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 $C^r$ open and there exists a $C^r$ open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.     

Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents

This is a survey-type article whose goal is to review some recent results on existence of hyperbolic dynamical systems with discrete time on compact smooth manifolds and on coexistence of hyperbolic and non-hyperbolic behavior. It also discusses two approaches to the study of genericity of systems with nonzero Lyapunov exponents.      

Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2, ℝ) cocycles

We consider the linear cocycle (T, A) induced by a measure preserving dynamical system T : X → X and a map A: X → SL(2, ℝ). We address the dependence of the upper Lyapunov exponent of (T, A) on the dynamics T when the map A is kept fixed. We introduce explicit conditions on the cocycle that allow to perturb the dynamics, in the weak and uniform topologies, to make the exponent drop arbitrarily close to zero. In the weak topology we deduce that if X is a compact connected manifold, then for a C^r (r ≥ 1) open and dense set of maps A, either (T, A) is uniformly hyperbolic for every T, or the Lyapunov exponents of (T, A) vanish for the generic measurable T. For the continuous case, we obtain that if X is of dimension greater than 2, then for a C^r (r ≥ 1) generic map A, there is a residual set of volume-preserving homeomorphisms T for which either (T, A) is uniformly hyperbolic or the Lyapunov exponents of (T, A) vanish. *Partially supported by CNPq-Profix and Franco-Brazilian cooperation program in Mathematics.     

Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles

The C ¹ density conjecture of Palis asserts that diffeomorphisms exhibiting either a homoclinic tangency or a heterodimensional cycle are C ¹ dense in the complement of the C ¹ closure of hyperbolic systems. In this paper we prove some results towards the conjecture.* Work supported by the National Natural Science Foundation and the Doctoral Education Foundation of China, and the Qiu Shi Science and Technology Foundation of Hong Kong.     

Surface billiards and nonvanishing Lyapunov exponents

We study in this paper the billiards on surfaces with mix-valued Gaussian curvature and the condition which gives nonvanishing Lyapunov exponents of the system. We introduce a criterion upon which a small perturbation of the surface will also produce a system with positive Lyapunov exponents. Some examples of such surfaces are given in this article.     

Lyapunov exponents for continuous random transformations

In this paper, the concept of Lyapunov exponent is generalized to random transformations that are not necessarily differentiable. For a class of random repellers and of random hyperbolic sets obtained via small perturbations of deterministic ones respectively, the new exponents are shown to coincide with the classical ones.     

Lyapunov exponents for one-dimensional cellular automata

Summary In the paper we give a mathematical definition of the left and right Lyapunov exponents for a one-dimensional cellular automaton (CA). We establish an inequality between the Lyapunov exponents and entropies (spatial and temporal).     

Lyapunov exponents and control sets near singular points

We consider control affine systems of the form

     

Lyapunov exponents and stationary solutions for affine stochastic delay equations

A certain class of affine delay equations is considered. Two cases for the forcingfunction M are treated: M locally integrable deterministic, and M a random process with stationaryincrements. The Lyapunov spectrum of the homogeneous equation is used to decompose the state spaceinto finite-dimensional and finite-codimensional subspaces. Using a suitable variation of constants representation, formulas for the projection of the trajectories onto the above subspaces are obtained. If the homogeneous equation is hyperbolic and M has stationary increments, existence and uniqueness of a stationary solution for the affine stochastic delay equation is proved. The existence of Lyapunov exponents for the affine equation and their dependence on initial conditions is als studied.     

Geometric expansion,Lyapunov exponents and foliations

We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an application, we show examples of systems with persistent non-absolute continuous center and weak unstable foliations. This generalizes the remarkable results of Shub and Wilkinson to cases where the center manifolds are not compact.     

Hyperbolicity and integral expression of the Lyapunov exponents for linear cocycles

Consider in this paper a linear skew-product system

     

Almost sure and moment Lyapunov exponents for a stochastic beam equation

The transverse vibrations of an Euler-Bernoulli beam with axial tension P and axial white noise forcing are given by

     

Approximate and efficient calculation of dominant Lyapunov exponents of high-dimensional nonlinear dynamic systems

This short communication presents an efficient method for calculating dominant Lyapunov exponents (LEs) of high-dimensional nonlinear dynamic systems based on their reduced-order models obtained from the linear model reduction theory. Mathematical derivation shows that the LEs of the reduced-order models correspond to the dominant LEs of the original systems. Two numerical examples are provided to demonstrate the effectiveness of the method.     

The relation between quenched and annealed Lyapunov exponents in random potential on trees

Our subject of interest is a simple symmetric random walk on the integers which faces a random risk to be killed. This risk is described by random potentials, which are defined by a sequence of independent and identically distributed non-negative random variables. To determine the risk of taking a walk in these potentials we consider the decay of the Green function. There are two possible tools to describe this decay: The quenched Lyapunov exponent and the annealed Lyapunov exponent. It turns out that on the integers and on regular trees we can state a precise relation between these two.     

Lyapunov Exponents for Position Dependent Random Maps: Formulae and Applications

In this article, we provide formulae for Lyapunov exponents of position dependent random maps of the interval. We then apply our results to a financial market model with short-lived assets.     

Root Configurations for Hyperbolic Polynomials of Degree 3, 4, and 5

A real polynomial of one real variable is (strictly) hyperbolic if it has only real (and distinct) roots. There are 10 (resp. 116) possible non-degenerate configurations between the roots of a strictly hyperbolic polynomial of degree 4 (resp. 5) and of its derivatives (i.e., configurations without equalities between roots). The standard Rolle theorem allows 12 (resp. 286) such configurations. The result is based on the study of the hyperbolicity domain of the family P(x,a)=x ⁿ+a ₁ x ^n-1+...+a _n for n=4,5 (i.e., of the set of values of aⁿ for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets Res(P ⁽ⁱ⁾,P ^(j))=0, 0 i < jn-1.     

Long-term dynamics of discrete-time predator–prey models: stability of equilibria,cycles and chaos

ABSTRACT

In [A.S. Ackleh, M.I. Hossain, A. Veprauskas, and A. Zhang, Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Differ. Equ. Appl. 25 (2019), pp. 1568–1603.], we established conditions for the persistence and local asymptotic stability of the interior equilibrium for two discrete-time predator–prey models (one without and with evolution to resist toxicants). In the current paper, we provide a more in-depth analysis of these models, including global stability of equilibria, existence of cycles and chaos. Our main focus is to examine how the speed of evolution ν may impact population dynamics. For both models, we establish conditions under which the interior equilibrium is global asymptotically stable using perturbation analysis together with the construction of Lyapunov functions. For small ν, we show that the global dynamics of the evolutionary system are nothing but a continuous perturbation of the non-evolutionary system. However, when the speed of evolution is increased, we perform numerical studies which demonstrate that evolution may introduce rich dynamics including cyclic and chaotic behaviour that are not observed when evolution is absent.