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We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on-off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map., 34 pages

We consider a class of step skew product systems of interval diffeomorphisms over shift operators, as a means to study random compositions of interval diffeomorphisms. The class is chosen to present in a simplified setting intriguing phenomena of intermingled basins, master-slave synchronization and on-off intermittency. We provide a self-contained discussion of these phenomena., Comment: 30 pages, accepted by DCDS-S

We consider a class of cubic stochastic operators that are motivated by models for evolution of frequencies of genetic types in populations. We take populations with three mutually exclusive genetic types. The long term dynamics of single maps, starting with a generic initial condition where in particular all genetic types occur with positive frequency, is asymptotic to equilibria where either only one genetic type survives, or where all three genetic types occur. We consider a family of independent and identically distributed maps from this class and study its long term dynamics, in particular its random point attractors. The long term dynamics of the random composition of maps is asymptotic, almost surely, to equilibria. In contrast to the deterministic system, for generic initial conditions these can be equilibria with one or two or three types present (depending only on the distribution)., 15 pages

We treat step skew products over transitive subshifts of finite type with interval fibers. The fiber maps are diffeomorphisms on the interval; we assume that the end points of the interval are fixed under the fiber maps. Our paper thus extends work by V. Kleptsyn and D. Volk who treated step skew products where the fiber maps map the interval strictly inside itself. We clarify the dynamics for an open and dense subset of such skew products. In particular we prove existence of a finite collection of disjoint attracting invariant graphs. These graphs are contained in disjoint areas in the phase space called trapping strips. Trapping strips are either disjoint from the end points of the interval (internal trapping strips) or they are bounded by an end point (border trapping strips). The attracting graphs in these different trapping strips have different properties.

We prove that any compact manifold without boundary admits a pair of diffeomorphisms that generates $C^1$ robustly minimal dynamics. We apply the results to the construction of blenders and robustly transitive skew product diffeomorphisms., 16 pages, 2 figures

We give an explicit construction of families of Dm-equivariant polynomial vector fields in possessing a codimension one homoclinic cycle. The homoclinic cycle consists of m homoclinic trajectories all connected to the equilibrium at the origin. The constructed vector fields can provide a setting for a (numerical) bifurcation study of these homoclinic cycles, in particular for m equal to a multiple of 4, where the bifurcations form an open problem. © 2013 © 2013 Taylor & Francis.

Shub & Wilkinson and Ruelle & Wilkinson studied a class of volume preserving diffeomorphisms on the three dimensional torus that are stably ergodic. The diffeomorphisms are partially hyperbolic and admit an invariant central foliation of circles. The foliation is not absolutely continuous, in fact, Ruelle & Wilkinson established that the disintegration of volume along central leaves is atomic. We show that in such a class of volume preserving diffeomorphisms the disintegration of volume along central leaves is a single delta measure. We also formulate a general result for conservative three dimensional skew product like diffeomorphisms on circle bundles, providing conditions for delta measures as disintegrations of the smooth invariant measure., 15 pages, accepted by Proc. Amer. Math. Soc

We review recent results from the theory of random differential equations with bounded noise. Assuming the noise to be “sufficiently robust in its effects” we discuss the feature that any stationary measure of the system is supported on a “Minimal Forward Invariant” (MFI) set. We review basic properties of the MFI sets, including their relationship to attractors in systems where the noise is small. In the main part of the paper we discuss how MFI sets can undergo discontinuous changes that we have called hard bifurcations. We characterize such bifurcations for systems in one and two dimensions and we give an example of the effects of bounded noise in the context of a Hopf–Andronov bifurcation.

We study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group. The relative homoclinic cycles we consider are not robust, but have codimension one. We assume real leading eigenvalues and connecting trajectories that approach the equilibria along leading directions. We show how suspensions of subshifts of finite type generically appear in the unfolding. Descriptions of the suspended subshifts in terms of the geometry and symmetry of the connecting trajectories are provided. © 2011 American Mathematical Society.

Lorenz-like attractors are known to appear in unfoldings from certain codimension two homoclinic bifurcations for differential equations in ℝ3 that possess a reflectional symmetry. This includes homoclinic loops under a resonance condition and the inclination-flip homoclinic loops. We show that Lorenz-like attractors also appear in the third possible codimension two homoclinic bifurcation (for homoclinic loops to equilibria with real different eigenvalues); the orbit-flip homoclinic bifurcation. We moreover provide a bifurcation analysis computing the bifurcation curves of bifurcations from periodic orbits and discussing the creation and destruction of the Lorenz-like attractors. Known results for the inclination flip are extended to include a bifurcation analysis.

A heteroclinic network for an equivariant ordinary differential equation is called switching if each sequence of heteroclinic trajectories in it is shadowed by a nearby trajectory. It is called forward switching if this holds for positive trajectories. We provide an elementary example of a switching robust homoclinic network and a related example of a forward switching asymptotically stable robust homoclinic network. The examples are for five-dimensional equivariant ordinary differential equations. © 2010 Taylor & Francis.

For random diffeomorphisms depending on a parameter, nearly invariant sets are described via an associated deterministic control system. Conditions are provided guaranteeing that the system leaves the support of an invariant measure under small perturbations of the parameter and estimates for the exit times are given. © 2010 Taylor & Francis.

We construct unbounded strange attractors for vector fields in R 3 that are robust transitive under uniformly small perturbations. Their geometry is reminiscent of geometric Lorenz and other singular hyperbolic attractors, but they contain no equilibria.

We discuss iterates of random circle diffeomorphisms with identically distributed noise, where the noise is bounded and absolutely continuous. Using arguments of B. Deroin, V.A. Kleptsyn and A. Navas, we provide precise conditions under which random attracting fixed points or random attracting periodic orbits exist. Bifurcations leading to an explosion of the support of a stationary measure from a union of intervals to the circle are treated. We show that this typically involves a transition from a unique random attracting periodic orbit to a unique random attracting fixed point.

In these notes, we discuss obstructions to the existence of local invariant manifolds of some smoothness class, near hyperbolic fixed points of diffeomorphisms. We present an elementary construction for continuously differentiable invariant manifolds that are not necessarily normally hyperbolic, near attracting fixed points. The analogous theory for invariant manifold near hyperbolic equilibria of differential equations is included. For differential equations, we construct one dimensional invariant manifolds of higher smoothness class.Keywords:Hyperbolic fixed points, Smoothness, Diffeomorphisms, Differential equations

We consider a family of differential equations that describes traveling waves in a reaction-diffusion equation modeling oxidation of carbon oxide on a platinum surface, near the onset of spatio-temporal chaos. The organizing bifurcation for the bifurcation structure with small carbon oxide pressures, turns out to be a codimension 3 bifurcation involving a homoclinic orbit to an equilibrium undergoing a transcritical bifurcation. We show how infinitely many T-point bifurcations of multi loop heteroclinic cycles occur in the unfolding.

We consider a system of differential equations proposed by Busse et al (1992 Physica D 61 94-105) to describe the development of spatio-temporal structures in Rayleigh-Bénard convection, near the skewed varicose instability. Numerical computations make it clear that the global bifurcations are organized by a codimension two bifurcation with heteroclinic cycles and a double principal stable eigenvalue at the origin. We carry out the bifurcation study and prove in particular the occurrence in the unfolding of robustly transitive strange attractors akin to Lorenz attractors. In contrast to the actual Lorenz attractors, these attractors contain two equilibria.

We describe algorithms for computing hyperbolic invariant sets of diffeomorphisms and their stable and unstable manifolds. This includes the calculation of Smale horseshoes and the stable and unstable manifolds of periodic points in any finite dimension.

We discuss dynamics near homoclinic orbits to saddle-focus equilibria in three-dimensional vector fields. The existence of periodic and strange attractors is investigated not in unfoldings, but in families for which each member has a homoclinic orbit. We consider how often, in the sense of measure, periodic and strange attractors occur in such families. We also discuss the fate of typical orbits, and establish that despite the possible existence of attractors, a large proportion of points from a small vicinity of the homoclinic orbit, lies outside the basin of an attractor.

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 ℝ3. 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.

Cascades of period doubling bifurcations are found in one parameter families of differential equations in ℝ3. When varying a second parameter, the periodic orbits in the period doubling cascade can disappear in homoclinic bifurcations. In one of the possible scenarios one finds cascades of homoclinic doubling bifurcations. Relevant aspects of this scenario can be understood from a study of interval maps close to x↦p+r(1 −xβ)2, β∈ (½,1). We study a renormalization operator for such maps. For values of β close to ½, we prove the existence of a fixed point of the renormalization operator, whose linearization at the fixed point has two unstable eigenvalues. This is in marked contrast to renormalization theory for period doubling cascades, where one unstable eigenvalue appears. From the renormalization theory we derive consequences for universal scalings in the bifurcation diagrams in the parameter plane.

We consider the dynamics of piecewise smooth interval maps $f$ with a nowhere vanishing derivative. We show that if $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If, in addition all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelling, then some iterate of $f$ is expanding. In this case, $f$ admits an absolutely continuous invariant probability measure.

This paper studies three-parameter unfoldings of resonant orbit flip and inclination flip homoclinic orbits. First, all known results on codimension-two unfoldings of homoclinic flip bifurcations are presented. Then we show that the orbit flip and inclination flip both feature the creation and destruction of a cusp horseshoe. Furthermore, we show near which resonant flip bifurcations a homoclinic-doubling cascade occurs. This allows us to glue the respective codimension-two unfoldings of homoclinic flip bifurcations together on a sphere around the central singularity. The so obtained three-parameter unfoldings are still conjectural in part but constitute the simplest, consistent glueings.

In a series of important papers [GS1,GS2] Gavrilov and Shilnikov established a topological conjugacy between a surface diffeomorphism having a dissipative hyperbolic periodic point with certain types of quadratic homoclinic tangencies and the full shift on two symbols, thus exhibiting horseshoes near a tangential homoclinic point. In this note, which should be viewed of as an addendum to [BW] we extend this result by showing that such a diffeomorphism with a one-sided isolated homoclinic tangency having any order contact, possible with infinite order contact, possesses a horseshoe near the homoclinic point.

We study one parameter families of vector fields that are defined on three dimensional manifolds and whose nonwandering sets are structurally stable. As families, these families may not be structurally stable; heteroclinic bifurcations that give rise to moduli can occur. Some but not all moduli are related to the geometry of stable and unstable manifolds. We study a notion of stability, weaker then structural stability, in which geometry and dynamics on stable and unstable manifolds are reflected. We classify the families from the above mentioned class of families that are stable in this sense.

We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of diffeomorphisms. The iterated function systems admit a description as skew product systems of diffeomorphisms on compact manifolds driven by shift operators. Under open conditions including transitivity and negative fiber Lyapunov exponents, we prove the existence of a unique attracting invariant graph for the skew product system. This explains the occurrence of synchronization. The result extends previous results for iterated function systems by diffeomorphisms on the circle, to arbitrary compact manifolds., Comment: 17 pages, to appear in Bulletin of the Brazilian Mathematical Society, New Series

We present a method for the numerical computation of invariant manifoids of hyperbolic and pseudohyperbolic fixed points of diffeomorphisms. The derivation of this algorithm is based on well-known properties of (almost) invariant foliations. Numerical results illustrate the performance of our method.

We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set., Comment: Color figure version of a manuscript to appear in Discrete and Continuous Dynamics Systems - A

In random differential equations with bounded noise minimal forward invariant (MFI) sets play a central role since they support stationary measures. We study the stability and possible bifurcations of MFI sets. In dimensions 1 and 2 we classify all minimal forward invariant sets and their codimension one bifurcations in bounded noise random differential equations.

We discuss the dynamics of skew product maps defined by circle diffeomorphisms forced by expanding circle maps. We construct an open class of such systems that are robustly topologically mixing and for which almost all points in the same fiber converge under iteration. This property follows from the construction of an invariant attracting graph in the natural extension, a skew product of circle diffeomorphisms forced by a solenoid homeomorphism.

Deng has demonstrated a mechanism through which a perturbation of a vector field having an inclination-flip homoclinic orbit would have a Smale horseshoe. In this article we prove that if the eigenvalues of the saddle to which the homoclinic orbit is asymptotic satisfy the condition 2λu > min{−λs, λuu} then there are arbitrarily small perturbations of the vector field which possess a Smale horseshoe. Moreover we analyze a sequence of bifurcations leading to the annihilation of the horseshoe. This sequence contains, in particular, the points of existence of n-homoclinic orbits with arbitrary n.

We construct iterated function systems on compact manifolds that are C1 robustly minimal. On the m-dimensional torus and on two-dimensional compact manifolds, examples are provided of C1 robustly minimal iterated function systems that are generated by just two diffeomorphisms.

Melbourne [An example of a nonasymptotically stable attractor, Nonlinearity 4(3) (1991), pp. 835-844] discusses an example of a robust heteroclinic network that is not asymptotically stable but which has the strong attracting property called essential asymptotic stability. We establish that this phenomenon is possible for homoclinic networks, where all heteroclinic trajectories are symmetry related. Moreover, we study a transverse bifurcation from an asymptotically stable to an essentially asymptotically stable homoclinic network. The essentially asymptotically stable homoclinic network turns out to attract all nearby points except those on codimension-one stable manifolds of equilibria outside the homoclinic network.

Differential equations that are equivariant under the action of a finite group can possess robust homoclinic cycles that can moreover be asymptotically stable. For differential equations in R 4 there exists a classification of different robust homoclinic cycles for which moreover eigenvalue conditions for asymptotic stability are known. We study resonance bifurcations that destroy the asymptotic stability of robust ‘simple homoclinic cycles’ in four-dimensional differential equations. We establish that typically a periodic trajectory near the cycle is created, asymptotically stable in the supercritical case.

In unfoldings of resonant homoclinic bellows interesting bifurcation phenomena occur: two suspensed Smale horseshoes can collide and disappear in saddle-node bifurcations (all periodic orbits disappear through saddle-node bifurcations, there are no other bifurcations of periodic orbits), or a suspended horseshoe can go through saddle-node and period-doubling bifurcations of the periodic orbits in it to create an additional ``doubled horseshoe''.

In unfoldings of resonant homoclinic bellows interesting bifurcation phenomena occur: two suspensed Smale horseshoes can collide and disappear in saddle-node bifurcations (all periodic orbits disappear through saddle-node bifurcations, there are no other bifurcations of periodic orbits), or a suspended horseshoe can go through saddle-node and period-doubling bifurcations of the periodic orbits in it to create an additional "doubled horseshoe".

We discuss one parameter families of unimodal maps, with negative Schwarzian derivative, unfolding a saddle-node bifurcation. We show that there is a parameter set of positive but not full Lebesgue density at the bifurcation, for which the maps exhibit absolutely continuous invariant measures which are supported on the largest possible interval. We prove that these measures converge weakly to an atomic measure supported on the orbit of the saddle-node point. Using these measures we analyze the intermittent time series that result from the destruction of the periodic attractor in the saddle-node bifurcation and prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the periodic attractor.

We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times.In conservative systems one finds one parameter families of suspended horseshoes, parameterized by the level of the first integral. A somewhat similar picture occurs in reversible systems, with two homoclinic orbits that are both symmetric. The lack of a first integral implies that complete horseshoes do not exist. We provide a description of orbits that necessarily do exist.A second possible configuration in reversible systems occurs if a non-symmetric homoclinic orbit exists and forms a bellows together with its symmetric image. We describe the nonwandering set in an unfolding. The nonwandering set is shown to simultaneously contain one-parameter families of periodic orbits, hyperbolic periodic orbits of different index, and heteroclinic cycles between these periodic orbits.

Random diffeomorphisms with bounded absolutely continuous noise are knownto possess a finite number of stationary measures. We discuss thedependence of stationary measures on an auxiliary parameter, thusdescribing bifurcations of families of random diffeomorphisms. Abifurcation theory is developed under mild regularity assumptions on thediffeomorphisms and the noise distribution (e.g. smooth diffeomorphismswith uniformly distributed additive noise are included). We distinguishbifurcations where the density function of a stationary measure variesdiscontinuously or where the support of a stationary measure variesdiscontinuously. We establish that generic random diffeomorphisms arestable. The densities of stable stationary measures are shown to be smoothand to depend smoothly on an auxiliary parameter, except at bifurcationvalues. The bifurcation theory explains the occurrence of transients andintermittency as the main bifurcation phenomena in random diffeomorphisms.Quantitative descriptions by means of average escape times from sets asfunctions of the parameter are provided. Further quantitative propertiesare described through the speed of decay of correlations as a function ofthe parameter. Random differentiable maps which are not necessarilyinjective are studied in one dimension; we show that stableone-dimensional random maps occur open and dense and that in one-parameterfamilies bifurcations are typically isolated. We classify codimension-onebifurcations for one-dimensional random maps; we distinguish threepossible kinds, the random saddle node, the random homoclinic and therandom boundary bifurcation. The theory is illustrated on families ofrandom circle diffeomorphisms and random unimodal maps.

We present an overview of the theory of homoclinic doubling cascades, describing bifurcation theory and discussing universal scaling properties obtained from a renormalization theory.

We study bifurcations from a singular heteroclinic cycle in ℝ3. This heteroclinic cycle contains a hyperbolic singularity and a saddle-node. At the saddlenode, the linearized vector field has two complex conjugate eigenvalues in addition to an eigenvalue 0. This implies the existence of a cascade of inclination-flip homoclinic bifurcations to the heteroclinic bifurcation.

In this article, originally published online at DOI: 10.1080/14689361003769770 and in this issue pp. 351–358, the authors note that: ‘It was brought to our attention that Holmes (1980) discussed dy...

We study the dynamics near transverse intersections of stable and unstable manifolds of sheets of symmetric periodic orbits in reversible systems. We prove that the dynamics near such homoclinic and heteroclinic intersections is not $C^1$ structurally stable. This is in marked contrast to the dynamics near transverse intersections in both general and conservative systems, which can be $C^1$ structurally stable. We further show that there are infinitely many sheets of symmetric periodic orbits near the homoclinic or heteroclinic orbits. We establish the robust occurrence of heterodimensional cycles, that is, heteroclinic cycles between hyperbolic periodic orbits of different index, near the transverse intersections. This is shown to imply the existence of hyperbolic horseshoes and infinitely many periodic orbits of different index, all near the transverse intersections.