Milnor Attractor Research Articles

Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

To identify and to explain coupling-induced phase transitions in coupled map lattices (CML) has been a lingering enigma for about two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, recent proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive. In this paper, we consider a family of CML composed by piecewise expanding individual map, global interaction and finite number $$N$$ of sites, in the weak coupling regime where the CML is uniformly expanding. We show, mathematically for $$N=3$$ and numerically for $$N\ge 3$$ , that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics. Despite that it only addresses finite-dimensional systems, to some extend, this result reconciles the previous ones as it shows that loss of ergodicity/symmetry breaking can occur in basic CML, independently of any decay in the Lyapunov dimension.

Wild Milnor attractors accumulated by lower-dimensional dynamics

AbstractWe present new examples of open sets of diffeomorphisms such that generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that all other classes except one are contained in periodic surfaces. This study allows us to obtain the existence of Milnor attractors as well as study ergodic properties of the diffeomorphisms in those open sets by using ideas and results from Bonatti and Viana [SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly contracting. Israel J. Math.115 (2000), 157–193] and Buzzi and Fisher [Entropic stability beyond partial hyperbolicity. Preprint, 2011, arXiv:1103:2707].

Relatively unstable attractors

There are different non-equivalent definitions of attractors in the theory of dynamical systems. The most common are two definitions: the maximal attractor and the Milnor attractor. The maximal attractor is by definition Lyapunov stable, but it is often in some ways excessive. The definition of Milnor attractor is more realistic from the physical point of view. The Milnor attractor can be Lyapunov unstable though. One of the central problems in the theory of dynamical systems is the question of how typical such a phenomenon is. This article is motivated by this question and contains new examples of so-called relatively unstable Milnor attractors. Recently I. Shilin has proved that these attractors are Lyapunov stable in the case of one-dimensional fiber under some additional assumptions. However, the question of their stability in the case of multidimensional fiber is still an open problem.

Unstable Orbits and Milnor Attractors in the Discontinuous Flat Top Tent Map

In this work we consider the discontinuous flat top tent map which represents an example for discontinuous piecewise-smooth maps, whereby the system function is constant on some interval. Such maps show several characteristics caused by this constant value which are still insufficiently investigated. In this work we demonstrate that in the discontinuous flat top tent map every unstable periodic orbit may become a Milnor attractor. Moreover, it turns out that there exists a strong connection between stable and unstable orbits and that the appearance of a single unstable orbit may cause an infinite number of stable orbits to appear. Based on this connection we provide a more precise explanation of the recently discovered self-similar bifurcation scenario occurring in the discontinuous flat top tent map denoted as the nested period incrementing scenario.

Thick attractors of boundary preserving diffeomorphisms

A diffeomorphism is said to have a thick attractor provided that its attractor has positive but not full Lebesgue measure. A set in a functional space is quasiopen, if it may be obtained from an open set by removing a countable number of hypersurfaces. We prove that there exists a quasiopen set in the space of boundary preserving diffeomorphisms of a compact manifold with boundary, such that any map in this set has a thick attractor. The meaning of the word “attractor” should be specified. In the above claim an “attractor” is, roughly speaking, a “topologically mixing maximal attractor”. We also conjecture that the claim is true for the Milnor attractor of diffeomorphisms and prove the claim for Milnor attractors of mild skew products. We reduce the conjecture above to a general conjecture about Milnor attractors of partially hyperbolic diffeomorphisms.

On local attraction properties and a stability index for heteroclinic connections

Some invariant sets may attract a nearby set of initial conditions but nonetheless repel a complementary nearby set of initial conditions. For a given invariant set with a basin of attraction N, we define a stability index σ(x) of a point x ∊ X that characterizes the local extent of the basin. Let Bϵ denote a ball of radius ϵ about x. If σ(x) > 0, then the measure of Bϵ ∖ N relative the measure of the ball is O(ϵ|σ(x)|), while if σ(x) < 0, then the measure of Bϵ ∩ N relative the measure of the ball is of this order. We show that this index is constant along trajectories, and we relate this orbit invariant to other notions of stability such as Milnor attraction, essential asymptotic stability and asymptotic stability relative to a positive measure set. We adapt the definition to local basins of attraction (i.e. where N is defined as the set of initial conditions that are in the basin and whose trajectories remain local to X).This stability index is particularly useful for discussing the stability of robust heteroclinic cycles, where several authors have studied the appearance of cusps of instability near cycles that are Milnor attractors. We study simple (robust heteroclinic) cycles in and show that the local stability indices (and hence local stability properties) can be calculated in terms of the eigenvalues of the linearization of the vector field at steady states on the cycle. In doing this, we extend previous results of Krupa and Melbourne (1995 Ergod. Theory Dyn. Syst. 15 121–48; 2004 Proc. R. Soc. Edinb. A 134 1177–97) and give criteria for simple heteroclinic cycles in to be Milnor attractors.

On statistical attractors and the convergence of time averages

AbstractThere are various notions of attractor in the literature, including measure (Milnor) attractors and statistical (Ilyashenko) attractors. In this paper we relate the notion of statistical attractor to that of the essential ω-limit set and prove some elementary results about these. In addition, we consider the convergence of time averages along trajectories. Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables. For non-ergodic systems, time averages may not exist even for almost all trajectories. However, averages of some observables may converge; we characterize conditions on observables that ensure convergence of time averages even in non-ergodic systems.

Noise-induced switching near a depth two heteroclinic network and an application to Boussinesq convection

We investigate the robust heteroclinic dynamics arising in a system of ordinary differential equations in R(4) with symmetry [Formula in text]. This system arises from the normal form reduction of a 1: squate root of 2 mode interaction for Boussinesq convection. We investigate the structure of a particular robust heteroclinic attractor with "depth two connections" from equilibria to subcycles as well as connections between equilibria. The "subcycle" is not asymptotically stable, due to nearby trajectories undertaking an "excursion," but it is a Milnor attractor, meaning that a positive measure set of nearby initial conditions converges to the subcycle. We investigate the dynamics in the presence of noise and find a number of interesting properties. We confirm that typical trajectories wind around the subcycle with very occasional excursions near a depth two connection. The frequency of excursions depends on noise intensity in a subtle manner; in particular, for anisotropic noise, the depth two connection may be visited much more often than for isotropic noise, and more generally the long term statistics of the system depends not only on the noise strength but also on the anisotropy of the noise. Similar properties are confirmed in simulations of Boussinesq convection for parameters giving an attractor with depth two connections.

Thick attractors of step skew products

A diffeomorphism is said to have a thick attractor provided that its Milnor attractor has positive but not full Lebesgue measure. We prove that there exists an open set in the space of boundary preserving step skew products with a fiber [0,1], such that any map in this set has a thick attractor.

Working memory dynamics and spontaneous activity in a flip-flop oscillations network model with a Milnor attractor

Many cognitive tasks require the ability to maintain and manipulate simultaneously several chunks of information. Numerous neurobiological observations have reported that this ability, known as the working memory, is associated with both a slow oscillation (leading to the up and down states) and the presence of the theta rhythm. Furthermore, during resting state, the spontaneous activity of the cortex exhibits exquisite spatiotemporal patterns sharing similar features with the ones observed during specific memory tasks. Here to enlighten neural implication of working memory under these complicated dynamics, we propose a phenomenological network model with biologically plausible neural dynamics and recurrent connections. Each unit embeds an internal oscillation at the theta rhythm which can be triggered during up-state of the membrane potential. As a result, the resting state of a single unit is no longer a classical fixed point attractor but rather the Milnor attractor, and multiple oscillations appear in the dynamics of a coupled system. In conclusion, the interplay between the up and down states and theta rhythm endows high potential in working memory operation associated with complexity in spontaneous activities.

Decay of geometry for unimodal maps: An elementary proof

We prove that a nonrenormalizable smooth unimodal interval map with critical order between 1 and 2 displays decay of geometry, by an elementary and purely real argument. This completes a real approach to Milnor's attractor problem for smooth unimodal maps with critical order not greater than 2.

Chaotic itinerancy and power-law residence time distribution in stochastic dynamical systems

Chaotic itinerant motion among varieties of ordered states is described by a stochastic model based on the mechanism of chaotic itinerancy. The model consists of a random walk on a half-line and a Markov chain with a transition probability matrix. The stability of attractor ruin in the model is investigated by analyzing the residence time distribution of orbits at attractor ruins. It is shown that the residence time distribution averaged over all attractor ruins can be described by the superposition of (truncated) power-law distributions if the basin of attraction for each attractor ruin has a zero measure. This result is confirmed by simulation of models exhibiting chaotic itinerancy. Chaotic itinerancy is also shown to be absent in coupled Milnor attractor systems if the transition probability among attractor ruins can be represented as a Markov chain.

Unstable attractors: existence and robustness in networks of oscillators with delayed pulse coupling

We consider unstable attractors: Milnor attractors A such that, for some neighbourhood U of A, almost all initial conditions leave U. Previous research strongly suggests that unstable attractors exist and even occur robustly (i.e. for open sets of parameter values) in a system modelling biological phenomena, namely in globally coupled oscillators with delayed pulse interactions.In the first part of this paper we give a rigorous definition of unstable attractors for general dynamical systems. We classify unstable attractors into two types, depending on whether or not there is a neighbourhood of the attractor that intersects the basin in a set of positive measure. We give examples of both types of unstable attractor; these examples have non-invertible dynamics that collapse certain open sets onto stable manifolds of saddle orbits.In the second part we give the first rigorous demonstration of existence and robust occurrence of unstable attractors in a network of oscillators with delayed pulse coupling. Although such systems are technically hybrid systems of delay differential equations with discontinuous ‘firing’ events, we show that their dynamics reduces to a finite dimensional hybrid system after a finite time and, hence, we can discuss Milnor attractors for this reduced finite dimensional system. We prove that for an open set of phase resetting functions there are saddle periodic orbits that are unstable attractors.

Product dynamics for homoclinic attractors

Heteroclinic cycles may occur as structurally stable asymptotically stable attractors if there are invariant subspaces or symmetries of a dynamical system. Even for cycles between equilibria, it may be difficult to obtain results on the generic behaviour of trajectories converging to the cycle. For more–complicated cycles between chaotic sets, the non–trivial dynamics of the ‘nodes’ can interact with that of the ‘connections’. This paper focuses on some of the simplest problems for such dynamics where there are direct products of an attracting homoclinic cycle with various types of dynamics. Using a precise analytic description of a general planar homoclinic attractor, we are able to obtain a number of results for direct product systems.We show that for flows that are a product of a homoclinic attractor and a periodic orbit or a mixing hyperbolic attractor, the product of the attractors is a minimal Milnor attractor for the product. On the other hand, we present evidence to show that for the product of two homoclinic attractors, typically only a small subset of the product of the attractors is an attractor for the product system.

Chaotic itinerancy generated by coupling of Milnor attractors.

We report the existence of chaotic itinerancy in a coupled Milnor attractor system. The attractor ruins consist of tori or local chaos generated from the original Milnor attractors. The chaotic behavior exhibited by a single orbit can be considered a "nonstationary" state, due to the extremely slow convergence of the Lyapunov exponents, but the behavior averaged over randomly chosen initial conditions is consistent with the limit theorem. We present as a possibly new indication of chaotic itinerancy the presence of slow decay of large fluctuations of the largest Lyapunov exponent.

A note about Milnor attractor and riddled basin

In this paper, we give some examples to show that the relation among Milnor attractor and the support of nature measure and riddling attractor.

Convergence to local random attractors

Random attractors allow one to classify qualitative and quantitative aspects of the long-time behaviourof stochastically forced systems viewed as random dynamical systems (RDS) in an analogous way to attractors for deterministic systems. We compare several notions of random attractor in RDS by examining convergence of trajectories to a random invariant set in different ways, including convergence of mean distance, convergence in probability and convergence on dense subsequences as well as pullback convergence. We give examples showing how these concepts of attraction are inequivalent. We also examine definitions for local attractors and possible methods of decomposition of random attractors into 'random Milnor' attractors. We point out some problems that remain in interpreting these. Finally, we examine numerical simulations of a stochastic van der Pol-Duffing equation and find cases where there appear to be Milnor attractors with positive top Lyapunov exponents. We explain bursting behaviour of the two-point motion for some parameter values in terms of the presence of non-trivial random Milnor attractors.

Dominance of Milnor attractors in globally coupled dynamical systems with more than 7+/-2 degrees of freedom.

The prevalence of Milnor attractors has recently been reported in a class of high-dimensional dynamical systems. We study how this prevalence depends on the number of degrees of freedom by using a globally coupled map and show that the basin fraction of Milnor attractors increases drastically around 5-10 degrees of freedom, saturating for higher numbers of degrees of freedom. It is argued that this dominance of Milnor attractors in the basin arises from a combinatorial explosion of the basin boundaries. In addition, the dominance is also found in a system without permutation symmetry, i.e., a coupled dynamical system of nonidentical elements. Possible relevance to the magic number 7+/-2 in psychology is briefly discussed.

Milnor strange nonchaotic attractor with complex basin of attraction.

The transcritical blowout bifurcation of a quasiperiodic torus on an invariant subspace is studied in this paper. We found that the strange nonchaotic attractor (SNA) beyond the blowout bifurcation is only a weak attractor in the sense of Milnor. This is different from the popularly studied case where the Milnor attractor is a chaotic one with a riddled basin of attraction. Characters of this Milnor SNA and the influence of a random noise are studied.

Milnor attractors and topological attractorsof a piecewise linear map

A very simple two-dimensional map is discussed. It is shown that forappropriate values of the parameters there is a two-dimensional subset ofthe plane on which the dynamics is transitive and periodic orbits aredense, but that this topological attractor contains a one-dimensional setwhich attracts almost all points(i.e. it is a Milnor attractor). This arises naturally as a precursor to ablowout bifurcation to on-off intermittency in this system, and confirmsa conjecture due to Pikovsky and Grassberger.