a discussion on the coexistence of heteroclinic orbit and saddle foci for third order systems abstract the coexistence of heteroclinic orbits and saddle foci is concerned with the basic assumption in shil nikov heteroclinic theorem two aspects of this discussion are conducted in the paper firstly many third order systems which possess exact heteroclinic orbits expressed by pure hyperbolic functions or the combination of hyperbolic and triangle functions and so on have been constructed at the same time the existence of saddle foci is tested and some problems are proposed secondly and more importantly the existence of heteroclinic orbits to saddle foci is studied the necessary condition for the coexistence of heteroclinic orbits and saddle foci is obtained finally an example is given to show the effectiveness of the results and some conclusions and problems are presented

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

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

existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems abstract this paper presents the existence of si lnikov orbits in two different chaotic systems belong to the class of lorenz systems more exactly in the lu system and in the zhou s system both systems have exactly two heteroclinic orbits which are symmetrical with respect to the z axis by using the undetermined coefficient method the existence of the homoclinic orbit for the zhou s system has been proven also by using the undetermined coefficient method as a result the si lnikov criterion along with some technical conditions guarantees that lu and zhou s systems have both smale horseshoes and horseshoe type of chaos moreover the geometric structures of attractors are determined by these heteroclinic orbits

finite dimensional dynamical system modeling thermal instabilities we describe a three dimensional dynamical system which is obtained as a pseudo spectral approximation to a free boundary problem modeling solid combustion and rapid solidification and is capable of generating its major dynamical patterns these patterns include a hopf bifurcation followed by a sequence of secondary period doubling and a transition to chaos reverse sequences and sequences followed by shilnikov type trajectories a computer assisted bifurcation analysis uncovers some novel mechanisms of stability exchange the most striking of them is an infinite period bifurcation which resembles the classical shilnikov bifurcation but instead of a funnel shaped spiral along which the period is continually increasing the continuation produced a series of isolas each isola is a closed branch of solutions of roughly the same period and with the same number of oscillations the isolas corresponding to consecutive numbers of low amplitude oscillations about the equilibrium are adjacent to each other and appear to accumulate on a saddle focus homoclinic connection of shilnikov type 2000 elsevier science b v all rights reserved

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

a construction of frobenius manifolds from stability conditions 2018 london mathematical society a finite quiver q without loops or 2 cycles defines a cy3 triangulated category d q and a finite heart a q d q we show that if q satisfies some strong conditions then the space of stability conditions a q supported on this heart admits a natural family of semisimple frobenius manifold structures constructed using the invariants counting semistable objects in d q in the case of an evaluating the family at a special point we recover a branch of the saito frobenius structure of the an singularity y2 xn 1 we give examples where applying the construction to each mutation of q and evaluating the families at a special point yields a different branch of the maximal analytic continuation of the same semisimple frobenius manifold in particular we check that this holds in the case of an n5

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

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

exploiting randomness on continuous sets abstract in this paper we introduce a new type of field continuous sets where we can exploit randomness in the non repeating decimal expansions of irrationals for cryptographical purposes and present two specific sets a real interval 0 1 and a functional space f 0 1 on 0 1 we propose ideal irrational random number generator iirng which generates non repeating random number sequence as a truly rng by computing the decimal expansion of an randomly chosen irrational on f 0 1 we propose integral encryption scheme ies with which we can encrypt an infinite message and obtain perfect security in one time encryption by computing the integration of the message on a randomly chosen function either the seeds of iirng or the keys of ies are sufficiently safe and immune to exhaustive key search both iirng and ies require the assumption that an element of 0 1 or f 0 1 can be uniformly randomly chosen though the assumption cannot be achieved in classical finite machine we present the discretization of the assumption i e randomly choosing an element of u or v the set of all possible methods of generating irrationals or functions the immunity of seeds or keys to exhaustive key search still exists since any finite search for a random element of u or v is inefficient this is the basic idea of implementing iirng and ies in finite machine two corresponding examples irng and ibc are also presented whose securities are guaranteed by the randomly chosen elements of u or v

heteroclinic bifurcations near hopf zero bifurcation in reversible vector fields in r 3 abstract we study the dynamics near a symmetric hopf zero also known as saddle node hopf or fold hopf bifurcation in a reversible vector field in r 3 with involutory an reversing symmetry whose fixed point subspace is one dimensional we focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle foci we study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets horseshoes independent of the eigenvalues of the saddle foci as well as cascades of bifurcations of periodic heteroclinic and homoclinic orbits finally we discuss the application of our results to the michelson system describing stationary states and travelling waves of the kuramoto sivashinsky pde

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

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

phase behavior under the averaging over disorder realizations abstract effects of the averaging over disorder realizations samples on the phase behavior are analyzed in terms of the mean field approximation for the random field ising model with infinite range interactions it is found that the averaging is equivalent to a drastic modification in the statistics of the quenched variables in its turn this lowers the critical temperature of a second order phase transition or depending on the sampling even suppresses the ordered phase possible first order transitions are shown to be softened by the sample averaging common issues and differences in the interpretation of these effects in the context of the simulation and experimental studies are discussed

chaos via torus breakdown from a four dimensional autonomous oscillator with two diodes abstract by using a remarkably simple and natural degenerate technique the mechanism of chaos via torus breakdown observed in a simple four dimensional autonomous circuit including two diodes is investigated rigorously this degenerate technique is uniquely comparable to the well known slow fast singular perturbation method the idealized case where the diodes are assumed to operate as switches is considered in this case the governing equation is represented by a constrained equation and the poincare mapping is derived rigorously as the circle map the torus breakdown in the four dimensional autonomous circuit is well explained by the poincare mapping the theoretical results are verified by laboratory experiment

on the behavior of the lorenz equation backward in time abstract the sets of solutions to the lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined the set associated with the middle eigenvalue is shown to project surjectively onto a plane thereby providing a lower estimate for its dimension specific bounds are also found for a cone containing this set

coherent structures in nonlocal dispersive active dissipative systems we analyze coherent structures in nonlocal dispersive active dissipative nonlinear systems using as a prototype the kuramoto sivashinsky ks equation with an additional nonlo cal term that contains stabilizing destabilizing and dispersive parts as for the local generalized kuramoto sivashinsky gks equation see e g t kawahara and s toh phys fluids 31 1988 pp 2103 2111 we show that sufficiently strong dispersion regularizes the chaotic dynamics of the ks equation and the solutions evolve into arrays of interacting pulses that can form bound states we analyze the asymptotic characteristics of such pulses and show that their tails tend to zero al gebraically but not exponentially as for the local gks equation since the shilnikov type approach is not applicable for analyzing bound states in nonlocal equations we develop a weak interaction theory and show that the standard first neighbor approximation is no longer applicable it is then essential to take into account long range interactions due to the algebraic decay of the tails of the pulses in addition we find that the number of possible bound states for fixed parameter values is always finite and we determine when there is long range attractive or repulsive force between the pulses finally we explain the regularizing effect of dispersion by showing that as dispersion is increased the pulses generally undergo a transition from absolute to convective instability we also find that for some nonlocal operators increasing the strength of the stabilizing destabilizing term can have a regularizing deregularizing effect on the dynamics

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

on selecting models for nonlinear time series abstract constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models or to choose between competing classes this paper discusses a method of building nonlinear models of possibly chaotic systems from data while maintaining good robustness against noise the models that are built are close to the simplest possible according to a description length criterion the method will deliver a linear model if that has shorter description length than a nonlinear model we show how our models can be used for prediction smoothing and interpolation in the usual way we also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series

embedding as a modeling problem abstract standard approaches to time delay embedding will often fail to provide an embedding that is useful for many common applications this happens in particular when there are multiple timescales in the dynamics we present a modified procedure non uniform embedding which overcomes such problems in many cases for more complex nonlinear dynamics we introduce variable embedding where in a suitable sense the embedding changes with the state of the system we show how to implement these procedures by combining embedding and modeling into a single procedure with a single optimization goal

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

approximate ground states of the random field potts model from graph cuts while the ground state problem for the random field ising model is polynomial and can be solved using a number of well known algorithms for maximum flow or graph cut the analog random field potts model corresponds to a multiterminal flow problem that is known to be np hard hence an efficient exact algorithm is very unlikely to exist as we show here it is nevertheless possible to use an embedding of binary degrees of freedom into the potts spins in combination with graph cut methods to solve the corresponding ground state problem approximately in polynomial time we benchmark this heuristic algorithm using a set of quasiexact ground states found for small systems from long parallel tempering runs for a not too large number q of potts states the method based on graph cuts finds the same solutions in a fraction of the time we employ the new technique to analyze the breakup length of the random field potts model in two dimensions

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

pulse dynamics in low reynolds number interfacial hydrodynamics experiments and theory abstract we analyze interaction of nonlinear pulses in active dispersive dissipative nonlinear media a particular example of such media is a viscous thin film coating a vertical fibre experiments for this system reveal that the interface evolves into a train of droplike solitary pulses in which numerous inelastic coalescence events take place in such events larger pulses catch up with smaller ones and annihilate them however for certain flow conditions and after a certain distance from the inlet no more coalescence is observed and the flow is described by quasi equilibrium solitary pulses interacting continuously with each other through attractions and repulsions and eventually they form bound states of groups of pulses in which the pulses travel with the same velocities as a whole this experimental study represents the first evidence of formation of bound states in low reynolds number interfacial hydrodynamics to gain theoretical insight into the interaction of the pulses and formation of bound states we derive a weakly nonlinear model for the flow the generalized kuramoto sivashinsky gks equation that retains the fundamental mechanisms of the wave evolution namely dominant nonlinearity instability stability and dispersion much like in the experiments the spatio temporal evolution of the gks equation is dominated by quasi stationary solitary pulses which continuously interact with each other through coalescence events or attractions repulsions to understand the latter case we utilize a weak interaction theory for the solitary pulses of the gks equation the theory is based on representing the solution of the equation as a superposition of the pulses and an overlap function and leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses or alternatively the evolution of the separation distances by analyzing the fixed points of this system we obtain bound states of interacting pulses for two pulses we provide a criterion for the existence of a countable infinite or finite number of bound states depending on the strength of the dispersive term in the equation the interaction theory and resulting bound states are corroborated by computations of the full equation we also find qualitative agreement between the theory and the experiments

global generic dynamics close to symmetry abstract our object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in s 3 the cycle involves two saddle foci of different type and is structurally stable within the class of z 2 z 2 symmetric vector fields the cycle contains a two dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry breaking perturbations gradually breaking the symmetry in a two parameter family we get a wide range of dynamical behaviour an attracting periodic trajectory other heteroclinic trajectories homoclinic orbits n pulses suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of shilnikov type we also show that generically the coexistence of linked homoclinic orbits at the two saddle foci has codimension 2 and takes place arbitrarily close to the symmetric cycle

robust heteroclinic cycles in the one dimensional complex ginzburg landau equation numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one dimensional complex ginzburg landau equation cgl on the unit spatially periodic domain these cycles connect difierent spatially and temporally inhomogeneous time periodic solutions as t 1 a careful analysis of the connections is made using a projection onto 5 complex fourier modes it is shown flrst that the time periodic solutions can be treated as relative equilibria after consideration of the symmetries of the cgl second the cycles are shown to be robust since the individual heteroclinic connections exist in invariant

spiralling dynamics near heteroclinic networks abstract there are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically we construct explicitly a two parameter family of vector fields on the three dimensional sphere s 3 whose flow has a spiralling attractor containing the following two hyperbolic equilibria heteroclinic trajectories connecting them transversely and a non trivial hyperbolic invariant and transitive set the spiralling set unfolds a heteroclinic network between two symmetric saddle foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols coexisting with newhouse phenomena the vector field is the restriction to s 3 of a polynomial vector field in r 4 in this article we also identify global bifurcations that induce chaotic dynamics of different types

bifurcations of an optically pumped three level laser model abstract we study the bifurcations in a sixth order model of a resonant homogeneously broadened three level optically pumped laser standard analytical and numerical techniques are used to obtain a complete unfolding of the model s equilibrium bifurcations in its four dimensional parameter space regions of stationary periodic and potentially chaotic behaviour are then readily identified turning to two specific cases we study the principal global bifurcations in a two parameter plane and show their origin in special degenerate bifurcation points of codimension two the different relative dispositions of the principal homoclinic and heteroclinic connections in these two examples result in different local bifurcation behaviours and chaotic dynamics which we illustrate in detail

heteroclinic cycles and periodic orbits for the o 2 equivariant 0 1 2 mode interaction we study the quadratic normal form describing the generic interaction of fourier modes of wavenumbers 0 1 and 2 under the symmetry group o 2 of rotations and reflections in the case that the homogeneous quadratic terms preserve energy the sum of the squares of absolute values of the complex variables this system is a generalization of the 1 2 mode interaction studied by dangelmayr g dangelmayr steady state mode interactions in the presence of o 2 symmetry dyn stab syst 1 2 1986 159 185 armbruster et al d armbruster j guckenheimer p holmes heteroclinic cycles and modulated travelling waves in systems with o 2 symmetry physica d 29 1988 257 282 and others and its restriction to the 1 2 subspace is a degenerate case of that system it displays all the classes of fixed points periodic orbits standing and travelling waves invariant tori modulated travelling waves and heteroclinic cycles found in the 1 2 interaction as well as new heteroclinic cycles connecting pure and mixed modes chaotic cycles and strange periodic orbits we describe the key dynamical features show that the degenerate 1 2 case possesses a second organizing center at which bifurcation curves coalesce provide representative bifurcation sets and diagrams for the 1 2 and 0 1 2 systems and use a conservative limit to understand the periodic orbits in the latter system 2005 elsevier b v all rights reserved

softening of first order transition in three dimensions by quenched disorder we study by extensive monte carlo simulations the effect of random bond dilution on the phase transition of the three dimensional four state potts model that is known to exhibit a strong first order transition in the pure case the phase diagram in the dilution temperature plane is determined from the peaks of the susceptibility for sufficiently large system sizes in the strongly disordered regime numerical evidence for softening to a second order transition induced by randomness is given here a large scale finite size scaling analysis made difficult due to strong crossover effects presumably caused by the percolation fixed point is performed

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

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

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

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

bifurcations in a model of magnetoconvection abstract bifurcations from oscillatory solutions are studied in a truncated model of two dimensional boussinesq magnetoconvection the fifth order system of nonlinear differential equations is integrated numerically and in certain parameter regimes there is a bifurcation from symmetrical to asymmetrical oscillations followed by a period doubling cascade after the accumulation point there is a semiperiodic cascade leading to chaotic behaviour then the semiperiodic cascade is repeated in reverse followed by a period halving cascade and a bifurcation back to symmetry finally the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit that connects two saddle foci the interval with aperiodic solutions contains many pairs of narrow windows with asymmetrical or symmetrical periodic solutions each with its own cascade this pattern of behaviour is likely to occur when a periodic orbit approaches a symmetrical pair of saddle foci with eigenvalues that satisfy shil nikov s inequality

multi bump orbits homoclinic to resonance bands we establish the existence of several classes of multi bump orbits homoclinic to resonance bands for completely integrable hamiltonian systems subject to small amplitude hamiltonian or dissipative perturbations each bump is a fast excursion away from the resonance band and the bumps are interspersed with slow segments near the resonance band the homoclinic orbits which include multi bump silnikov orbits connect equilibria and periodic orbits in the resonance band the main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands

critical behavior of the pure and random bond two dimensional triangular ising ferromagnet we investigate the effects of quenched bond randomness on the critical properties of the two dimensional ferromagnetic ising model embedded in a triangular lattice the system is studied in both the pure and disordered versions by the same efficient two stage wang landau method in the first part of our study we present the finite size scaling behavior of the pure model for which we calculate the critical amplitude of the specific heat s logarithmic expansion for the disordered system the numerical data and the relevant detailed finite size scaling analysis along the lines of the two well known scenarios logarithmic corrections versus weak universality strongly support the field theoretically predicted scenario of logarithmic corrections a particular interest is paid to the sample to sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality

on existence of homoclinic orbits for some types of autonomous quadratic systems of differential equations abstract the new existence conditions of homoclinic orbits for the system of ordinary quadratic differential equations are founded further the realization of these conditions together with the shilnikov homoclinic theorem guarantees the existence of a chaotic attractor at 3d autonomous quadratic system examples of the chaotic attractors are given

the non transverse shil nikov hopf bifurcation uncoupling of homoclinic orbits and homoclinic tangencies abstract it is known that in a neighbourhood of a codimension two shil nikov hopf bifurcation primary periodic orbits lie on a single wiggly curve in period parameter space with accumulation points at parameter values of a pair of homoclinic tangencies to a periodic orbit in contrast it has recently been shown by hirschberg and laing that primary periodic orbits lie on an infinity of isolas in a neighbourhood of certain degenerate homoclinic tangency to a periodic orbit this paper analyses the codimension three bifurcation caused by a non transverse i e degenerately parametrically unfolded shil nikov hopf bifurcation which contains nearby dynamics akin to both degeneracies two cases are classified as being downward pointing or upward pointing depending on whether the variation of a third parameter causes either the annihilation of a locus of saddle focus homoclinic orbits to equilibria or the uncoupling of this locus from the locus of hopf bifurcations we undertake a heuristic analysis of the unfolding showing that in both cases it contains codimension two non transversal homoclinic orbits to equilibria and non transversal homoclinic tangencies to periodic orbits unfolding the former non transverse orbit is shown to cause two wiggly curves to coalesce and leave finitely many isolas of periodic orbits unfolding the latter causes two wiggly curves to coalesce into first infinitely many and then finitely many isolas asymptotic expressions are given for the accumulation of two types of isola forming bifurcations the implications of z 2 equivariance on the unfolding is discussed finally numerical evidence is presented for both upward and downward pointing non transverse shil nikov hopf bifurcations occuring in a model of an autonomous non linear electronic circuit both with respect to a z 2 equivariant and a z 2 non equivariant equilibrium numerical computation of curves of periodic orbits homoclinic orbits and homoclinic tangencies to periodic orbits are shown to agree broadly with the theory but to uncover extra complications

potts model on directed small world voronoi delaunay lattices abstract the critical properties of the potts model with q 3 and 4 states in two dimensions on directed small world voronoi delaunay random lattices with quenched connectivity disorder are investigated this disordered system is simulated by applying the monte carlo update heat bath algorithm the potts model on these directed small world random lattices presents in fact a second order phase transition with new critical exponents for q 3 and value of the rewiring probability p 0 01 but for q 4 the system exhibits only a first order phase transition independent of p 0 p 1

the diffusionless lorenz equations shil nikov bifurcations and reduction to an explicit map abstract a simplified one parameter version of the lorenz model is obtained in the limit of high rayleigh and prandtl numbers physically corresponding to diffusionless convection it is argued that the bifurcation structure of this simplified lorenz model essentially involves only shil nikov bifurcations an exact solution to this simplified dynamical system is given which serves as the limit for strong forcing and appears to be a new integrable case of the lorenz equations for small values of the bifurcation parameter an approximate analytical and multipeaked map is obtained which gives successive periods of the pulse like motion this map leads to self similar behaviour in parameter space

heteroclinic dynamics in the nonlocal parametrically driven nonlinear schrodinger equation faraday waves are described under appropriate conditions by a damped nonlocal parametrically driven nonlinear schrodinger equation as the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics the origin of these transitions is explained using a careful study of a two mode galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves these connections are associated with cascades of gluing and symmetry switching bifurcations such bifurcations are located in the partial differential equations as well

homoclinic orbits in a parametrized saddle focus system abstract we consider a parametrized dynamical system with a saddle focus equilibrium point and which for one value of the parameter has a homoclinic orbit conditions on the eigenvalues for the equilibrium point together with transversality conditions imply the existence of an infinite discrete set of parameter values for which the system has a homoclinic orbit such systems arise in the study of nerve axon equations where a homoclinic orbit corresponds to a finite train of nerve impulses traveling at a velocity identified by the associated parameter value

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

chaos in a three dimensional single loop feedback system with a piecewice linear feedbackk function abstract this paper describes chaotic behaviour in a single loop feedback system of low order the system is particularly simple and because it is piecewise linear and hence can be undersdtood globally a description of the flow much more complete than usual can be given an orbit structure is suggested for the system after a minor approximation is considered the approximation gives two equivalent discrete one dimensional maps of the interval to itself and these maps are discussed in the light of recent work the validity of the approximation is confirmed by computer simulation

stability conditions and stokes factors let mathcal a be the category of modules over a complex finite dimensional algebra we show that the space of stability conditions on mathcal a parametrises an isomonodromic family of irregular connections on 1 with values in the hall algebra of mathcal a the residues of these connections are given by the holomorphic generating function for counting invariants in mathcal a constructed by d joyce in geom topol 11 667 725 2007

criticality in the randomness induced second order phase transition of the triangular ising antiferromagnet with nearest and next nearest neighbor interactions abstract using a wang landau entropic sampling scheme we investigate the effects of quenched bond randomness on a particular case of a triangular ising model with nearest j n n and next nearest neighbor j n n n antiferromagnetic interactions we consider the case r j n n n j n n 1 for which the pure model is known to have a columnar ground state where rows of nearest neighbor spins up and down alternate and undergo a weak first order phase transition from the ordered to the paramagnetic state with the introduction of quenched bond randomness we observe the effects signaling the expected conversion of the first order phase transition to a second order phase transition and using the lee kosterlitz method we quantitatively verify this conversion the emerging under random bonds continuous transition shows a strongly saturating specific heat behavior corresponding to a negative exponent and belongs to a new distinctive universality class with 1 135 11 1 744 9 and 0 124 8 thus our results for the critical exponents support an extensive but weak universality and the emerged continuous transition has the same magnetic critical exponent but a different thermal critical exponent as a wide variety of two dimensional 2d systems without and with quenched disorder

new type of complex dynamics in the 1 2 spatial resonance abstract the amplitude equations describing the 1 2 spatial resonance in o 2 symmetric systems are re examined complex dynamical behavior associated with a new class of global connections is identified and analyzed these connections form further from the mode interaction point where the structurally stable heteroclinic cycles described by armbruster et al physica d 29 1988 257 no longer exist the resulting behavior is characteristic of systems with o 2 z 2 symmetry in which the symmetry z 2 is weakly broken

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

homoclinic and heteroclinic orbits in a modified lorenz system this paper presents a mathematically rigorous proof for the existence of chaos in a modified lorenz system using the theory of shil nikov bifurcations of homoclinic and heteroclinic orbits together with its dynamical behaviors which have been extensively studied the chaotic dynamics of the modified lorenz system are now much better understood providing a rigorous theoretic foundation to support studies and applications of this important class of chaotic systems

stability conditions on a non compact calabi yau threefold we study the space of stability conditions stab x on the non compact calabi yau threefold x which is the total space of the canonical bundle of mathbb p 2 we give a combinatorial description of an open subset of stab x and state a conjecture relating stab x to the frobenius manifold obtained from the quantum cohomology of mathbb p 2 we give some evidence from mirror symmetry for this conjecture

an intelligent noise reduction method for chaotic signals based on genetic algorithms and lifting wavelet transforms time series observed in real world is often nonlinear even chaotic however observed data is often contaminated by noise of various types to effectively extract desired information from observed data it is vital to preprocess data to reduce noise for both the analysis of dynamical systems and many potential applications of these systems in this paper we present a noise reduction approach to the problem of additive source separation characterized by wide band power spectra when one of the sources is chaotic the algorithm is based on a center based genetic algorithm cbga in lifting wavelet framework in which the cbga is used for threshold optimization this method intelligently adapts itself to various types of noise and it weighs preservation of dynamics and denoising through signal to noise ratio snr and root mean square error rmse computer simulations show that the approach is very effective in diminishing different kinds of noise and performs better in terms of visual quality as well as quantitative metrics than existing algorithms

implicit one dimensional discrete maps and their connection with existence problem of chaotic dynamics in 3 d systems of differential equations abstract new types of chaotic attractors for some 3 d autonomous systems of ordinary quadratic differential equations are founded examples are given

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

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

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

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

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