on the lebesgue measure of li yorke pairs for interval maps we investigate the prevalence of li yorke pairs for c2 and c3 multimodal maps f with non flat critical points we show that every measurable scrambled set has zero lebesgue measure and that all strongly wandering sets have zero lebesgue measure as does the set of pairs of asymptotic but not asymptotically periodic points

boundary effects on localized structures in spatially extended systems abstract we present a general method of analyzing the influence of finite size and boundary effects on the dynamics of localized solutions of non linear spatially extended systems the dynamics of localized structures in infinite systems involve solvability conditions that require projection onto a goldstone mode our method works by extending the solvability conditions to finite sized systems by incorporating the finite sized modifications of the goldstone mode and associated nonzero eigenvalue we apply this method to the special case of non equilibrium domain walls under the influence of dirichlet boundary conditions in a parametrically forced complex ginzburg landau equation where we examine exotic nonuniform domain wall motion due to the influence of boundary conditions

slow motion manifolds far from the attractor in multistable reaction diffusion equations abstract we consider a scalar reaction diffusion equation with multistable nonlinearity with a particular symmetry by reduction to a family of transmission problems in r and by contraction arguments a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed an approximation for the associated vector field is also provided this shows that the motion on those manifolds is exponentially slow as in the well known case of the bistable equation however in opposition to the bistable case some of these manifolds are far from the attractor since these manifolds correspond to metastable patterns this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion it is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds

wave propagation in a fitzhugh nagumo type model with modified excitability we examine a generalized fitzhugh nagumo fhn type model with modified excitability derived from the diffusive morris lecar equations for neuronal activity we obtain exact analytic solutions in the form of traveling waves using a piecewise linear approximation for the activator and inhibitor reaction terms we study the existence and stability of waves and find that the inhibitor species exhibits different types of wave forms fronts and pulses while the activator wave maintains the usual kink front shape the nonequilibrium ising bloch bifurcation for the wave speed that occurs in the fhn model where the control parameter is the ratio of inhibitor to activator time scales persists when the strength of the inhibitor nonlinearity is taken as the bifurcation parameter

positive filtering for continuous time positive systems under l 1 performance in this paper we address the positive filtering problem for positive continuous time systems under the l1 induced performance a pair of positive filters with error bounding feature is proposed to estimate the output of positive systems a novel characterisation is first obtained to ensure that the filtering error system is asymptotically stable with a prescribed l1 induced performance then necessary and sufficient conditions for the existence of required filters are presented and the obtained results are expressed in terms of linear programming problems which can be easily checked by standard software finally a numerical example is given to illustrate the effectiveness of the proposed design procedures

on the spectral and combinatorial structure of 2d positive systems abstract the dynamics of a 2d positive system depends on the pair of nonnegative square matrices that provide the updating of its local states in this paper several spectral properties such as finite memory separability and property l which depend on the characteristic polynomial of the pair are investigated under the nonnegativity constraint and in connection with the combinatorial structure of the matrices some aspects of the perron frobenius theory are extended to the 2d case in particular conditions are provided guaranteeing the existence of a common maximal eigenvector for two nonnegative matrices with irreducible sum finally some results on 2d positive realizations are presented

attractors for graph critical rational maps we call a rational map f graph critical if any critical point either belongs to an invariant finite graph g or has minimal limit set or is non recurrent and has limit set disjoint from g we prove that for any conformal measure either for almost every point of the julia set j f its limit set coincides with j f or for almost every point of j f its limit set coincides with the limit set of a critical point of f

drift bifurcation of dissipative solitons due to a change of shape experiment and theory abstract the subject of this paper is self organized solitary current filaments being observed experimentally in a planar dc gas discharge system with high ohmic barrier we report on the dynamical behavior of these objects which we refer to as dissipative solitons an application of stochastic data analysis methods enables the separation of intrinsic dynamics from stochastic contributions using the externally applied voltage as control parameter we observe a drift bifurcation that is accompanied by a change of shape of the bifurcating dissipative soliton in order to interpret the experimental finding a theoretical model in form of a three component reaction diffusion system is considered allowing for a qualitative understanding of the obtained results we discuss an analytical expression for the velocity of the dissipative soliton as a function of an appropriate control parameter and demonstrate that the change of the shape can induce the drift bifurcation this analytical finding is supported by numerical simulations of the model equations

higher order resonances and instability of high frequency wkb solutions abstract this paper focuses on the destabilizing role of resonances in high frequency wkb solutions specifically we study higher order resonances associated with higher order harmonics generated by nonlinearities we give examples of systems and solutions for which such resonances generate instantaneous instabilities even though the equations linearized around the leading wkb terms are initially stable meaning in particular that the key destabilizing terms are not present in the data

a linear framework on the distributed model predictive control of positive systems abstract this paper investigates the distributed model predictive control dmpc for positive systems with interval and polytopic uncertainties respectively different from the traditional quadratic dmpc a new linear dmpc framework is established for positive systems first a linear cooperative cost function performance index function is introduced for the dmpc of positive systems accordingly some linear constraint conditions are imposed on the systems then a linear co positive lyapunov function is constructed using the linear lyapunov function the dmpc controllers are designed for interval positive systems and polytopic positive systems respectively the controller of each subsystem contains not only the state information of the subsystem but also the state information of other correlated subsystems linear programming based optimization algorithm is addressed to compute the controller parameters to guarantee the feasibility and stability of the systems a cone invariant set is proposed finally two case studies are provided to illustrate the effectiveness of the theoretical findings

l1 induced norm and controller synthesis of positive systems in this paper the problem of 1 1 induced controller design for discrete time positive systems is investigated with the use of linear lyapunov function an analytical method to compute the exact value of 1 1 induced norm is first presented then a novel characterization for stability and 1 1 induced performance is proposed based on the characterization a necessary and sufficient condition for the existence of desired controllers is derived and an iterative convex optimization approach is developed to solve the condition in addition the synthesis of the state feedback controller for single input multiple output simo positive systems is investigated for this special case an analytic solution is established to show how the optimal 1 1 induced controller can be designed and some links to the spectral radius of the closed loop systems are provided finally the theoretical results are illustrated through a numerical example

remarks on excitability stability and sign of equilibria in cooperative systems the strict positivity of equilibria is known to be equivalent to asymptotic stability in excitable positive linear systems in contrast with intuition this property does not hold for nonlinear cooperative systems however weaker results are pointed out and proved to be effective in the analysis of several nonlinear cooperative systems

motion and pinning of discrete interfaces we describe the motion of interfaces in a two dimensional discrete environment by coupling the minimizing movements approach by almgren taylor and wang and a discrete to continuous analysis we show that below a critical ratio of the time and space scalings we have no motion of interfaces pinning while above that ratio the discrete motion is approximately described by the crystalline motion by curvature on the continuum described by almgren and taylor the critical regime is much richer exhibiting a pinning threshold small setsmove large sets are pinned partial pinning portions of interfaces may not move pinning after an initial motion possibly to a non convex limit set quantization of the interface velocity and non uniqueness effects

unsteady fronts in an autocatalytic system travelling waves in a model for autocatalytic reactions have for some parameter regimes been suggested to have oscillatory instabilities this instability is confirmed by various methods including linear stability analysis exploiting evan9s function and direct numerical simulations the front instability sets in when the order of the reaction m exceeds some threshold m c that depends on the inverse of the lewis number the stability boundary m m c is found numerically for m order one in the limit m 1 in which the system becomes similar to combustion systems with arrhenius kinetics the method of matched asymptotic expansions is employed to find the asymptotic front speed and show that m c 1 1 as 1 just beyond the stability boundary the unstable rocking of the front saturates supercritically if the order is increased still further period doubling bifurcations occur and for small tau there is a transition to chaos through intermittency after the disappearance of a period 4 orbit

criteria of spontaneous interconversions of traveling and static arbitrary dimensional dissipative structures abstract the general criteria of spontaneous transformation of traveling dissipative structures dss autowaves into static dss are derived on the basis of translational symmetry of the problem these criteria determine the system parameters in particular the excitation level a am when the velocity of autowaves of arbitrary dimension and shape becomes zero it is demonstrated that the criteria coincide with the instability condition of static dss with respect to the aperiodic increase of a certain fluctuation describing a small translation of the dss along arbitrary axis i e with the condition of spontaneous transformation of static dss into traveling ones it is shown that the velocity of autowaves depends on parameter a near am v a m a 1 2 the models of fitzhugh nagumo type are analyzed and it is shown that the particular conditions obtained earlier follow from the general criteria derived here some scenarios of interconversion of traveling and static dss are discussed

wavy fronts and speed bifurcation in excitable systems with cross diffusion a bifurcation of excitation fronts induced by cross diffusion in two component bistable reaction diffusion systems of activator inhibitor type is discovered this bifurcation is similar to the nonequilibrium ising bloch bifurcation a different type of fronts whose spatial profiles are characterized by oscillating tails are associated with this bifurcation these fronts are described using exact analytical solutions of piecewise linear reaction diffusion equations

spatially structured activity in synaptically coupled neuronal networks ii lateral inhibition and standing pulses we consider the existence and stability of standing pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled networks of excitatory and inhibitory neurons in a single spatial domain assuming an arrangement of synaptic connections described by lateral inhibition previous formal arguments have demonstrated the existence of both stable and unstable standing pulses s amari biol cybern 27 1977 pp 77 87 these results have formed the basis for several recent hypotheses regarding the generation of sustained activity patterns in prefrontal cortex and other brain regions implicit in the lateral inhibition arrangement however is the assumption that the dynamics of inhibition are instantaneous here we present two arguments demonstrating the loss of stability of standing pulse solutions through a hopf bifurcation when more realistic inhibitory dynamics are considered the first argument parallels amari s formal presentation while the seco

on the invertibility of leslie systems the problem of reconstructing the stocking rate of exploited renewable resources from an output time series is considered the resource is described by a linear age structured model of the leslie type and the measured output is a linear and positive combination of the number of individuals in each reproductive class the problem can be formally answered by using a well known result of system theory a linear system is invertible if and only if the zeros of its transfer function are all smaller than one in modulus in the present case this would imply working with a polynomial of degree n 1 n being the maximum age of the population by explicitly computing the transfer function we show that the invertibility of the system can indeed be established by checking if the zeros of a polynomial of degree r 1 r being the number of reproductive age classes are smaller than one in modulus moreover we investigate two particular measurement schemes and we show that the first one which is possibly the most commonly used to sample animal populations gives rise to an invertible system on the contrary the second scheme can generate noninvertible systems although two single conditions for invertibility that we point out in the paper seem to be often satisfied in the case of real populations

an asymptotic study of the inductive pattern formation mechanism in drosophila egg development the complexity and nonlinear dynamics of patterning networks in development make modeling an important approach for the evaluation of the experimentally derived pattern formation mechanisms as a rule mechanistic models of patterning networks have large number of uncertain parameters model analysis requires extensive computational searches of the parameter space analytical techniques can circumvent these difficulties and offer important insights into the networks functional capabilities here we present an asymptotic analysis of the multiple steady states and transitions between them in a mechanistic model of patterning events specifying the formation of a pair organ in drosophila oogenesis the model describes the interaction between the spatially nonuniform inductive signal and a network of spatially distributed feedback loops our approach dramatically reduces the complexity of the problem and provides an explicit analytical method for the construction and parametric analysis of the patterned states responsible for signaling the analysis reveals a skeleton structure for the patterning capability of the considered regulatory module and demonstrates how a single regulatory network can be used to generate a variety of developmental patterns

analytic solutions for monotonic and oscillating fronts in a reaction diffusion system under external fields abstract a piecewise linear reaction diffusion system including an external field is considered analytic solutions are obtained for the propagating front the front velocity the perturbed front and the growth rate of perturbations it is found that when the ratio of the time scales null cline parameter and the external field are small enough the moving front has an oscillating behaviour similar to that of a front in an oscillatory medium a comparison with an oscillatory front invading an unstable state at small velocity is made and it is argued that there are crucial differences between the two cases in particular we find that for fronts oscillating about a stable state both concentrations must oscillate in two variable models whereas fronts oscillating about an unstable state can approach it with one of the two variables being monotonic

heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems abstract we apply a general heteroclinic and homoclinic bifurcation theory to the study of bifurcations of travelling waves of bistable reaction diffusion systems using the notion of separation we first prove the existence of a cusp point of the set of travelling front solutions in the parameter space this as well as the symmetry of the system yields a coexisting pair of front and back solutions which undergoes the homoclinic bifurcation producing a pulse solution all the hypotheses imposed on the general heteroclinic and homoclinic bifurcation theorem are rigorously verified for a system of bistable reaction diffusion equations containing a small parameter e by using singular perturbation techniques especially the slep method a relation between the stability of front or back solutions and the intersecting manner of the stable and unstable manifolds is also given by means of the separation

biased random walks and propagation failure the critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non markovian biased random walks with reactions from the hamilton jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation if the reaction rate is larger than the jump frequency invasion always succeeds even in the case of maximal bias numerical simulations support our analytical predictions

stability of front solutions in inhomogeneous media abstract we use the fitzhugh nagumo f n model to study front solutions in inhomogeneous media inhomogeneity is modeled by adding a space dependent term z x in the equations changes in z shift the cubic nullcline up and down in the phase plane of the f n model this is used to mimic shifts in the cubic shaped nullcline of a bidomain model of calcium waves in egg cells physica d in press caused by changes in the total calcium concentration conditions for the existence and stability of stationary front solutions for a general class of z functions are obtained analytically when the dynamics are piecewise linear we show that spatial inhomogeneities cause pinning and oscillations of front solutions this is best demonstrated when z x is a linear ramp when the slope is large a linear ramp breaks the translational invariance of the stationary front and stabilizes the front the front becomes less stable as the slope decreases at a critical slope the front becomes unstable through a hopf bifurcation beyond which oscillations in the front occur this also occurs for nonlinear spatial inhomogeneities including a step increase with varying magnitude and steepness this bifurcation has been found previously physica d 106 1997 270 phys rev e 55 1997 366 in a local analysis of the f n model near a co dimension two point sometimes called a non equilibrium ising bloch nib bifurcation and with inhomogeneities of small magnitude the analysis presented here applies globally in parameter space and is valid for inhomogeneities of arbitrary magnitude we show that this bifurcation need not occur in conjunction with the nib point it is caused exclusively by spatial inhomogeneity a rich variety of wave phenomena including front oscillation front pinning and front reflection are shown to occur in inhomogeneous media

oscillations of fronts and front pairs in two and three component reaction diffusion systems abstract we investigate multi co ponent reaction diffusion systems on a finite one dimensional domain the first component is assumed both to react on a short time scale and to have a small diffusion length leading to patterns with sharp fronts to analyze stability we apply the slep method by nishiura and fujii this method is superior to standard perturbation methods we illustrate this using a two component system in the three component system in a situation where both the influence of the first component on the other components and the interaction of the latter is only weak we show that there is a unique hopf destabilization of stationary fronts when changing time constants as an example we treat a system with global coupling used e g for semi conductor devices for filaments front pairs we find two types of hopf destabilizations breathing and swinging the global coupling controls which type occurs this corresponds to recent experimental results and is confirmed via numerical calculations

stimulus locked traveling waves and breathers in an excitatory neural network we analyze the existence and stability of stimulus locked traveling waves in a one dimensional synaptically coupled excitatory neural network the network is modeled in terms of a nonlocal integro differential equation in which the integral kernel represents the spatial distribution of synaptic weights and the output firing rate of a neuron is taken to be a heaviside function of activity given an inhomogeneous moving input of amplitude i0 and velocity v we derive conditions for the existence of stimulus locked waves by working in the moving frame of the input we use this to construct existence tongues in v i0 parameter space whose tips at i0 0 correspond to the intrinsic waves of the homogeneous network we then determine the linear stability of stimulus locked waves within the tongues by constructing the associated evans function and numerically calculating its zeros as a function of network parameters we show that as the input amplitude is reduced a stimulus locked wave within the tongue of

julia fatou sullivan theory for real one dimensional dynamics our aim is to show that the julia fatou sullivan structure theory for the dynamics of rational maps is also valid for smooth endomorphisms of the circle and of the interval under extremely mild smoothness and non flatness conditions in order to stress the similarity between real and complex one dimensional dynamics let us recall the main results from the fatou julia sullivan theory if f is a rational map then there is a dynamical decomposition of the riemann sphere into the disjoint union of two totally invariant i e both forward and backward invariant sets j f f f here f f is the domain of normality of the family of iterates of f and is called the fatou set its complement which is called the julia set off is a compact set which contains all the complications of the dynamics off the connected components of the open set f f are mapped onto each other by f hence the orbit of a component of f f is the union of some components of f f julia proved in the beginning of the century that if a component of f f is periodic and contains an attracting periodic point then the orbit of this component must contain a critical point sullivan in the remarkable paper su showed via quasi conformal deformations that the components are eventually periodic and fall into finitely many orbits let n be either the circle s j or a compact interval of the real line and f n n be a smooth endomorphism a critical point of f is a point where the derivative vanishes a critical point is non flat if some higher derivative is non zero a critical point is an inflection point if it has a neighbourhood where f is monotone otherwise it is called a turning point assume f is not a homeomorphism i f f is a homeomorphism then these maps correspond to degree 1 rational maps of the riemann sphere and the situation is

positive filtering for positive takagi sugeno fuzzy systems under l 1 performance in this paper the positive filtering problem is addressed for positive takagi sugeno t s fuzzy systems under the 1 induced performance to estimate the output of positive t s fuzzy systems error bounding positive filters are constructed a new performance characterization is first established to guarantee the asymptotic stability of the filtering error system with the 1 induced performance moreover sufficient conditions expressed by linear programming problems are derived to design the required filters finally a numerical example is presented to show the effectiveness of the derived theoretical results

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

a strong unique continuation theorem for parabolic equations the unique continuation is best understood for second order elliptic operators the classical paper by carleman 7 established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients this was remarkable since it indicates that even a non analytic solution of an elliptic equation can behave somewhat in an analytic manner the powerful technique he used the so called carleman weighted inequality has played a central role in later developments in 1950 s aronszajn 4 and cordes 11 generalized his result to higher dimensions in recent years this classical subject caught new attention from a great number of people much efforts have been made to relax the smoothness hypotheses on the coefficients see 6 17 43 1 22 a seminal paper by jerison and kenig 26 has shown the unique continuation property for operators of the form c x with c l 2 loc where n 3 is the space dimension further improvements have been carried out in considering other classes of coefficients fefferman phong class and kato class 42 8 12 29 in extending the result to operators with first derivative terms and variable leading coefficients 46 47 51 52 and etc see 25 15 16 28 5 27 and the references therein for second order linear parabolic operators with time independent coefficients the strong unique continuation property along with detailed estimates on the hausdorff measures of nodal sets was reduced in 32 33 to the previously established elliptic counterparts the reduction from time independent parabolic

spherically symmetric internal layers for activator inhibitor systems i existence by a lyapunov schmidt reduction abstract reaction diffusion systems of activator inhibitor type are studied on an n dimensional ball with the homogeneous neumann boundary conditions under the condition that the activator diffuses slowly reacts rapidly and the inhibitor diffuses rapidly reacts moderately we show that the system admits a family of spherically symmetric internal transition layer equilibria the method of proof consists of rigorous asymptotic expansions and a lyapunov schmidt reduction

a hopf bifurcation in a parabolic free boundary problem abstract the occurrence of a hopf bifurcation in a free boundary problem for a parabolic partial differential equation is investigated the bifurcation parameter is contained in the equation which describes the evolution of the free boundary the problem investigated in this paper arises as the singular limit of a system of reaction diffusion equations with mckean reaction dynamics numerical evidence is examined which shows the emergence of periodic steady states for small positive values of a regularization of the problem is introduced making it possible to deal with the heaviside discontinuity in the reaction term and well posedness of the free boundary problem is obtained by application of results from the theory of nonlinear evolution equations to the regularized problem it is then shown that a pair of complex eigenvalues of the linearized problem crosses the imaginary axis as 0 and the existence of a hopf bifurcation is proved using an implicit function theorem argument

diffusive effects on dispersion in excitable media perturbation and numerical methods are used to study traveling and standing wave solutions for a system of reaction diffusion equations modeled after excitable media and the effect of variation of the diffusion coefficients it is shown that traveling waves and standing waves can coexist and that as a certain diffusion coefficient is varied a branch of traveling waves can bifurcate from a branch of standing waves providing a transition between only traveling solutions in one limit and only standing waves in the other limit key words traveling waves standing waves bifurcation dispersion reaction diffusion equations singular perturbation

wave block in excitable media due to regions of depressed excitability we study propagation failure using the one dimensional scalar bistable equation with a passive gap region by applying comparison principles for this type of equation the problem of finding conditions for block is reduced to finding conditions for the existence of steady state solutions we present a geometrical method that allows one to easily compute the critical gap length above which a steady state solution and thus block first occurs the method also helps to uncover the general bifurcation structure of the problem including the stability of the steady state solutions in obtaining these results we characterize the relationship between the properties of the system and propagation failure the method can easily be extended to other gap dynamics we use it to show that block associated with any local inhomogeneity must be associated with a limit point bifurcation

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

higher dimensional localized patterns in excitable media abstract the excitable reaction diffusion equation model of the form e u t e 2 2 u u v v t 2 v u v is considered when u is assumed to be of mckean s piecewise linear type the interfacial approach can be applied to the stability of various localized patterns in higher dimensional spaces it is shown that a band shaped localized pattern is destabilized into a zig zag mode or a varicose mode and that a disk shaped localized pattern is destabilized into static modes and when is small into an oscillatory mode like a breather motion numerical simulations are performed to confirm such destabilizations for more general nonlinear functions u

pattern selection in a general model of convection diffusion and catalytic reaction abstract we study the process of pattern selection in a large class of heterogeneous models of catalytic reactor by analyzing the behavior of a simple condensed model which obeys reflection and inversion symmetries and which captures the main features of a detailed model the three variable model incorporates a very fast and long ranged variable which may describe the fluid phase in a mixed plug flow or an axial dispersion reactor or the pore phase in a catalytic particle and two variables a medium ranged activator and a localized inhibitor that describe the solid phase patterns are classified according to their symmetry properties patterns may emerge already with simple bistable kinetics in a plug flow reactor i e a single integrodifferential equation pattern selection is determined by the phase planes spanned by the reactor by the ratio of front residence time to the period of oscillations and by the ratio of convection to fluid phase diffusions terms patterns include stationary or oscillatory fronts or pulses antiphase oscillations which appear at the transition from homogeneous solution unidirectional pulses that are the dominant patterns when mixing is weak and stationary or oscillatory waves that emerge with strong fluid mixing the simple form of the model allows us to suggest a general classification of emerging patterns and to identify the corresponding conditions in term of realistic models of reactors

onset of unidirectional pulse propagation in an excitable medium with asymmetric heterogeneity heterogeneity is one of the most important and ubiquitous types of external perturbations in dissipative systems to know the behaviors of pulse waves in such media is closely related to studying the collision process between the pulse and the heterogeneity induced ordered pattern in particular we focus on unidirectional propagation of pulses in a medium with an asymmetric bump heterogeneity this topic has attracted much interest recently with respect to potential computational aspects of chemical pulse propagation as well as with respect to pulse propagation in biological signal processing we employ a three component reaction diffusion system with one activator and two inhibitor species to illustrate these issues the reduced dynamics near a drift bifurcation describes the phenomena in the full partial differential equations by ordinary differential equations such a reduced dynamics is able to capture unidirectional propagation properties of pulses near an asymmetric heterogeneity in a qualitatively correct way a remarkable feature is that such unidirectional behavior is linked to the imperfection of global bifurcation structure and the resulting asymmetric locations of critical points

front bifurcations in reaction diffusion systems with inhomogeneous parameter distributions abstract fronts between two different stable homogeneous domains are considered in the framework of two component reaction diffusion rd systems with fitzhugh nagumo type dynamics the dominance principle ensuring that for a system with homogeneous parameters the direction of front propagation is uniquely determined by the relative strengths of the two domains fails when the inhibitor s controller s time scale is slowed down sufficiently this leads to a regime of bi directional propagation where for a given set of parameters either of the two domains may win the competition bi directional propagation may be viewed as being organized by a bifurcation of the stationary front obtained for equally strong domains which is destabilized when a critical value of the relative time constant is crossed although generically one would expect a transcritical scenario this bifurcation turns out to be of supercritical pitchfork type for arbitrary nonlinearity i e the bifurcation is independent of an accidental inflection symmetry of the dynamics for inhomogeneous parameter distributions the behavior becomes richer leading to stationary and oscillating pinning as well as to reflections of incoming fronts this situation is analyzed performing a multiple scale perturbation expansion up to third order in the vicinity of the above bifurcation leading to a two dimensional reduced dynamics globally valid in space the two degrees of freedom relevant for the description are the position of the front and a characteristic feature of its shape the latter is reminiscent of the momentum of classical particle physics results are compared with full system numerics

hausdorff dimension and conformal measures of feigenbaum julia sets 1 1 statement of the results one of the first questions usually asked about a fractal subset of r is whether it has the maximal possible hausdorff dimension n it certainly happens if the set has positive lebesgue measure on the other hand it is easy to construct fractal sets of zero measure but of dimension n moreover this phenomenon is often observable for fractal sets produced by conformal dynamical systems iterated rational functions or kleinian groups in particular the analogy with kleinian groups suggested that the julia sets of feigenbaum maps should have hausdorff dimension two in this paper we will show that this is not always the case a quadratic polynomial or more generally a quadratic like map is called feigenbaum if it is infinitely renormalizable of bounded combinatorial type with a priori bounds see 2 for a precise definition feigenbaum polynomials are remarkable dynamical systems whose geometry has been a focus of research for the past 25 years by analogy with kleinian groups it was anticipated that their julia sets have dimension 2 and this problem has been around for a while however in this paper we will show that this is not always the case

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

internal layer oscillations in fitzhugh nagumo equation a controller propagator system with a fitzhugh nagumo equation can be reduced to a free boundary problem when a layer parameter e is equal to zero we shall show the existence of solutions and the occurence of a hopf bifurcation for this free boundary problem as the controlling parameter varies

stationary and oscillatory fronts in a two component genetic regulatory network model abstract we investigate a two component gene network model originally used to describe the spatiotemporal patterning of the gene products in early drosophila development by considering a particular mode of interaction between the two gene products denoted proteins a and b we find both stable stationary and time oscillatory fronts can occur in the reaction diffusion system we reduce the system by replacing b with its spatial average shadow system and assume an abrupt on and off switch for the genes in doing so explicit formula are obtained for all steady state solutions and their linear eigenvalues using the diffusion of a d a and the basal production rate r as bifurcation parameters we explore ranges in which a monotone stationary front is stable and show it can lose stability through a hopf bifurcation giving rise to oscillatory fronts we also discuss the existence and stability of steady state and time oscillatory solutions with multiple extrema an intuitive explanation for the occurrence of stable stationary and oscillatory front solutions is provided based on the behavior of a in the absence of b and the opposite regulation between a and b such behavior is also interpreted in terms of the biological parameters in the model including those governing the connection of the gene network

attractors and recurrence for dendrite critical polynomials we call a rational map f dendrite critical if all its recurrent critical points either belong to an invariant dendrite d or have minimal limit sets we prove that if f is a dendrite critical polynomial then for any conformal measure either for almost every point its limit set coincides with the julia set of f or for almost every point its limit set coincides with the limit set of a critical point c of f moreover if is non atomic then c can be chosen to be recurrent a corollary is that for a dendrite critical polynomial and a non atomic conformal measure the limit set of almost every point contains a critical point

a free boundary problem arising in some reacting diffusing system we prove the well posedness for a one dimensional free boundary problem arising from some reaction diffusion system the interfacial point hits a boundary point in finite time or remains inside for all time in the large diffusion limit the system is reduced to ordinary differential equations of finite dimension

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

bloch front turbulence theory and experiments abstract we report on experimental and theoretical findings of front destabilization that causes spontaneous spiral vortex nucleation and produces a state of spatio temporal disorder the experiments were carried out on an oscillatory photosensitive belousov zhabotinsky reaction that is periodically forced in time numerical studies were carried out on a modified complex ginzburg landau equation and on the fitzhugh nagumo model using velocity curvature relations for fronts we associate the onset of spatio temporal disorder with the nonequilibrium ising bloch nib bifurcation and study the generic patterns that form on both sides of the bifurcation as the distance from it is increased

structural properties of controlled population models abstract the classical leslie model used for describing the natural growth of age structured populations is adapted to the case of controlled populations such a model is a nonlinear positive system with non negative inputs and enjoyes remarkable structural properties the non trivial equilibria are critically stable but can be stabilized by a state variable feedback the set of states reachable from any given initial state is a positive cone generated by n reachability vectors b ab a n 1 b the system is completely observable and its state can be reconstructed provided the input sequence can be suitably programmed all these properties constitute an important theoretical framework for population control problems

backward stability for polynomial maps with locally connected julia sets we study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability then we deduce from this that a polynomial f with a locally connected julia set is backward stable outside any neighborhood of its attracting and neutral cycles for a conformal measure this easily implies that one of the following holds 1 for a e x e j f x j f 2 for a e x j f x c x for a critical point c x depending on x

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

the existence and stability of spike equilibria in the one dimensional gray scott model the pulse splitting regime abstract the existence stability and pulse splitting behavior of spike patterns in the one dimensional gray scott model on a finite domain is analyzed in the semi strong spike interaction regime this regime is characterized by a localization of one of the components of the reaction near certain spike locations while the other component exhibits a more global spatial variation across the domain the method of matched asymptotic expansions is then used to construct k spike equilibria in terms of a certain core problem this core problem is studied numerically and asymptotically for each integer k 1 it is shown that there are two branches of k spike equilibria that meet at a saddle node bifurcation value for small values of the diffusivity d of the second component these saddle node bifurcation points occur at approximately the same value a combination of asymptotic and numerical methods is used to analyze the stability of these branches of k spike equilibria with respect to both drift instabilities associated with the small eigenvalues and oscillatory instabilities of the spike profile in this way the key bifurcation and spectral conditions of ei et al s ei y nishiura k ueda 2 n splitting or edge splitting a manner of splitting in dissipative systems jpn j ind appl math 18 2001 181 205 believed to be essential for pulse splitting behavior in a reaction diffusion system are verified by having verified these conditions a simple analytical criterion for the occurrence of pulse splitting is then formulated and confirmed with full numerical simulations of the gray scott model this criterion verifies a conjecture based on numerics and topological arguments reported in a doelman r a gardner t j kaper stability analysis of singular patterns in the 1d gray scott model a matched asymptotics approach physica d 122 1 4 1998 1 36 the analytical results are compared with previously obtained results for pulse splitting behavior

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

excitability stability and sign of equilibria in positive linear systems abstract the relationship between sign and stability of equilibria in positive linear systems is investigated in this paper in particular it is proved that the well known result stability and irreducibility imply strict positivity can be better specified by introducing the notion of excitability in fact stability excitability and strict positivity of the equilibria are three perfectly balanced properties since any one of them is implied by the two others

global bifurcation phenomena of traveling wave solutions for some bistable reaction diffusion systems on considere le systeme bistable e u t e 2 u xx f u v v t v xx g u v ou e est petit on etudie les solutions onde de propagation qui tendent vers deux etats stables quand x

on the existence of conformal measures a general notion of conformal measure is introduced and some basic properties are studied sufficient conditions for the existence of these measures are obtained using a general construction principle the geometric properties of conformal measures relate equilibrium states and hausdorff measures this is shown for invariant subsets of s under expanding maps

conservative exact rational maps of the sphere abstract in this paper we investigate conditions for conservativity and exactness ofd to 1 endomorphisms we prove that all rational maps of degree greater than 1 are either conservative and exact with respect to normalized lebesgue measure on the sphere or have an attracting property inside the julia set we provide examples of rational maps which can be shown to be conservative and exact by using the previous result which was not formerly known to have this property

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

effects on global coupling of an interface problem in an activator inhibitor system in this paper we shall explore the dynamics of interfaces in the problem with global coupling in order to investigate the existence of time periodic solutions as the global coupling intensity and the bifurcation parameter vary

necessity of internal and boundary bulk balance law for existence of interfaces for an elliptic system with nonlinear boundary condition abstract for a system of stationary solutions to a reaction diffusion equations with small diffusion coefficient and nonlinear flux boundary condition we prove that the bulk balance law not only on the domain but on its boundary as well is a necessary condition for formation of internal and boundary layers

wild attractors of polymodal negative schwarzian maps we study wild attractors of polymodal negative schwarzian interval maps and prove that they are it super persistently recurrent a polymodal version of persistent recurrence we also prove that if a map has an attractor which is a cycle of intervals then at almost every point of this cycle the map has properties similar to the markov property introduced by martens thus the lack of super persistent recurrence at a critical point c can be considered as a mild topological expanding property and this expansion prevents c from being a wild attractor in the previous paper we have shown that it also prevents the map from being c2 stable

stability of position control of traveling waves in reaction diffusion systems we consider the stability of position control of traveling waves in reaction diffusion systems as proposed in l ober and engel phys rev lett 112 148305 2014 instead of analyzing the controlled reaction diffusion system stability is studied on the reduced level of the equation of motion for the position over time of perturbed traveling waves we find an interval of perturbations of initial conditions for which position control is stable this interval can be interpreted as a localized region where traveling waves are susceptible to perturbations for stationary solutions of reaction diffusion systems with reflection symmetry this region does not exist analytical results are in qualitative agreement with numerical simulations of the controlled schl ogl model

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

wavy front dynamics in a three component reaction diffusion system with one activator and two inhibitors an analytic description for traveling waves in a one dimensional reaction diffusion system with one activator and two inhibitors and with equal diffusion constants is developed using a piecewise linear approximation for the nonlinear activator reaction term the case of front waves is examined in more detail the monotonic and oscillating fronts being separately considered the corresponding wave profiles are constructed and the speed equation is obtained and discussed it is found that the fronts in the three component model propagate faster than the fronts in the two component system the front interaction is studied using numerical calculations the results show that at head on collisions two oscillating fronts produce a wavy domain which spreads in space with time

equilibria stability and reachability of leslie systems with nonnegative inputs leslie systems a particular class of positive systems used in population dynamics and control are defined and analyzed a simple property relating sign and stability of their equilibria and some geometric characteristics of their reachability set in cases of bounded and unbounded inputs are proved all the properties do not hold in general in other classes of positive systems it is shown that leslie systems have strictly positive equilibria if and only if they are asymptotically stable their reachability set in the case of unbounded inputs is the positive cone generated by the reachability vectors while in the case of bounded inputs it is a polyhedron whose vertices can be easily computed all of these properties follow from the nonpositiveness of the coefficients of the characteristic polynomial of the system