generic bifurcation of hamiltonian systems with symmetry we study generic bifurcations of equilibria in one parameter hamiltonian systems with symmetry group on the generalized eigenvalues of the linearized system go through zero theorem 3 3 classifies expected actions of on the generalized eigenspace of this zero eigenvalue generic one degree of freedom symmetric systems is section 4 remarks concerning systems with more degrees of freedom are given in section 5

chaos in a low order model of magnetoconvection abstract in the limit of tall thin rolls weakly nonlinear convection in a vertical magnetic field is described by an asymptotically exact third order set of ordinary differential equations these equations are shown here to have three codimension two bifurcation points a takens bogdanov bifurcation at which a gluing bifurcation is created a point at which the gluing bifurcation is replaced by a pair of homoclinic explosions between which there are lorenz like chaotic trajectories and a new type of bifurcation point at which the first of a cascade of period doubling bifurcation lines originates the last two bifurcation points are analysed in terms of a one dimensional map the equations also have a t point at which there is a heteroclinic connection between a saddle and a pair of saddle foci emerging from this point is a line of sil nikov bifurcations involving homoclinic connections to a saddle focus

homoclinic bifurcations in chua s circuit abstract in this paper we study the possible relationship between the birth of the double scroll l o chua et al ieee cas 33 11 1986 1073 and the homoclinic bifurcations in the traditional chua s equations using a one dimensional poincare map we determine the existence of secondary symmetric homoclinic orbits of shil nikov type born with the chua s attractor connecting unstable and stable manifolds of the trivial equilibrium point in addition taking into account the presence of other homoclinic orbits for the asymmetric attractor and heteroclinic orbits for the symmetric attractor connecting unstable and stable manifold of the non trivial equilibrium points we suggest a hypothesis about the birth of double scroll structure on the plane

hopf bifurcation for equivariant conservative and time reversible systems we study the bifurcation of small periodic solutions at a non semi simple 1 1 resonance in equivariant conservative or equivariant time reversible systems by using an equivariant liapunov schmidt method and restricting to solutions with an appropriate isotropy we reduce the problem to a scalar bifurcation equation the analysis of this equation shows a bifurcation behaviour similar to that found for the hamiltonian hopf bifurcation

bifocal homoclinic bifurcations homoclinic orbits to bifocus type stationary points have been studied theoretically by a number of authors but up until now only one analytic example has been found in this paper we summarise and extend the known theory regarding bifocal homoclinic bifurcations and present numerical verification of some of the more interesting theoretical predictions that have been made

sil nikov saddle node interaction near a codimension 2 bifurcation laser with injected signal abstract we describe a sil nikov saddle node interaction in the proximity of a hopf saddle node bifurcation both from numerical and theoretical viewpoints inspired by numerical observations on a three dimensional model of a laser with injected signal we describe the interaction near the codimension 2 point using a return map controlled by the parameters of the bifurcation the theoretical predictions compare well with numerical experiments finally we discuss the reasons for the existence of the sil nikov saddle node bifurcation in the laser with injected signal and correspondingly the characteristics of other systems that might display this bifurcation

probabilistic approach to homoclinic chaos three dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues i and giving rise to homoclinic chaos when the shil nikov condition is satisfied are studied the 2d poincare map and its 1d contractions capturing the essential features of the flow are given at homoclinicity these 1d maps are found to be piecewise linear this property allows one to reduce the frobenius perron equation to a master equation whose solution is analytically known the probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived

sbil nikov hopf bifurcation abstract we derive a 2 parameter poincare map modeling a 3 dimensional vector flow near the codimension 2 point defined by a sil nikov homoclinic orbit to a fixed point undergoing a hopf bifurcation in one parameter regime the map reproduces glendinning and sparrow s results for the sil nikov bifurcation while in another regime it provides new results for a small hopf cycle with a global reinjection mechanism the map is analyzed finding primary homoclinic tangencies to the small hopf cycle parameter dependence of periodic orbits associated with these tangencies is described global behavior of the tube like unstable manifold of the hopf cycle is studied in depth analytically and numerically locating its simplest subsidiary tangencies

long tailed maps as a representation of mixed mode oscillatory systems mixed mode oscillatory mmo systems are known to exhibit generic features such as the reversal of period doubling sequences and crossover to period adding sequences as bifurcation parameters are varied in addition they exhibit a nearly one dimensional unimodal poincare map with a long tail the numerical results of a map with a unique critical point map l show that these dynamical features are reproduced we show that a few generic conditions extracted from the map l are adequate to explain the reversal of period doubling sequences and crossover to period adding sequences we derive scaling relationsthat determine the parameter widths of the dominant windows of periodic orbits sandwiched between two successive states of rlk sequence and verify the same with the map l as the conditions used to derive the scaling relations do not depend on the form of map we suggest that the analysis is applicable to a family of two parameter one dimensional maps that satisfy these conditons

homoclinic doubling cascades cascades of period doubling bifurcations have attracted much interest from researchers of dynamical systems in the past two decades as they are one of the routes to onset of chaos in this paper we consider routes to onset of chaos involving homoclinic doubling bifurcations we show the existence of cascades of homoclinic doubling bifurcations which occur persistently in two parameter families of vector fields on 3 the cascades are found in an unfolding of a codimension three homoclinic bifurcation which occur an orbit flip at resonant eigenvalues we develop a continuation theory for homoclinic orbits in order to follow homoclinic orbits through infinitely many homoclinic doubling bifurcations

homoclinic phenomena in opto thermal bistability with localized absorption abstract time dynamics of a family of opto thermal nonlinear devices is described by means of a system of linear partial differential equations subjected to a nonlocal and nonlinear boundary condition and a rich variety of homoclinic phenomena is numerically found linear stability analysis shows that the effective dynamical dimension is determined by the device structure i e by the number of layers between two mirrors and then it may be easily varied a variety of local and global bifurcations observed in bilayer systems are described in detail showing that the dynamics is in effect two dimensional except for subtle features appearing in a gluing bifurcation where two homoclinic connections occur almost simultaneously complex behaviour is shown to occur in the case of trilayer systems with a very similar dynamics to the one of the well known rossler model of third order ordinary differential equations two different families of aperiodic phase portraits are described in detail and their association with homoclinic connections to saddle invariant sets of different configurations is pointed out the occurrence of complex dynamics is demonstrated by means of first return 1d maps obtained in proper poincare sections

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

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

characterization of a two parameter mixed mode electrochemical behavior regime using neural networks abstract we use nonlinear signal processing techniques based on artificial neural networks to construct a continuous time model set of ordinary differential equations odes from experimental observations of mixed mode oscillations during the galvanostatic oxidation of hydrogen on platinum in the presence of bismuth and chloride ions the data was in the form of time series of the potential for different values of the applied current and chloride ion concentration we use the model to reconstruct the experimental dynamics and to explore the associated bifurcation structures in phase space using numerical bifurcation techniques we locate stable and unstable periodic solutions calculate eigenvalues and identify bifurcation points this approach constitutes a promising data post processing procedure for investigating phase space and parameter space of real experimental systems it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings for example the fitted model suggests the existence of a sub critical hopf bifurcation near the range of parameters probed in the experiments this might explain the experimental difficulty in locating small amplitude simple oscillations

linear hamiltonian hopf bifurcation for point group invariant perturbations of the 1 1 1 resonance we consider g x r invariant hamiltonians h on complex projective 2 space where g is a point group and r is the time reversal group we find the symmetry induced stationary points of h and classify them in terms of their linear stability we then determine those points that can undergo a linear hamiltonian hopf bifurcation

experimental demonstration of subtleties in subharmonic intermittency abstract experiments on the belousov zhabotinsky reaction in a well stirred flow reactor elucidate one of the routes to chaos subharmonic intermittency type iii intermittency measurements conducted as a function of two independent control parameters demonstrate how difficult it is in practice to distinguish this route to chaos from subcritical period doubling transitions with similar chaotic time signatures necessary criteria for the establishment of a second order continuous transition such as the absence of hysteresis are discussed in the context of one dimensional maps

the hopf bifurcation with symmetry for the navier stokes equations in l p ω n with application to plane poiseuille flow the existence of periodic solutions of the navier stokes equations in function spaces based upon lp nis proved the paper has three parts a a proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of lp n b the evolution equation is restricted to a space of time periodic functions and a fredholm integral equation on this space is formed the lyapunov schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry c the theory is applied to the bifurcation of periodic solutions from planar poiseuille flow in the presence of symmetry so 2 x o 2 x s1 yielding new results for this classic problem the o 2 invariance is in the spanwise direction with the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction there are however points of degeneracy on the neutral surface a numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi periodic solutions

representations of hermitian kernels by means of krein spaces ii invariant kernels in this paper we study hermitian kernels invariant under the action of a semigroup with involution we characterize those hermitian kernels that realize the given action by bounded operators on a krein space this is motivated by the gns representation of algebras associated to hermitian functionals the dilation theory of hermitian maps on c algebras as well as others we explain the key role played by the technique of induced krein spaces and a lifting property associated to them

qualitative analysis of the rossler equations bifurcations of limit cycles and chaotic attractors in this paper we study different aspects of the paradigmatic rossler model we perform a detailed study of the local and global bifurcations of codimension one and two of limit cycles this provides us a global idea of the three parametric evolution of the system we also study the regions of parameters where we may expect a chaotic behavior by the use of different chaos indicators the combination of the different techniques gives an idea of the different routes to chaos and the different kinds of chaotic attractors we may found in this system

equivalent low order model for a nonlinear diffusion equation abstract we present an equivalent low order model for a simple pde system that exhibits interesting low dimensional dynamics with a rich variety of homoclinic phenomena and whose effective dimension may be gradually increased by means of system parameters the system is a linear heat equation subject to a nonlinear and nonlocal boundary condition and the reduction procedure is based on a finite element method we will show that both the pde and ode systems have indentical stationary solution with a very similar linear stability behaviour and exhibit also very similar dynamics at least within parameter ranges corresponding to physical devices

chaos from hopf bifurcation in a fluid flow experiment results of an experimental study of a hopf bifurcation with broken translation symmetry that organizes chaotic homoclinic dynamics from a t2 torus in a fluid flow as a direct consequence of physical boundaries are presented it is shown that the central features of the theory of hopf bifurcation in o 2 symmetric systems where the translation symmetry is broken are robust and are appropriate to describe the appearance of modulated waves homoclinic bifurcation takens bogdanov point and chaotic dynamics in a fluid flow experiment

a crisis in the belousov zhabotinskii reaction experiment and simulation an abrupt transition that has the character of aninterior crisis was observed in an experiment on the belousov zhabotinskii reaction as a control parameter was varied a crisis is a qualitative change in the dynamics of a system observed when an attractor collides with the stable manifold of a fixed point the interpretation of the observed behavior as a crisis is corroborated by a numerical analysis of a seven variable model of the reaction the waveforms attractors and maps obtained in the simulation are remarkably similar to those obtained in the laboratory experiment the simulation indicates that the crisis is a consequence of a multivalued first return map

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

global bifurcation of polygonal relative equilibria for masses vortices and dnls oscillators given a regular polygonal arrangement of identical objects turning around a central object masses vortices or dnls oscillators this paper studies the global bifurcation of relative equilibria in function of a natural parameter central mass central circulation or amplitude of the oscillation the symmetries of the problem are used in order to find the irreducible representations the linearization and with the help of a degree theory the symmetries of the bifurcated solutions

the origins of chaos in a modified van der pol oscillator abstract an experimental and theoretical investigation of a modified van der pol oscillator has led to the discovery of degenerate bifurcations which control the global dynamics of the system a two dimensional parameter space for the oscillator is studied and the measured loci of critical points where a qualitative change of behaviour is observed are given in detail it is shown explicitly by analysis of the experimental data that chaos arises through the silnikov mechanism

minimal irreversible quantum mechanics the decay of unstable states abstract brownian motion is modelled by a harmonic oscillator brownian particle interacting with a continuous set of uncoupled harmonic oscillators the interaction is linear in the coordinates and the momenta the model has an analytical solution that is used to study the time evolution of the reduced density operator it is derived in a closed form in the one particle sector of the model the irreversible behavior of the brownian particle is described by a reduced density matrix

recognizing chaotic time waveforms in terms of a parametrized family of nonlinear predictors abstract consider a chaotic dynamical system which exhibits a variety of chaotic time waveforms with a change in the bifurcation parameters this paper presents an algorithm for estimating the underlying bifurcation parameters of the chaotic time waveforms in experimental situation in which no a priori analytical knowledge of the dynamical system is available first we construct qualitatively similar parametrized family of nonlinear predictors only from several sets of chaotic time waveforms qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original chaotic time waveforms are then characterized in terms of the qualitatively similar bifurcation parameters of the nonlinear predictors we call the characterization of chaotic time waveforms in terms of the underlying bifurcation parameters chaotic time waveform recognition several numerical experiments using the rossler equations show the efficiency of the algorithm the effect of observational noise included in chaotic time waveforms is also considered

voltage interval mappings for activity transitions in neuron models for elliptic bursters abstract we performed a thorough bifurcation analysis of a mathematical elliptic bursting model using a computer assisted reduction to equationless one dimensional poincare mappings for a voltage interval using the interval mappings we were able to examine in detail the bifurcations that underlie the complex activity transitions between tonic spiking and bursting bursting and mixed mode oscillations and finally mixed mode oscillations and quiescence in the fitzhugh nagumo rinzel model we illustrate the wealth of information qualitative and quantitative that was derived from the poincare mappings for the neuronal models and for similar electro chemical systems

belyakov homoclinic bifurcations in a tritrophic food chain model complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper first it is shown that the model admits a sequence of pairs of belyakov bifurcations codimension two homoclinic orbits to a critical node then fold and period doubling cycle bifurcation curves associated to each pair of belyakov points are computed and analyzed the overall bifurcation scenario explains why stable limit cycles and strange attractors with different geometries can coexist the analysis is conducted by combining numerical continuation techniques with theoretical arguments

a case study for homoclinic chaos in an autonomous electronic circuit a trip taken from takens bogdanov to hopf sbil nikov abstract the dynamics of an autonomous electronic oscillator is analysed both theoretical study with methods of local bifurcation theory and numerical simulation by using continuation methods to detect global behaviours are carried out comment is made on the presence of homoclinic connections which organise the periodic and nonperiodic behaviour the richness and complexity of the periodic oscillations are described and the presence of isolated branches of odd period orbits are emphasized further attention is paid to the chaotic attractor coexisting with sil nikov homoclinicity even in the presence of a hopf degeneracy the hopf sil nikov singularity

computation of the equivariant 1 stem abstract let g be a compact lie group in this paper combining a short exact sequence obtained by balanov and krawcewicz with some additional topological techniques we complete the computation of the first equivariant stem 1 g st using the exact sequence and a property of nonabelian connected compact lie groups whose proof was suggested to us by r oliver we show that this group is finite if and only if g is finite

relative equilibria near stable and unstable hamiltonian relative equilibria for a symmetric hamiltonian system lower bounds for the number of relative equilibria surrounding stable and formally unstable relative equilibria on nearby energy levels are given

some remarks on the gelfand naimark segal representations of topological algebras after an appropriate restatement of the gelfand naimark segal construction for topological algebras we prove that there exists an isomorphism among the set cycl a of weakly continuous strongly cyclic representations of a barreled dual separable algebra with unit a the space hilba a of the hilbert spaces that are continuously embedded in a and are invariant under the dual left regular action of a and the set of the corresponding reproducing kernels we show that these isomorphisms are cone morphisms and we prove many interesting results that follow from this fact we discuss how these results can be used to describe cyclic representations on more general inner product spaces

possible new strange attractors with spiral structure we define a class of three dimensional differential equations which might present strange attractors of a new kind this is illustrated by numerical observations on an explicit example

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

bifurcations of mixed mode oscillations in a three variable autonomous van der pol duffing model with a cross shaped phase diagram abstract the bifurcation structure of a three variable van der pol duffing type model is studied in some detail with special attention to the mixed mode solutions a type of complex periodic behavior frequently encountered in oscillating chemical reactions the mixed mode oscillations in the model occur close to two hopf bifurcations which are arranged with the saddle node bifurcations in a so called cross shaped phase diagram a bifurcation diagram also typical for chemical reactions the mixed mode oscillations are shown to lie on isolated bifurcation curves which are all born in a single codimension two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch with the introduction of an additional slow time scale the same model can exhibit more complex mixed mode oscillations and torus bifurcations

on gns representations on inner product spaces a generalization of the gns construction to hermitian linear functionals w defined on a unital algebra is considered along these lines a continuity condition h upon w is introduced such that h proves to be necessary and sufficient for the existence of a j representation on a krein space the property whether or not the gram operator j leaves the common and invariant domain of the representation invariant is characterized as well by properties of the functional w as by those of furthermore the interesting class of positively dominated functionals is introduced and investigated some applications to tensor algebras are finally discussed

instabilities of front patterns in reaction diffusion systems abstract recent experiments in chemical reaction diffusion systems with externally imposed concentration gradients may provide access to a host of spatio temporal pattern formation phenomena these systems tend to form steep reaction fronts in response to the external gradient we use singular perturbation techniques normal form calculations and numerical simulations to investigate the existence and the stability of such sustained fronts in one dimensional systems the theoretical predictions are found in quantitative agreement with direct simulations of the hopf bifurcation from steady to periodically oscillating front structures observed in the couette flow reactor also conditions are found under which oscillations of the spatial structure become chaotic in two dimensional systems we address the issue of realizing an experimental situation hitherto unattained a one dimensional chain of coupled oscillators at the onset of the hopf destabilization of the front structure we point out the intimate relationship between the frequency of oscillation of the homogeneous front pattern and the characteristic wavelength of the turing pattern that can develop along the front 2 d 1 2 we comment on subsequent bifurcations that may result from the nonlinear interaction between hopf and turing instabilities as the precursors to spatio temporal chaos in conclusion we emphasize the possibility of probing the transition to mediated defect turbulence in thin film gel reactors

irreducible representations of the heisenberg algebra in krein spaces the representations of the heisenberg algebra in krein spaces more generally in weakly complete inner product spaces are classified under general regularity and irreducibility conditions besides the fock representation two other types appear one with negative the other with a two sided discrete spectrum of the number operator

bifurcation to spatially induced chaos in a reaction diffusion system a one dimensional reaction diffusion equation is used to model a class of experimental open chemical systems in which spatiotemporal patterns can be sustained indefinitely numerical simulations for the model in parameter regimes corresponding to experiment reveal only low dimensional behavior the observed bifurcation sequence leads from steady state concentration profiles to temporally chaotic patterns the physical mechanism that causes this behavior is deduced from analysis of the local dynamics localized oscillators appear at each end of the reactor and one of the oscillators acts as a periodic forcing for the other the numerical results are in good qualitative agreement with the experiments

towards a definition of the quantum ergodic hierarchy ergodicity and mixing abstract in a previous paper we have given a general framework for addressing the definition of quantum chaos by identifying the conditions that a quantum system must satisfy to lead to non integrability in its classical limit in this paper we will generalize those results with the purpose of defining the two lower levels of the quantum ergodic hierarchy ergodicity and mixing we will also argue for the physical relevance of this approach by considering a particular example where our formalism has been successfully applied

effects of coarse graining on the scaling behavior of long range correlated and anti correlated signals abstract we investigate how various coarse graining signal quantization methods affect the scaling properties of long range power law correlated and anti correlated signals quantified by the detrended fluctuation analysis specifically for coarse graining in the magnitude of a signal we consider i the floor ii the symmetry and iii the centro symmetry coarse graining methods we find that for anti correlated signals coarse graining in the magnitude leads to a crossover to random behavior at large scales and that with increasing the width of the coarse graining partition interval this crossover moves to intermediate and small scales in contrast the scaling of positively correlated signals is less affected by the coarse graining with no observable changes when 1 while for 1 a crossover appears at small scales and moves to intermediate and large scales with increasing for very rough coarse graining 3 based on the floor and symmetry methods the position of the crossover stabilizes in contrast to the centro symmetry method where the crossover continuously moves across scales and leads to a random behavior at all scales thus indicating a much stronger effect of the centro symmetry compared to the floor and the symmetry method for coarse graining in time where data points are averaged in non overlapping time windows we find that the scaling for both anti correlated and positively correlated signals is practically preserved the results of our simulations are useful for the correct interpretation of the correlation and scaling properties of symbolic sequences

homoclinic hyperchaos an explicit example abstract a four variable 2 domain piecewise linear c 1 ordinary differential equation is indicated explicitly two parameter sets and initial conditions implying a homoclinic return to a generalized saddle focus are given with finite precision the numerically larger eigenvalue is negative the two positive eigenvalues one for the complex conjugate pair the other real are equal only under this singular condition does the extended neighborhood in parameter space in which topological hyperchaos survives include the homoclinic case itself this does not diminish the power of the homoclinicity criterion the second set of parameters and initial conditions is valid for arbitrarily high dimension numbers it implies existence of maximum hyperchaos in an extended neighborhood analytical and numerical studies on both the c 1 and the c version of the equation are possible

applications of topological algebras of unbounded operators in this paper we discuss some physical applications of topological algebras of unbounded operators the first example is a simple system of free bosons then we consider other models related to this one finally we discuss the time evolution of two interacting models of matter and bosons we show that for all these systems it is possible to build up a common mathematical framework where the existence of the thermodynamical limit of the algebraic dynamics can be conveniently analyzed

probing a subcritical instability with an amplitude expansion an exploration of how far one can get abstract we explore methods to locate subcritical branches of spatially periodic solutions in pattern forming systems with a nonlinear finite wavelength instability we do so by means of a direct expansion in the amplitude of the linearly least stable mode about the appropriate reference state which one considers this is motivated by the observation that for some equations fully nonlinear chaotic dynamics has been found to be organized around periodic solutions that do not simply bifurcate from the basic laminar state we apply the method to two model equations a subcritical generalization of the swift hohenberg equation and a novel extension of the kuramoto sivashinsky equation that we introduce to illustrate the abovementioned scenario in which weakly chaotic subcritical dynamics is organized around periodic states that bifurcate from infinity and that can nevertheless be probed perturbatively we explore the reliability and robustness of such an expansion with a particular focus on the use of these methods for determining the existence and approximate properties of finite amplitude stationary solutions such methods obviously are to be used with caution the expansions are often only asymptotic approximations and if they converge their radius of convergence may be small nevertheless expansions to higher order in the amplitude can be a useful tool to obtain qualitatively reliable results

periodic solutions of nonlinear parabolic equations abstract in this paper we study the periodic neumann boundary value problem for semilinear parabolic equations we present some existence results and study the structure of the set of solutions giving sufficient conditions under which that solution set is compact connected or acyclic

homoclinic chaos in chemical systems abstract we first focus on complex dynamical phenomena generated by chemical kinetics in homogeneous media we comment on the alternating sequences of periodic and chaotic states observed in some experiments on the belousov zhabotinsky reaction when conducted in a continuously stirred tank reactor we present numerical results obtained with a seven variable oregonator model which reproduce most of the features of the experimental sequences we discuss the origin and the nature of the chemical chaos encountered along these sequences in terms of the chaotic orbits which exist on the way to homoclinicity we construct poincare map models which reduce to either multi humped or multi branched 1d maps in the strong area contraction limit we report on a recent experiment which provides evidence that the iteration scheme deduced from the time series of spiral type strange attractors satistifies the symbolic dynamics predicted by sil nikov theory of homoclinic chaos we then investigate spatio temporal pattern forming phenomena in a one dimensional reaction diffusion system with equal diffusion coefficients when imposing a concentration gradient through the system this model mimics the sustained stationary and periodically oscillating front structures observed in recent experiments conducted in an open couette flow reactor we emphasize the possibility that the oscillations of the spatial structure become chaotic we report on numerical simulations of a diffusion induced intermittent bursting phenomenon that is likely to be observed in bench experiments we elaborate on the interpretation of this intermittent occurrence of spatially localized structures in an extended system in terms of sil nikov s homoclinic chaos

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

incomplete approach to homoclinicity in a model with bent slow manifold geometry the dynamics of a model originally proposed for a type of instability in plastic flow has been investigated in detail the bifurcation portrait of the system in two physically relevant parameters exhibits a rich variety of dynamical behavior including period bubbling and period adding or farey sequences the complex bifurcation sequences characterized by mixed mode oscillations exhibit partial features of shilnikov and gavrilov shilnikov scenario utilizing the fact that the model has disparate time scales of dynamics we explain the origin of the relaxation oscillations using the geometrical structure of the bent slow manifold based on a local analysis we calculate the maximum number of small amplitude oscillations s in the periodic orbit of l s type for a given value of the control parameter this further leads to a scaling relation for the small amplitude oscillations the incomplete approach to homoclinicity is shown to be a result of the finite rate of softening of the eigenvalues of the saddle focus fixed point the latter is a consequence of the physically relevant constraint of the system which translates into the occurrence of back to back hopf bifurcation

imperfect homoclinic bifurcations experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations these results are discussed in the light of a theory of imperfect global bifurcations it is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system

analysis of a six component atmospheric spectral model chaos predictability and vacillation abstract the properties of a low order atmospheric spectral model are investigated by using a bifurcation analysis of its steady states and periodic solutions the time asymptotic behaviour of the model is either stationary periodic quasi periodic or chaotic depending on the parameter values and initial conditions different scenarios are found leading to the generation of strange attractors some include the occurrence of homoclinic orbits such that for nearby parameter values chaotic orbits exist moving in small tubes around the homoclinic orbits the chaotic motion describes an irregular flow predictable on a time scale given by the reciprocal of its positive lyapunov exponent however the model cannot describe transitions between different preferent flow regimes this is due to the severe truncation of the spectral expansions

chaos in a class of reacting systems induced by robust asymptotic feedback chaotic behavior in a controlled reacting system is discussed the open loop dynamics of the reactor i e without control actions does not display chaos the feedback control has a unique adjust parameter l for all l l 0 the reactor under control actions converges to a prescribed point l is a parameter then the above condition implies that if l is larger than any constant l the reactor is stabilized at the prescribed point however homoclinic chaos can be induced by the continuous robust asymptotic feedback if 0 l l indeed the closed loop system i e the reactor under feedback ations displays four regions in the first one a limit cycle with large amplitude is found in the second one chaotic behavior is displayed small amplitude limit cycle is found in the third region finally in the fourth region the feedback yields asymptotic convergence of the trajectories to the prescribed point

homoclinic tangencies in the dynamics of articulated pipes conveying fluid abstract numerical evidence is presented showing homoclinic tangencies to periodic orbits occuring in a four dimensional system of ordinary differential equations that describes the dynamics of a pair of articulated pipes conveying fluid bifurcation sequences of mixed mode oscillations are discovered in two separate regions of parameter space in one region the equations have z 2 symmetry and a symmetric pair of heteroclinic tangencies is implied in both cases satisfactory agreement is found between the observed bifurcation sequences and the local theory of homoclinic tangencies global branches of mixed mode orbits are computed it is discovered that some orbits are generated by either sil nikov or figure of eight homoclinic orbits to stationary points whereas others form isolas that feature characteristic towers of saddle node and period doubling bifurcations

global structure of periodicity hubs in lyapunov phase diagrams of dissipative flows infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow namely a r ossler s oscillator we show that spirals associated with periodicity hubs emerge and accumulate at the folding of certain fractal like sheaves of shilnikov homoclinic bifurcations of a common saddle focus equilibrium the specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor

direct observation of homoclinic orbits in human heart rate variability homoclinic trajectories of the interbeat intervals between contractions of ventricles of the human heart are identified the interbeat intervals are extracted from 24 h holter ecg recordings three such recordings are discussed in detail mappings of the measured consecutive interbeat intervals are constructed in the second and in some cases in the fourth iterate of the map of interbeat intervals homoclinic trajectories associated with a hyperbolic saddle are found the homoclinic trajectories are often persistent for many interbeat intervals sometimes spanning many thousands of heartbeats several features typical for homoclinic trajectories found in other systems were identified including a signature of the gluing bifurcation the homoclinic trajectories are present both in recordings of heart rate variability obtained from patients with an increased number of arrhythmias and in cases in which the sinus rhythm is dominant the results presented are a strong indication of the importance of deterministic nonlinear instabilities in human heart rate variability

singular hopf bifurcations and mixed mode oscillations in a two cell inhibitory neural network abstract recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators r curtu a shpiro n rubin j rinzel mechanisms for frequency control in neuronal competition models siam j appl dyn syst 7 2 2008 609 649 showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics several types of behavior were identified such as fusion equivalent to a steady state of identical activity levels for both neural units oscillations due to either an escape or a release mechanism and a winner take all state of bistability the model consists of two neural populations interacting through reciprocal inhibition each endowed with a slow negative feedback process in the form of spike frequency adaptation in this paper we report the occurrence of another complex oscillatory pattern the mixed mode oscillations mmos they exist in the model at the transition between the relaxation oscillator dynamical regime and the winner take all regime the system distinguishes itself from other neuronal models where mmos were found by the following interesting feature there is no autocatalysis involved as in the examples of voltage gated persistent inward currents and or intrapopulation recurrent excitation and therefore the two cells in the network are not intrinsic oscillators the oscillations are instead a combined result of the mutual inhibition and the adaptation we prove that the mmos are due to a singular hopf bifurcation point situated in close distance to the transition point to the winner take all case we also show that in the vicinity of the singular hopf other types of bifurcations exist and we construct numerically the corresponding diagrams

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

bifurcation analysis of a system of coupled chemical oscillators bromate chlorite iodine abstract we examine the dynamical behavior of two chemical oscillators bromate iodide and chlorite iodide as well as the coupled system bromate chlorite iodide study of the period and amplitude of oscillation at the bifurcation points yields phase portraits for the uncoupled systems these are combined to give insights into the nature of the higher dimensional coupled system which exhibits a wide variety of dynamical phenomena including multiple stationary states birhythmicity and chaos analysis of the topology of the phase portraits and of the experimentally determined return map suggests that the coupled system may possess a lorenz type attractor

models of local relativistic quantum fields with indefinite metric in all dimensions a condition on a set of truncated wightman functions is formulated and shown to permit the construction of the hilbert space structure included in the morchio strocchi modified wightman axioms the truncated wightman functions which are obtained by analytic continuation of the truncated schwinger functions of euclidean scalar random fields and covariant vector quaternionic random fields constructed via convoluted generalized white noise are then shown to satisfy this condition as a consequence such random fields provide relativistic models for indefinite metric quantum field theory in dimension 4 vector case respectively in all dimensions scalar case

relaxation oscillations and negative strain rate sensitivity in the portevin le chatelier effect a characteristic feature of the portevin le chatelier effect or the jerky flow is the stick slip nature of stress strain curves which is believed to result from the negative strain rate dependence of the flow stress the latter is assumed to result from the competition of a few relevant time scales controlling the dynamics of jerky flow we address the issue of time scales and its connection to the negative strain rate sensitivity of the flow stress within the framework of a model for the jerky flow which is known to reproduce several experimentally observed features including the negative strain rate sensitivity of the flow stress we attempt to understand the above issues by analyzing the geometry of the slow manifold underlying the relaxational oscillations in the model we show that the nature of the relaxational oscillations is a result of the atypical bent geometry of the slow manifold the analysis of the slow manifold structure helps us to understand the time scales operating in different regions of the slow manifold using this information we are able to establish connection with the strain rate sensitivity of the flow stress the analysis also helps us to provide a proper dynamical interpretation for the negative branch of the strain rate sensitivity

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

experimental evidence for homoclinic chaos in an electrochemical growth process abstract experimental investigations of galvanostatic electrochemical growth processes in steady thin layer electrolytes are reported for some control parameter values the measurement of the cell potential versus time provides periodic and aperiodic oscillations the deterministic character of these potential signals is demonstrated using standard dynamical systems theory tools such as time delay phase portrait reconstructions poincare maps and one dimensional maps a transient period doubling transition is described the homoclinic nature of the chaotic behavior is emphasized the origin of the spatio temporal behavior is discussed in terms of the competition between the electrochemical process reduction of metallic ions and the transport processes migration diffusion and convection analysis of video images shows that the oscillations in growth rate are in phase across the entire cell width up to 40 mm and are in phase with the oscillations in the cell potential

oscillators with chaotic behavior an illustration of a theorem by shil nikov using an explicit one parameter family of differential equations describing oscillators with feedback effects we prove the existence of values of the parameters such that there exist infinitely many unstable periodic orbits of saddle type the proof relies on a theorem by shil nikov which we propose as an explanation for the origin and structure of the chaotic behavior displayed by many well known third order differential systems

invariant tori for commuting hamiltonian pdes we generalize to some pdes a theorem by eliasson and nekhoroshev on the persistence of invariant tori in hamiltonian systems with r integrals of motion and n degrees of freedom r n the result we get ensures the persistence of an r parameter family of r dimensional invariant tori the parameters belong to a cantor like set the proof is based on the lyapunov schmidt decomposition and on the standard implicit function theorem some of the persistent tori are resonant we also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations in the first case we construct 2 dimensional tori while in the second case we construct 3 dimensional tori

reconstructing bifurcation diagrams only from time waveforms abstract a simple algorithm is proposed for reconstructing bifurcation diagrams only from time waveforms of a dynamical system without knowing an explicit form of the dynamics and its dependence on the parameters reconstruction means that the obtained bifurcation diagram is qualitatively similar to that of the original system the algorithm consists of two steps first a nonlinear predictor is sought using a particular class of functions with many parameters second the karhunen loeve transform is used to find only significant parameters contributing the bifurcations the algorithm is tested against two families of dynamical system the henon family and the coupled logistic delayed logistic family

what can we learn from homoclinic orbits in chaotic dynamics state diagrams of two model systems involving three variables are constructed the parameter dependence of different forms of complex nonperiodic behavior and particularly of homoclinic orbits is analyzed it is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system

homoclinic tangencies in an autocatalytic model of interfacial processes at the bone surface abstract the periodic chaotic sequences occuring with a three variable autocatalytic model which represents the calcium dynamics at the liquid solid bone interface were analysed the numerically observed accumulation of some periodic oscillations were found to result from a homoclinic tangency to a periodic orbit the bifurcation structure was analysed in greater detail by modelling the first return map of the continuous model using a piecewise linear one dimensional map this map accounts for the fractal bifurcation structure of the model and indicates the underlying existence of a homoclinic tangency to a smale horseshoe the relevance of the autocatalytic model was examined by a qualitative comparison of these overall dynamic features with those observed in electrodissolution experiments

a hierarchy of transitions to mixed mode oscillation in an electrochemical system abstract the dynamic behavior of the electrocatalytic oxidation of hydrogen on a platinum anode in the presence of cu 2 and cl ions under galvanostatic conditions has been studied experimentally in two control parameters current density and copper ion concentration this parameter plane can roughly be divided into three broad regions a steady state is found at low values of the current density at medium values the system shows small amplitude oscillations whereas at high current densities typical mixed mode oscillations mmos are found depending on the cu 2 concentration the transition from the small amplitude to the mixed mode regime takes place from a simple periodic period doubled or chaotic attractor arising from a feigenbaum route in this paper we focus on the characteristics of these different transitions to mmos and try to construct a bifurcation diagram the bifurcation from the small chaotic attractor to mmos very likely constitutes an interior crisis a bifurcation at which the small chaotic attractor is destroyed by colliding with a stable manifold of a saddle type limit set at lower copper ion concentrations the transition to mmos moved through the period double cascade we compile different properties of the bifurcation diagram and discuss a scenario which fits the observations in a consistent way

time series analysis of physical systems possessing homoclinicity abstract homoclinic orbits and heteroclinic cycles play an important role in the study of dynamical systems there is now an established theoretical framework for investigating the often complicated sequences of bifurcations that can exist within small regions of parameter space close to the conditions for homoclinicity it is the eigenvalues of the saddle point that determine this behaviour in this paper a method is presented whereby an experimentalist with access to a single experimental time series taken from a system close to homoclinicity can estimate these eigenvalues appropriate nonlinear models may then be constructed and analysed that give predictions for the local behaviour of the system

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

hamiltonian hopf bifurcation with symmetry in this paper we study the appearance of branches of relative periodic orbits in hamiltonian hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework the theoretical study is illustrated with several examples

poincare versus boltzmann in sbil nikov phenomena abstract by suitable adjustment of the control parameters in a co 2 laser with feedback we show experimental evidence of sil nikov chaos characterized by intensity pulses almost equal in shape but irregularly separated in time the times of return to a poincare section are statistically spread however their iteration map is one dimensional and in close agreement with that arising from sil nikov theory thus the iteration map of the time intervals becomes the most appropriate indicator of this chaos the residual width of the experimentally measured maps is due to a transient fluctuation enhancement peculiar to macroscopic systems which is absent in low dimensional chaotic dynamics this appears as an inestricable mixture of deterministic unpredictability as introduced by poincare and stochastic fluctuations as considered in boltzmann statistical mechanics