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

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

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

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 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

characterization of robust controllers abstract this paper presents a procedure for designing a robust controller which is able to stabilize a multivariable system with interval uncertainties in the system parameters the uncertain parameters are assumed to lie within known intervals the fundamental idea behind the procedure is to characterize the set of robust controllers as a convex set by the new matrix inequalities result a suitable controller can then be chosen by means of an optimization algorithm the controller thus chosen will be able to ensure that the closed loop matrix lies within desired intervals for all the prescribed interval uncertainties in the system parameters

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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