suspended system around a divergence hopf bifurcation systematic experimental investigation of the finite amplitude dynamics of a multiple internally resonant suspended cable mass subjected to anti phase support motion at primary resonance is accomplished upon getting hints from a basic system configuration assumed as reference setup about the multiple bifurcation event possibly governing transition to complex dynamics an improved experimental apparatus is used to make it technically accessible results obtained by varying three control parameters namely the frequency and amplitude of excitation and the temperature of a thermostatic chamber embedding the experimental system allow us to characterize in depth various occurring classes of motion in terms of time and spatial complexity to describe peculiar and or persistent features of transition to nonregular dynamics and to trace them back to a canonical scenario from bifurcation theory variable response paths are detected via bifurcation diagrams and spectra of singular values of measurement results and overall behaviour charts are built in the excitation parameter space considering the temperature as a controllable parameter shows to be fundamental for i indirectly setting cable material properties to values for which the conjectured codimension 2 bifurcation becomes apparent ii qualitatively referring the experimental unfolding of regular and nonregular cable dynamics to the theoretical unfolding of the divergence hopf bifurcation normal form and iii determining system response not only in the strict neighbourhood of the organizing divergence hopf bifurcation but also in the ensuing postcritical regions where the dependence of material damping on temperature affects secondary bifurcations to low dimensional homoclinic chaos

switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in odes the paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n dimensional odes we discuss the implementation and the performance of the algorithm in several examples including an extended lorenz 84 model and a laser system

merging of resonance tongues abstract this paper considers the resonance behavior found near a saddle node hopf bifurcation in a flow the global structure of resonance tongues is examined and it is shown that in the absence of certain symmetries different resonance tongues can merge with one another the position and shape of the resonance tongues are seen to be related to the presence of homoclinic and heteroclinic bifurcations of two saddle foci the implications of these results for the normal form analysis of the saddle node hopf bifurcation are discussed

a geometric transversals approach to sensor motion planning for tracking maneuvering targets this paper presents a geometric transversals approach for representing the probability of track detection as an analytic function of time and target motion parameters by this approach the optimization of the detection probability subject to sensor kinodynamic constraints can be formulated as an optimal control problem using the proposed detection probability function the necessary conditions for optimality can be derived using calculus of variations and solved numerically using a variational iteration method vim the simulation results show that sensor state and control trajectories obtained by this approach bring about a significant increase in detection probability compared to existing strategies and require a computation that is significantly reduced compared to direct methods

dynamics in the 1 2 spatial resonance with broken reflection symmetry the dynamics near the 1 2 spatial resonance in systems with broken o 2 symmetry are described the system is assumed to remain so 2 invariant and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the o 2 symmetric system the symmetry breaking destroys these cycles and introduces a variety of new bifurcations into the system some which are associated with complex behavior the possible scenarios with increasing symmetry breaking are described and possible applications of the results are discussed

distributed optimal control for multi agent trajectory optimization this paper presents a novel optimal control problem referred to as distributed optimal control that is applicable to multiscale dynamical systems comprised of numerous interacting agents the system performance is represented by an integral cost function of the macroscopic state that is optimized subject to a hyperbolic partial differential equation known as the advection equation the microscopic control laws are derived from the optimal macroscopic description using a potential function approach the optimality conditions of the distributed optimal control problem are first derived analytically and then demonstrated numerically through a multi agent trajectory optimization problem

single and multiple pulse waves for the fitzhugh nagumo the fitzhugh nagumo equations describe the propagation of impulses along a nerve axon we study traveling wave solutions which satisfy a set of ordinary differential equations with a cubic nonlinearity f v v 1 v v a our emphasis is on solutions which tend to zero at pm infty an improved proof of a previous result that there are at least two such solutions is presented further by allowing the middle root v a of the nonlinearity to be a small parameter better estimates of the parameter range where such solutions exist are given in view of recent work of evans fenichel feroe and sil nikov our results imply that there are a great many more bounded traveling waves than was known previously in particular there is an infinite family of periodic waves all traveling with the same speed a reasonable transversality hypothesis introduced by evans fenichel and feroe implies also that there are multiple pulse waves at speeds close to that of a single pulse

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

brief paper optimal control problems with multiple characteristic time points in the objective and constraints in this paper we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points these time points can be either fixed or variable using the control parametrization technique and a time scaling transformation this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems each of these approximate problems can be viewed as a finite dimensional optimization problem new gradient formulae for the cost and constraint functions are derived with these gradient formulae standard gradient based optimization methods can be applied to solve each approximate optimal parameter selection problem for illustration two numerical examples are solved

geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction diffusion systems a periodic travelling wave propagating steadily in a spatially homogeneous reaction diffusion system corresponds one to one to a closed orbit in a certain finite dimensional dynamical system and hence the travelling wave can be characterized qualitatively by using topological characteristics of the corresponding closed orbit we introduce a sequence of characteristics delta gamma l k k 1 2 cdots of the closed orbit and show that they are closely associated with the stability and bifurcations of the travelling wave in this paper we mainly consider the first characteristic of the closed orbit the value of the characteristic can be determined if the form of the dispersion relation curve for the travelling wave i e the propagation speed c l as a function of the spatial period l is known we give three theorems to determine the value of the characteristic for three qualitatively different types of periodic travelling waves as an example we compute numerically the dispersion rel

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

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

diversity in interaction strength promotes rich dynamical behaviours in a three species ecological system abstract asymmetrical interactions are thought to have an important influence on multi species competitive systems and understanding this ecological process is a challenging task because of uncertain roles that asymmetrical interactions can have in maintaining species biodiversity to address this problem we employ a simple ecological model for multiple species with asymmetrical competitive strength and we analyse this model using the techniques from dynamical systems and bifurcation analysis in particular we examine the stability properties of steady states and also its bifurcation structures under parameter variation we demonstrate that there exists threshold values for asymmetrical competitive strength which correspond to transcritical and supercritical hopf bifurcations the occurrence of these bifurcations give rise to numerous dynamics in the model e g multi species coexistence species exclusion alternative stable state communities and oscillatory dynamics among multiple species additionally we observe the conditions under which co dimension two gavrilov guckenheimer bifurcation occurs as a result of the interaction between transcritical and hopf bifurcations which can lead to complex dynamics we also investigate how diversity in interaction strength between multiple competing species causes counterintuitive observations on one hand asymmetrical interactions promote the maintenance of biodiversity through multi species coexistence state and also coexistence via stable limit cycles corresponding to oscillatory dynamics on the other hand we also observe that the occurrence of stable limit cycles can lead to destabilisation of multi species communities

the entwined wiggling of homoclinic curves emerging from saddle node hopf instabilities abstract an analysis is presented of the homoclinic bifurcations occurring in a generic unfolding of a saddle node hopf singularity also known as a gavrilov guckenheimer point specifically an explanation is given of previously numerically observed oscillations of loci of homoclinic orbits to two different saddle focus equilibria these oscillations occur within an exponentially thin wedge of parameter space that emerges from the codimension two point the frequency of oscillation tends to zero as the codimension two point is approached earlier theory by gaspard showed that homoclinic orbits must exist inside the parameter wedge this result is here extended to give the frequency and amplitude of the oscillations of the homoclinic loci within the wedge it is also shown how the two loci are related to each other and in the case of only cubic perturbations of the normal form that they are precisely out of phase the analysis is shown to agree with numerical results on perturbed normal forms and in two model systems arising in applications to atmospheric dynamics and to calcium wave propagation

propagation phenomena in a bistable reaction diffusion system consideration is given to a system of reaction diffusion equations which have qualitative significance for several applications including nerve conduction and distributed chemical biochemical systems these equations are of the fitzhugh nagumo type and contain three parameters for certain ranges of the parameters the system exhibits two stable singular points a singular perturbation construction is given to illustrate that there may exist both pulse type and transition type traveling waves a complete rigorous description of which of these waves exist for a given set of parameters and how the velocities of the waves vary with the parameters is given for the case when the system has a piecewise linear nonlinearity numerical results of solutions to these equations are also presented these calculations illustrate how waves are generated from initial data how they interact during collisions and how they are influenced by local disturbances and boundary conditions in one example a righltward moving

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

two new reductions methods for polynomial differential equations and applications to nonlinear pdes abstract for ordinary differential equations of the form p y v v v n 0 which are polynomial in the variables v v v n two new reduction methods to first order equations are considered the reduced equations are of the forms v q 1 y v 1 q and v q 2 y v 1 q where q 1 and q 2 are two polynomials of degree p in v 1 q and v respectively whose coefficients depend on y in contrast to most of the known reduction methods of these types which use either q 1 or q 2 in this paper the values of the positive integers p q are not predetermined a procedure to obtain the possible values of the integers for which a reduction of any of these types may exist is provided as a consequence new reductions that cannot be obtained by other known methods may be found the new methods have been applied to obtain some reductions and consequently new solutions for three polynomial ordinary differential equations related to well known equations in mathematical physics the kuramoto sivashinsky equation a generalized benney equation and a 5 th order kdv equation some pieces of computer algebra code written in maple and implementing the underlying algorithms to derive the reductions are also included

existence and stability of multiple impulse solutions of a nerve equation a piecewise linear partial differential equation which models the propagation of a nerve impulse along an axon is studied for the equation parameters used there are two solutions which represent solitary impulses traveling with fixed form one with a slower velocity c s1 than the velocity c f1 of the other we show that near these velocities there exist multiple impulse solutions and exhibit four such solutions each of which has two pulses the local stability of these solutions is analyzed the two solutions whose velocities are near c s1 demonstrate instabilities which indicate growth or decay of the separate pulses for one of the multiple impulse solutions with velocity near c s1 and for one of those with velocity near c f1 the second pulse deomonstrates an instability which indicates a change in the spacing between the two pulses the other solution with velocity near c f1 is stable

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 optimal coordination for multiple heterogeneous euler lagrangian systems abstract in this paper we consider the distributed optimal coordination doc problem for multi agent systems with the agents in the form of euler lagrangian el dynamics we propose two different distributed protocols for the heterogeneous continuous time el agents to achieve the optimization task by constructing suitable lyapunov functions we prove the global convergence to the optimal coordination of the el systems in the case with parametric uncertainties and the global exponential convergence in the case without parametric uncertainties furthermore we estimate the regret bound for an uncertain doc problem with time varying cost functions and inexact gradients finally we provide a numerical example to validate the theoretical results

local birth of homoclinic chaos abstract local generation of chaotic behaviours associated with a sil nikov homoclinic orbit is studied near the codimension two bifurcation at the double stationary oscillatory instability of eigenvalues 0 we develop a perturbative calculation of the sil nikov homoclinic orbit and of its bifurcation locus starting from a conservative vector field flat nonanalytic behaviours due to non axisymmetric perturbations of the normal form are discussed and we show that the codimension one locus of sil nikov homoclinicity is unavoidable near bicriticality under certain conditions the results are applied to rossler s model quasiperiodic regimes born at the double instability are also studied and illustrated in the rossler model

using lin s method to solve bykov s problems abstract we consider nonwandering dynamics near heteroclinic cycles between two hyperbolic equilibria the constituting heteroclinic connections are assumed to be such that one of them is transverse and isolated such heteroclinic cycles are associated with the termination of a branch of homoclinic solutions and called t points in this context we study codimension two t points and their unfoldings in r n in our consideration we distinguish between cases with real and complex leading eigenvalues of the equilibria in doing so we establish lin s method as a unified approach to re gain and extend results of bykov s seminal studies and related works to a large extent our approach reduces the study to the discussion of intersections of lines and spirals in the plane

homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit we analyze homoclinic orbits near codimension 1 and 2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions the main motivation for this study is a self organized periodic replication process of travelling pulses which has been observed in reaction diffusion equations we establish conditions for existence and uniqueness of countably infinite families of curve segments of 1 homoclinic orbits which accumulate at codimension 1 or 2 heteroclinic cycles the main result shows the bifurcation of a number of curves of 1 homoclinic orbits from such codimension 2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points in addition a leading order expansion of the associated curves in parameter space is derived its coefficients are periodic with one frequency from the imaginary part of the leading stable floquet exponents of the periodic orbit and one from the winding number