Discrete and Continuous Dynamical Systems
December 2020 , Volume 40 , Issue 12
Special issue on dynamics at DEA 2019
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We prove local limit theorems for a cocycle over a semiflow by establishing topological, mixing properties of the associated skew product semiflow. We also establish conditional rational weak mixing of certain skew product semiflows and various mixing properties including order 2 rational weak mixing of hyperbolic geodesic flows on the tangent spaces of cyclic covers.
Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the set of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We expect our approach to be extendable to other holomorphic families of dynamical systems.
A conjecture of Arnold, Kozlov and Neishtadt on the exponentially small measure of the "non-torus" set in analytic systems with two degrees of freedom is discussed.
This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain
We consider externally forced equations in an evolution form. Mathematically, these are skew systems driven by a finite dimensional dynamical system. Two very common cases included in our treatment are quasi-periodic forcing and forcing by a stochastic process. We allow that the evolution is a PDE and even that it is not well-posed and that it does not define a flow (not all initial conditions lead to a solution).
We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a "time-dependent invariant manifold" (TDIM). These manifolds evolve with the forcing. They are such that the original equation is always tangent to a vector field in the manifold. Hence, for initial data in the TDIM, the original equation is equivalent to an ordinary differential equation. This allows us to define families of solutions of the full equation by studying the solutions of a finite dimensional system. Note that this strategy may apply even if the original equation is ill posed and does not admit solutions for arbitrary initial conditions (the TDIM selects initial conditions for which solutions exist). It also allows that the TDIM is infinite dimensional.
Secondly, we construct the center manifold for skew systems driven by the external forcing.
Thirdly, we present concrete applications of the abstract result to the differential equations whose linear operators are exponential trichotomy subject to quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.
We define globally hypoelliptic smooth
We study the regularity of the Green current for semi-extremal endomorphisms of
We investigate the plane dynamical system given by the secant map applied to a polynomial
We describe a recent method to show instability in Hamiltonian systems. The main hypothesis of the method is that some explicit transversality conditions – which can be verified in concrete systems by finite calculations – are satisfied.
In particular, for several types of perturbations of integrable Hamiltonian systems, the hypothesis can be verified by just checking that some Melnikov-type integrals have non-degenerate zeros. This holds for Baire generic sets of perturbations in the
Provided that the transversality conditions are verified, one concludes the existence of orbits which change the action coordinate by a quantity independent of the size of the perturbation. In fact, one can obtain orbits that follow any path in action space, up to an error decreasing with the size of the perturbation.
A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Hénon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.
Gauss-type iterates for random means are considered and their limit behaviour is studied. Among others the invariance of the limit with respect to the given random mean-type mapping
in the class of (deterministic) means in
We prove that if a finitely generated random group action is robustly expansive and has the shadowing property, then it is rigid. We apply this result to analyze the rigidity of certain iterated function systems or actions of the discrete Heisenberg group.
The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice
Given a finite alphabet
We discuss dynamical aspects of an analysis of the two–centre problem started in [
We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich–Aberth. Both are known to have found all roots of high degree polynomials with good complexity. Our goal is to determine in which cases which of the two algorithms is more efficient. We come to the conclusion that Newton is faster when the polynomials are given by recursion so they can be evaluated in logarithmic time with respect to the degree, or when all the roots are all near the boundary of their convex hull. Conversely, Ehrlich–Aberth has the advantage when no fast evaluation of the polynomials is available, and when roots are in the interior of the convex hull of other roots.
In this paper we study a functional equation of linear combination of nonautonomous iterations, which can be reduced from invariance of an interval homeomorphism under nonautonomous iteration. We discuss existence, uniqueness and continuous dependence for increasing Lipschitzian solutions on a compact interval, and also discuss for bounded increasing Lipschitzian solutions and unbounded ones on the whole
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