Successive approximation: A survey on stable manifold of fractional differential systems

This paper outlines a reliable strategy to approximate the local stable manifold near a hyperbolic equilibrium point for nonlinear systems of differential equations of fractional order. Furthermore, the local behavior of these systems near a hyperbolic equilibrium point is investigated based on the fractional Hartman-Grobman theorem. The fractional derivative is described in the Caputo sense. The solution existence, uniqueness, stability and convergence of the proposed scheme is discussed. Finally, the validity and applicability of our approach is examined with the use of a solvable model method.

Superstable manifolds of invariant circles and codimension-one Böttcher functions

Let $f: X~\dashrightarrow ~X$ be a dominant meromorphic self-map, where $X$ is a compact, connected complex manifold of dimension $n\gt 1$. Suppose that there is an embedded copy of ${ \mathbb{P} }^{1} $ that is invariant under $f$, with $f$ holomorphic and transversally superattracting with degree $a$ in some neighborhood. Suppose that $f$ restricted to this line is given by $z\mapsto {z}^{b} $, with resulting invariant circle $S$. We prove that if $a\geq b$, then the local stable manifold ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition $a\geq b$ cannot be relaxed without adding additional hypotheses by presenting two examples with $a\lt b$ for which ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is not real analytic in the neighborhood of any point.

ABSOLUTE CONTINUITY THEOREM FOR RANDOM DYNAMICAL SYSTEMS ON Rd

In this paper we provide a proof of the so-called absolute continuity theorem for random dynamical systems on Rd which have an invariant probability measure. First we present the construction of local stable manifolds in this case. Then the absolute continuity theorem basically states that for any two transversal manifolds to the family of local stable manifolds, the induced Lebesgue measures on these transversal manifolds are absolutely continuous under the map that transports every point on the first manifold along the local stable manifold to the second manifold, the so-called Poincaré map or holonomy map. In contrast to known results, we have to deal with the non-compactness of the state space and the randomness of the random dynamical system.

INVARIANT MANIFOLDS FOR STOCHASTIC MODELS IN FLUID DYNAMICS

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle [22] (cf. [20, 21]). The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao ([17–19]).

Fatou maps inℙndynamics

We study the dynamics of a holomorphic self-mapfof complex projective space of degreed>1by utilizing the notion of a Fatou map, introduced originally by Ueda (1997) and independently by the author (2000). A Fatou map is intuitively like an analytic subvariety on which the dynamics offare a normal family (such as a local stable manifold of a hyperbolic periodic point). We show that global stable manifolds of hyperbolic fixed points are given by Fatou maps. We further show that they are necessarily Kobayashi hyperbolic and are always ramified byf(and therefore any hyperbolic periodic point attracts a point of the critical set off). We also show that Fatou components are hyperbolically embedded inℙnand that a Fatou component which is attracted to a taut subset of itself is necessarily taut.