A link between topological entropy and Lyapunov exponents

We prove a Cr closing lemma for a class of partially hyperbolic symplectic diffeomorphisms. We show that for a generic Cr symplectic diffeomorphism, r = 1, 2, …, with a two-dimensional centre and close to a product map with a bunching condition, the set of all periodic points is dense.

Topological sequence entropy and topological dynamics of tree maps

This paper discusses the bifurcation of periodic points of a generic symplectic diffeomorphism of a two manifold that depends on a parameter.A complete classification of the types of bifurcation that can occur in the generic case is given.Introduction.This paper discusses the bifurcation of periodic points of an area preserving diffeomorphism that depends on a parameter.In the spirit of Thorn and Smale only the generic case is considered.As in Smale's work on discrete dynamical systems the present problem was suggested by problems in ordinary differential equations (see [1]).Whereas Smale modeled his theory on differential equations with dissipation the present problem was suggested by conservative differential equations.Throughout this paper smooth will always mean C.Let M be a smooth, compact, two dimensional, symplectic manifold, S=SX the circle considered as a smooth manifold and F the space of all smooth mappings <p: MxS-> M with the property that for each s e S the map <ps=<p(-, s): M-> M is a symplectic diffeomorphism.Let F have the usual topology of C maps.The space F is complete and therefore is a Baire space.A point (x, s) e M x S is called a periodic point of least period m if <p(x)=x and m is the smallest positive integer for which this holds.Any multiple of m will be called a period of (x, s).If (x, s) is a periodic point of <p of least period m then each point (<p\ix), s) e M x S is a distinct periodic point of least period m for i = 1,..., m and the set U= i (<pl(x), s) is called the orbit of <p through (x, s).We wish to discuss the nature of the set of periodic points for <p e F and in particular the manner in which these periodic points depend on the parameter seS.Instead of discussing the general element y e F we define a subset ^S^F and discuss the periodic points for tpe'S.Then by showing 'S is a residual subset of F insures that the discussion dealt with the generic case.The problem discussed here is closely related to the following problems in the theory of differential equations.Consider Duffing's equation d2xjdt2 + x + x3

Topological entropy of a dynamical system is an upper bound for the sum of positive Lyapunov exponents; in practice, it is strongly indicative of the presence of mixing in a subset of the domain. Topological entropy can be computed by partition methods, by estimating the maximal growth rate of material lines or other material elements, or by counting the unstable periodic orbits of the flow. All these methods require detailed knowledge of the velocity field that is not always available, for example, when ocean flows are measured using a small number of floating sensors. We propose an alternative calculation, applicable to two-dimensional flows, that uses only a sparse set of flow trajectories as its input. To represent the sparse set of trajectories, we use braids, algebraic objects that record how trajectories exchange positions with respect to a projection axis. Material curves advected by the flow are represented as simplified loop coordinates. The exponential rate at which a braid stretches loops over a finite time interval is the Finite-Time Braiding Exponent (FTBE). We study FTBEs through numerical simulations of the Aref Blinking Vortex flow, as a representative of a general class of flows having a single invariant component with positive topological entropy. The FTBEs approach the value of the topological entropy from below as the length and number of trajectories is increased; we conjecture that this result holds for a general class of ergodic, mixing systems. Furthermore, FTBEs are computed robustly with respect to the numerical time step, details of braid representation, and choice of initial conditions. We find that, in the class of systems we describe, trajectories can be re-used to form different braids, which greatly reduces the amount of data needed to assess the complexity of the flow.

We propose a deterministic algorithm for approximating a generating partition from a time series using tessellations. Using data generated by Hénon and Ikeda maps, we demonstrate that the proposed method produces partitions that uniquely encode all the periodic points up to some order, and provide good estimates of the metric and topological entropies. The algorithm gives useful results even with a short noisy time series.

In the present paper, we introduce a natural extension of Adler, Konheim and MacAndrew topological entropy for proper maps of locally compact separable metrizable spaces and prove a variational principle which states that this topological entropy, the supremum of the Kolmogorov–Sinai entropies and the minimum of the Bowen entropies always coincide. We apply this variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the Bowen formula for the Bowen entropy of an automorphism of a non-compact Lie group with respect to some invariant metric is just an upper bound for its topological entropy.

On the complexity of economic dynamics: An approach through topological entropy

This work examines self-mixing in active nematics, a class of fluids in which mobile topological defects drive chaotic flows in a system comprised of biological filaments and molecular motors. We present experiments that demonstrate how geometrical confinement can influence the braiding dynamics of the defects. Notably, we show that confinement in cardioid-shaped wells leads to realization of the golden braid, a maximally efficient mixing state of exactly three defects with no defect creation or annihilation. We characterize the golden braid state using different measures of topological entropy and the Lyapunov exponent. In particular, topological entropy measured from the stretching rate of material lines agrees well with an analytical computation from braid theory. Increasing the size of the confining cardioid produces a transition from the golden braid, to the fully chaotic active turbulent state.

Dripping water from a faucet is a typical example exhibiting rich nonlinear phenomena. For such a system, the time stamps at which water drops separate from the faucet can be directly observed in real experiments, and the time series of intervals τn between drop separations becomes a subject of analysis. Even if the mass mn of a drop at the onset of the nth separation, which is difficult to observe experimentally, exhibits perfectly deterministic dynamics, it may be difficult to obtain the same information about the underlying dynamics from the time series τn. This is because the return plot τn-1 vs. τn may become a multivalued relation (i.e., it doesn't represent a function describing deterministic dynamics). In this paper, we propose a method to construct a nonlinear coordinate which provides a "surrogate" of the internal state mn from the time series of τn. Here, a key of the proposed approach is to use isomap, which is a well-known method of manifold learning. We first apply it to the time series of τn generated from the numerical simulation of a phenomenological mass-spring model for the dripping faucet system. It is shown that a clear one-dimensional map is obtained by the proposed approach, whose characteristic quantities such as the Lyapunov exponent, the topological entropy, and the time correlation function coincide with the original dripping faucet system. Furthermore, we also analyze data obtained from real dripping faucet experiments, which also provide promising results.

On the total variation, topological entropy and sensitivity for interval maps

We present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. The main tool is local ergodicity theory for non-uniformly hyperbolic systems.

We consider piecewise monotone (not necessarily, strictly) piecewise C2 maps on the interval with positive topological entropy. For such a map f we prove that its topological entropy htop(f) can be approximated (with any required accuracy) by restriction on a compact strictly f-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.

In this paper we consider symplectic and volume-preserving diffeomorphisms of compact manifolds, and prove that, C^{r} generically, 1\leq r\leq\infty, the invariant manifolds of hyperbolic periodic points contain a dense subset of recurrent points. As a consequence, each of these invariant manifolds is contained in its omega limit set, and its closure is a chain transitive set.

Exploring abundance and nonlacunarity of hyperbolic times for endomorphisms preserving an ergodic probability with positive Lyapunov exponents, we obtain that there are periodic points of period growing sublinearly with respect to the length of almost every dynamical ball. In particular, we conclude that any ergodic measure with positive Lyapunov exponents satisfies the nonuniform specification property. As consequences, we (re)obtain estimates on the recurrence to a ball in terms of the Lyapunov exponents, and we prove that any expanding measure is the limit of Dirac measures on periodic points.

While structures and bifurcations controlling tracer particle transport and mixing have been studied extensively for systems with only stretching-and-folding, and to a lesser extent for systems with only cutting-and-shuffling, few studies have considered systems with a combination of both. We demonstrate two bifurcations for nonmixing islands associated with elliptic periodic points that only occur in systems with combined cutting-and-shuffling and stretching-and-folding, using as an example a map approximating biaxial rotation of a less-than-half-full spherical granular tumbler. First, we characterize a bifurcation of elliptic island containment, from containment by manifolds associated with hyperbolic periodic points to containment by cutting line tangency. As a result, the maximum size of the nonmixing region occurs when the island is at the bifurcation point. We also demonstrate a bifurcation where periodic points are annihilated by the cutting-and-shuffling action. Chains of elliptic and hyperbolic periodic points that arise when invariant tori surrounding an elliptic point break up [according to Kolmogorov-Arnold-Moser (KAM) theory] can annihilate when they meet a cutting line. Consequently, the Poincaré index (a topological invariant of smooth systems) is not preserved.