Hyperbolicity and Invariant Manifolds for Planar Nonautonomous Systems on Finite Time Intervals

Abstract

The method of invariant manifolds was originally developed for hyperbolic rest points of autonomous equations. It was then extended from fixed points to arbitrary solutions and from autonomous equations to nonautonomous dynamical systems by either the Lyapunov–Perron approach or Hadamard's graph transformation. We go one step further and study meaningful notions of hyperbolicity and stable and unstable manifolds for equations which are defined or known only for a finite time, together with matching notions of attraction and repulsion. As a consequence, hyperbolicity and invariant manifolds will describe the dynamics on the finite time interval. We prove an analog of the Theorem of Linearized Asymptotic Stability on finite time intervals, generalize the Okubo–Weiss criterion from fluid dynamics and prove a theorem on the location of periodic orbits. Several examples are treated, including a double gyre flow and symmetric vortex merger.

Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

“…The paper of Weiss and Infante (1965) and rigorous discussion on finite time stability. More recently, Duc and Siegmund (2008) developed basic building blocks (i.e. hyperbolic trajectories and their stable and unstable manifolds) for two-dimensional, time-dependent Hamiltonian systems defined only for a finite time interval.…”

Finite-Time Lagrangian Transport Analysis: Stable and Unstable Manifolds of Hyperbolic Trajectories and Finite-Time Lyapunov Exponents

“…Another approach to characterising the stability properties originates from the so-called EPH-partition due to Haller (see the Appendix and [38,28]). This criterion relies upon considering the characteristics of the so-called rate of strain tensor,Ŝ(t) (cf Definition A.7), and the strain acceleration tensor,M (t) (cf Definition A.9), derived for a flow linearised about the considered trajectory.…”

“…As already noted in §3.2.2, if the system (69) is only known (or defined) on a bounded interval I ⊂ IR, it is not possible to define the stable and unstable manifolds of ξ ξ ξ(t) = 0 in the traditional 'inifinite-time' sense even if ξ ξ ξ(t) = 0 is hyperbolic (in the inifinite-time sense) for the system (70) considered on I = IR. However, if ξ ξ ξ(t) = 0 is finite-time hyperbolic on I, one can define (cf [28]) the following two flow-invariant, 'stable' and 'unstable' sets: The finite-time stable set of γ γ γ(t) = 0 on I is given by…”

A Remark on Finite‐Time Hyperbolicity

“…With these quantities, recall two notions of hyperbolicity (see [1][2][3] for details, as well as [5,6] for more general background information on finite-time dynamics and its applications). Definition 1.1 (D-Hyperbolicity) System (1) is called D-hyperbolic on I if, for all t ∈ I, S(t) is indefinite and non-degenerate, and ξ, M (t)ξ > 0 for all ξ ∈ Z(t).…”

“…With these quantities, recall two notions of hyperbolicity (see [1][2][3] for details, as well as [5,6] for more general background information on finite-time dynamics and its applications). (1) is called M-hyperbolic on I if there exist constants α < 0 < β and an invariant projector P , i.e., the mapping P : I → R d×d is projection-valued with P (t)Φ(t, s) = Φ(t, s)P (s) for all t, s ∈ I, such that…”