Exploring invariant sets and invariant measures

We propose a method to explore invariant measures of dynamical systems. The method is based on numerical tools which directly compute invariant sets using a subdivision technique, and invariant measures by a discretization of the Frobenius-Perron operator. Appropriate visualization tools help to analyze the numerical results and to understand important aspects of the underlying dynamics. This will be illustrated for examples provided by the Lorenz system. (c) 1997 American Institute of Physics.     

Suspension Flows Over Countable Markov Shifts

We introduce the notion of topological pressure for suspension flows over countable Markov shifts, and we develop the associated thermodynamic formalism. In particular, we establish a variational principle for the topological pressure, and an approximation property in terms of the pressure on compact invariant sets. As an application we present a multifractal analysis for the entropy spectrum of Birkhoff averages for suspension flows over countable Markov shifts. The domain of the spectrum may be unbounded and the spectrum may not be analytic. We provide explicit examples where this happens. We also discuss the existence of full measures on the level sets of the multifractal decomposition.     

Some relations between dimension and Lyapounov exponents

We consider differentiable maps and compact invariant sets. We introduce dimensional quantities related to the ergodic invariant measures, and prove some simple relations.     

Multifractal Analysis for Lyapunov Exponents on Nonconformal Repellers

For nonconformal repellers satisfying a certain cone condition, we establish a version of multifractal analysis for the topological entropy of the level sets of the Lyapunov exponents. Due to the nonconformality, the Lyapunov exponents are averages of nonadditive sequences of potentials, and thus one cannot use Birkhoff’s ergodic theorem nor the classical thermodynamic formalism. We use instead a nonadditive topological pressure to characterize the topological entropy of each level set. This prevents us from estimating the complexity of the level sets using the classical Gibbs measures, which are often one of the main ingredients of multifractal analysis. Instead, we avoid even equilibrium measures, and thus in particular g-measures, by constructing explicitly ergodic measures, although not necessarily invariant, which play the corresponding role in our work.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, through FCT by Program POCTI/FEDER and the grant SFRH/BPD/12108/2003.     

Invariant Probability Measures and Non-wandering Sets for Impulsive Semiflows

We consider impulsive dynamical systems defined on compact metric spaces and their respective impulsive semiflows. We establish sufficient conditions for the existence of probability measures which are invariant under such impulsive semiflows. Under these conditions we also deduce the forward invariance of their non-wandering sets except the discontinuity points.     

Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hènon maps. For this class of systems we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Hènon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism. 2000 Mathematics Subject Classification. Primary: 37C45, 37D20, 37D35, Secondary: 37A35, 37E30     

Non-uniform Specification and Large Deviations for Weak Gibbs Measures

We establish bounds for the measure of deviation sets associated to continuous observables with respect to not necessarily invariant weak Gibbs measures. Under some mild assumptions, we obtain upper and lower bounds for the measure of deviation sets of some non-uniformly expanding maps, including quadratic maps and robust multidimensional non-uniformly expanding local diffeomorphisms. For that purpose, a measure theoretical weak form of specification is introduced and proved to hold for the robust classes of multidimensional non-uniformly expanding local diffeomorphisms and Viana maps.     

A method for visualization of invariant sets of dynamical systems based on the ergodic partition

We provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure. The algorithm is based on a constructive proof of the ergodic partition theorem for automorphisms of compact metric spaces. The ergodic partition of a compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A. The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets. The method is applied to the three-dimensional ABC map for which the dynamics was visualized by other methods in Feingold et al. [J. Stat. Phys. 50, 529 (1988)]. (c) 1999 American Institute of Physics.     

Generalized dimensions,entropies, and Liapunov exponents from the pressure function for strange sets

For conformal mixing repellers such as Julia sets and nonlinear one-dimensional Cantor sets, we connect the pressure of a smooth transformation on the repeller with its generalized dimensions, entropies, and Liapunov exponents computed with respect to a set of equilibrium Gibbs measures. This allows us to compute the pressure by means of simple numerical algorithms. Our results are then extended to axiom-A attractors and to a nonhyperbolic invariant set of the line. In this last case, we show that a first-order phase transition appears in the pressure.     

Machine failure forewarning via phase-space dissimilarity measures

We present a model-independent, data-driven approach to quantify dynamical changes in nonlinear, possibly chaotic, processes with application to machine failure forewarning. From time-windowed data sets, we use time-delay phase-space reconstruction to obtain a discrete form of the invariant distribution function on the attractor. Condition change in the system's dynamic is quantified by dissimilarity measures of the difference between the test case and baseline distribution functions. We analyze time-serial mechanical (vibration) power data from several large motor-driven systems with accelerated failures and seeded faults. The phase-space dissimilarity measures show a higher consistency and discriminating power than traditional statistical and nonlinear measures, which warrants their use for timely forewarning of equipment failure.     

Compact invariant sets of the Bianchi VIII and Bianchi IX Hamiltonian systems

In this Letter we prove that all compact invariant sets of the Bianchi VIII Hamiltonian system are contained in the set described by several simple linear equalities and inequalities. Moreover, we describe invariant domains in which the phase flow of this system has no recurrence property and show that there are no periodic orbits and neither homoclinic, nor heteroclinic orbits contained in the zero level set of its Hamiltonian. Similar results are obtained for the Bianchi IX Hamiltonian system.     

Phase resetting effects for robust cycles between chaotic sets

In the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated, owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena, including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. In this paper we introduce and discuss an instructive example of an ordinary differential equation where one can observe and analyze robust cycling behavior. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a R?ssler system), and/or saddle equilibria. For this model, we distinguish between cycling that includes phase resetting connections (where there is only one connecting trajectory) and more general non(phase) resetting cases, where there may be an infinite number (even a continuum) of connections. In the nonresetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability, whereas more general cases may give rise to "stuck on" cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase resetting and connection selection.     

Volume-preserving maps with an invariant

Several families of volume-preserving maps on R(3) that have an integral are constructed using techniques due to Suris. We study the dynamics of these maps as the topology of the two-dimensional level sets of the invariant changes. (c) 2002 American Institute of Physics.     

A rigorous partial justification of Greene's criterion

We prove several theorems that lend support to Greene's criterion for the existence or not of invariant circles in twist maps. In particular, we show that some of the implications of the criterion are correct when the Aubry-Mather sets are smooth invariant circles or uniformly hyperbolic. We also suggest a simple modification that can work in the case that the Aubry-Mather sets have nonzero Lyapunov exponents. The latter is based on a closing lemma for sets with nonzero Lyapunov exponents, which may have several other applications.     

Set-based corral control in stochastic dynamical systems: making almost invariant sets more invariant

We consider the problem of stochastic prediction and control in a time-dependent stochastic environment, such as the ocean, where escape from an almost invariant region occurs due to random fluctuations. We determine high-probability control-actuation sets by computing regions of uncertainty, almost invariant sets, and Lagrangian coherent structures. The combination of geometric and probabilistic methods allows us to design regions of control, which provide an increase in loitering time while minimizing the amount of control actuation. We show how the loitering time in almost invariant sets scales exponentially with respect to the control actuation, causing an exponential increase in loitering times with only small changes in actuation force. The result is that the control actuation makes almost invariant sets more invariant.     

Traveling patterns in cellular automata

A method to identify the invariant subsets of bi-infinite configurations of cellular automata that propagate rigidly with a constant velocity nu is described. Causal traveling configurations, propagating at speeds not greater than the automaton range, mid R:numid R:相似文献   

Multifractal structure of a riddled basin

For a two-dimensional piecewise linear map with a riddled basin, a multifractal spectrum f(gamma), which characterizes the "skeletons" of the riddled basin, is introduced. With f(gamma), the uncertainty exponent is obtained by a variational principle, which enables us to introduce a concept of a "boundary" for the riddled basin. A conjecture on the relation between f(gamma) and the "stable sets" of various ergodic measures, which coexist with the natural invariant measure on the chaotic attractor, is also proposed. (c) 2001 American Institute of Physics.     

Absolutely Continuous Invariant Measures¶for Piecewise Real-Analytic Expanding Maps¶on the Plane

We prove the existence of absolutely continuous invariant measures for piecewise real-analytic expanding maps on bounded regions in the plane. Received: 5 June 1998 / Accepted: 11 May 1999