Center Lyapunov Exponents Research Articles

Dynamical analysis of a novel 2D Lyapunov exponent controllable memristive chaotic map.

The proposal of discrete memristors has made memristive chaotic maps based on them an important research topic. In this study, a new two-dimensional chaotic map without fixed points is constructed, and numerical simulation results display its rich dynamical behaviors. The analysis reveals the map's center inversion symmetry and Lyapunov exponent controller. The map exhibits complex dynamical behaviors, including memristor initial-boosting and single-parameter-offset boosting. Embedding the absolute value function within the memristor results in the emergence of localized boosting-free regions. Moreover, a class of multicavity transients is captured that greatly enhances the system's complexity. Ultimately, this map is implemented on the STM32 platform, demonstrating its practical applicability in potential practical application scenarios.

Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity * *Y Cao is partially supported by National Key R&D Program of China(2022YFA1005802) and NSFC 12371194. Z Mi is partially supported by National Key R&D Program of China(2022YFA1007800) and NSFC 12271260. J Yang was partially supported by CNPq, FAPERJ, PRONEX and NNSFC Grants 12271538, 12071202 and 12161141002.

We study the partially hyperbolic diffeomorphisms whose centre direction admits the u-definite property in the sense that all the central Lyapunov exponents of each ergodic Gibbs u-state are either all positive or all negative. We prove that for this kind of partially hyperbolic diffeomorphisms, there are finitely many physical measures, whose basins cover a full Lebesgue measure subset of the ambient space. We also provide examples of diffeomorphisms whose centres are u-definite, where the supports of different ergodic Gibbs u-states have nontrivial intersections.

Rigidity of center Lyapunov exponents for Anosov diffeomorphisms on 3-torus

Let f f and g g be two Anosov diffeomorphisms on T 3 \mathbb {T}^3 with three-subbundles partially hyperbolic splittings where the weak stable subbundles are considered as center subbundles. Assume that f f is conjugate to g g and the conjugacy preserves the strong stable foliation, then their center Lyapunov exponents of corresponding periodic points coincide. This is the converse of the main result of Gogolev and Guysinsky [Discrete Contin. Dyn. Syst. 22 (2008), pp. 183–200]. Moreover, we get the same result for partially hyperbolic diffeomorphisms derived from Anosov on T 3 \mathbb {T}^3 .

Maximal entropy measures of diffeomorphisms of circle fiber bundles

We characterize the maximal entropy measures of partially hyperbolic C^2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons. In the special case of 3-dimensional nilmanifolds other than the torus, this entails the following dichotomy: either the diffeomorphism is a rotation extension of an Anosov diffeomorphism -- in which case there is a unique maximal measure, with full support and zero center Lyapunov exponents -- or there exist exactly two ergodic maximal measures, both hyperbolic and whose center Lyapunov exponents have opposite signs. Moreover, the set of maximal measures varies continuously with the diffeomorphism.

Rigidity of center Lyapunov exponents and $su$-integrability

Let f be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism A on \mathbb{T}^3 . We show that the stable and unstable bundles of f are jointly integrable if and only if every periodic point of f admits the same center Lyapunov exponent with A . This implies every conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism on \mathbb{T}^3 , is ergodic. This proves the Ergodic Conjecture proposed by Hertz–Hertz–Ures on \mathbb{T}^3 .

Statistical stability of mostly expanding diffeomorphisms

We study how physical measures vary with the underlying dynamics in the open class of Cr, r>1, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs u-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics.A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs cu-states. Both of these may be of independent interest.The non-transitive case is also treated: here the number of physical measures varies upper semi-continuously with the diffeomorphism, and physical measures vary continuously whenever possible.

Empirical Measures of Partially Hyperbolic Attractors

In this paper, we study the limit measures of the empirical measures of Lebesgue almost every point in the basin of a partially hyperbolic attractor. They are strongly related to a notion named Gibbs u-state, which can be defined in a large class of diffeomorphisms with less regularity and which is the same as Pesin–Sinai’s notion for partially hyperbolic attractors of $$C^{1+\alpha }$$ diffeomorphisms. In particular, we prove that for partially hyperbolic $$C^{1+\alpha }$$ diffeomorphisms with one-dimensional center, and for Lebesgue almost every point: (1) the center Lyapunov exponent is well defined, but (2) the sequence of empirical measures may not converge. In order to prove (2), we build a diffeomorphism with historical behavior which is transitive (contrary to the well-known example of Bowen). We also give some consequences on SRB measures and large deviations.

-openness of non-uniform hyperbolic diffeomorphisms with bounded -norm

We study the $C^{1}$-topological properties of the subset of non-uniform hyperbolic diffeomorphisms in a certain class of $C^{2}$ partially hyperbolic symplectic systems which have bounded $C^{2}$ distance to the identity. In this set, we prove the stability of non-uniform hyperbolicity as a function of the diffeomorphism and the measure, and the existence of an open and dense subset of continuity points for the center Lyapunov exponents. These results are generalized to the volume-preserving context.

Pathological center foliation with dimension greater than one

We are considering partially hyperbolic diffeomorphims of the torus, with \begin{document}${\rm{dim}}(E^c) > 1.$\end{document} We prove, under some conditions, that if the all center Lyapunov exponents of the linearization \begin{document}$A,$\end{document} of a DA-diffeomorphism \begin{document}$f,$\end{document} are positive and the center foliation of \begin{document}$f$\end{document} is absolutely continuous, then the sum of the center Lyapunov exponents of \begin{document}$f$\end{document} is bounded by the sum of the center Lyapunov exponents of \begin{document}$A.$\end{document} After, we construct a \begin{document}$C^1-$\end{document} open class of volume preserving DA-diffeomorphisms, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each \begin{document}$f$\end{document} in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of \begin{document}$f$\end{document} is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of \begin{document}$f$\end{document} is non absolutely continuous. We still build an example of a DA-diffeomorphism, such that the disintegration of volume along the two dimensional, non compact center foliation is neither Lebesgue nor atomic.

Dynamics analysis of 2-DOF complex planar mechanical system with joint clearance and flexible links

Joint clearance and flexible links are two important factors that affect the dynamic behaviors of planar mechanical system. Traditional dynamics studies of planar mechanism rarely take into account both influence of revolute clearance and flexible links, which results in lower accuracy. And many dynamics studies mainly focus on simple mechanism with clearance, the study of complex mechanism with clearance is a few. In order to study dynamic behaviors of two-degree-of-freedom (DOF) complex planar mechanical system more precisely, the dynamic analyses of the mechanical system with joint clearance and flexibility of links are studied in this work. Nonlinear dynamic model of the 2-DOF nine-bar mechanical system with revolute clearance and flexible links is built by Lagrange and finite element method (FEM). Normal and tangential force of the clearance joint is built by the Lankarani–Nikravesh and modified Coulomb’s friction models. The influences of clearance value and driving velocity of the crank on the dynamic behaviors are researched, including motion responses of slider, contact force, driving torques of cranks, penetration depth, shaft center trajectory, Phase diagram, Lyapunov exponents and Poincare map of clearance joint and slider are both analyzed, respectively. Bifurcation diagrams under different clearance values and different driving velocities of cranks are also investigated. The results show that clearance joint and flexibility of links have a certain impact on dynamic behavior of mechanism, and flexible links can partly decrease dynamic response of the mechanical system with clearance relative to rigid mechanical system with clearance.

Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle

We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are Cr open and there exists a Cr open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.

Stability by linear approximation for time scale dynamical systems

We study systems on time scales that are generalizations of classical differential or difference equations and appear in numerical methods. In this paper we consider linear systems and their small nonlinear perturbations. In terms of time scales and of eigenvalues of matrices we formulate conditions, sufficient for stability by linear approximation. For non-periodic time scales we use techniques of central upper Lyapunov exponents (a common tool of the theory of linear ODEs) to study stability of solutions. Also, time scale versions of the famous Chetaev's theorem on conditional instability are proved. In a nutshell, we have developed a completely new technique in order to demonstrate that methods of non-autonomous linear ODE theory may work for time-scale dynamics.

Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures

We give explicit C 1-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The conditions of the criterion are met on a C 1-dense and open subset of the set of diffeomorphisms having a robust cycle. As a corollary, there exists a C 1-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with nonhyperbolic ergodic measures with positive entropy. The criterion is based on a notion of a blender defined dynamically in terms of strict invariance of a family of discs.

New Derived from Anosov Diffeomorphisms with Pathological Center Foliation

In this paper we focused our study on derived from Anosov diffeomorphisms (DA diffeomorphisms ) of the torus $$\mathbb {T}^3,$$ it is, an absolute partially hyperbolic diffeomorphism on $$\mathbb {T}^3$$ homotopic to a linear Anosov automorphism of the $$\mathbb {T}^3.$$ We can prove that if $$f: \mathbb {T}^3 \rightarrow \mathbb {T}^3 $$ is a volume preserving DA diffeomorphism homotopic to a linear Anosov A, such that the center Lyapunov exponent satisfies $$\lambda ^c_f(x) > \lambda ^c_A > 0,$$ with x belongs to a positive volume set, then the center foliation of f is non absolutely continuous. We construct a new open class U of non Anosov and volume preserving DA diffeomorphisms, satisfying the property $$\lambda ^c_f(x) > \lambda ^c_A > 0$$ for $$m-$$ almost everywhere $$x \in \mathbb {T}^3.$$ Particularly for every $$f \in U,$$ the center foliation of f is non absolutely continuous.

Center Lyapunov exponents in partially hyperbolic dynamics

In this survey, we discuss the problem of removing zero Lyapunov exponents of smooth invariant measures along the center direction of a partially hyperbolic diffeomorphism and various related questions. In particular, we discuss disintegration of a smooth invariant measure along the center foliation. We also simplify the proofs of some known results and include new questions and conjectures.

The local $C^1$-density of stable ergodicity

In this paper, we prove that stable ergodicity is $C^1$-dense amongconservative partially hyperbolic systems which, in a stable way,have two ergodic measuressuch that one has all center Lyapunov exponents non-negative andthe other one has all center Lyapunov exponents non-positive.

Maximizing measures for partially hyperbolic systems with compact center leaves

AbstractWe obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of three-dimensional manifolds having compact center leaves: either there is a unique entropy-maximizing measure, this measure has the Bernoulli property and its center Lyapunov exponent is 0, or there are a finite number of entropy-maximizing measures, all of them with non-zero center Lyapunov exponents (at least one with a negative exponent and one with a positive exponent), that are finite extensions of a Bernoulli system. In the first case of the dichotomy, we obtain that the system is topologically conjugated to a rotation extension of a hyperbolic system. This implies that the second case of the dichotomy holds for an open and dense set of diffeomorphisms in the hypothesis of our result. As a consequence, we obtain an open set of topologically mixing diffeomorphisms having more than one entropy-maximizing measure.

Rich phase transitions in step skew products

We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exponents. It is associated with the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behaviour of different types. However, in some sense the expanding behaviour is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent.

Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents

We establish stable ergodicity for diffeomorphisms with partially hyperbolic attractors whose Lyapunov exponents along the center direction are all positive with respect to SRB measures.

Robust transitivity and almost robust ergodicity

In this paper, we prove a relation between robust transitivity and robust ergodicity for conservative diffeomorphisms. In dimension two, robustly transitive diffeomorphisms are robustly ergodic. In higher dimensions, we work with partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. We define almost robust ergodicity and prove that in the three-dimensional case robustly transitive diffeomorphisms are approximated by almost robustly ergodic ones. In higher dimensions under the condition of the central Lyapunov exponents being of the same sign, we prove that robustly transitive diffeomorphisms are robustly ergodic.