Compactification for asymptotically autonomous dynamical systems: theory, applications and invariant manifolds

Abstract

We develop a general compactification framework to facilitate analysis of nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyze the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework, and their analysis can be greatly simplified by the compactification.

Critical Transitions in Piecewise Uniformly Continuous Concave Quadratic Ordinary Differential Equations

“…The previous description of these cases shows the link between this type of tipping points and a simple nonautonomous saddle-node bifurcation pattern [30]: a transversal critical transition occurs when the attractor-repeller pair collides in just one bounded solution. Such a collision has been explored analytically and numerically in several contexts: in one-dimensional systems [5,26]; in higher-dimensional systems [1,36,44,45]; in set-valued dynamical systems [11]; in random dynamical systems [20]; in regards to earlywarning signals [35,36]; and in the nonautonomous formulation [29]. There are other points of connection between the two considered cases.…”

Rate-induced tipping to metastable Zombie fires

“…To analyse and understand non-obvious dynamical mechanisms that are responsible for the R-tipping instabilities in figures 3 and 4, we: Consider external inputs Tafalse(rtfalse) that tend exponentially to a constant at infinity.Reformulate the ensuing two-dimensional nonautonomous system (2.3)–(2.4) as a three-dimensional autonomous compactified system [37]. Identify three different timescales in the soil-carbon system.We choose to work with external inputs Tafalse(rtfalse) that decay exponentially to a constant Ta± as time t tends to ±normal∞.…”

“…Reformulate the ensuing two-dimensional nonautonomous system (2.3)–(2.4) as a three-dimensional autonomous compactified system [37].…”

“…Firstly, we consider external inputs that decay to a constant at infinity. Secondly, we compactify the problem to include the equilibrium base states for the autonomous limit systems from infinity [37]. These two strategies alone work for R-tipping due to crossing regular thresholds that are anchored at infinity by unstable compact invariant sets called regular R-tipping edge states ([8], Sec.4).…”

Transverse instability of concentric soliton waves

“…A convenient way to understand the structure of solution variety of (2.54-2.56) is through a dynamical systems point of view (Jones & Küpper 1986;Newton & Watanabe 1993). The idea is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at both ends by gluing in invariant subspaces that carry autonomous dynamics of the limit systems (Wieczorek et al 2021). Namely, reducing, for example, (2.54,2.55) to an non-autonomous system of first-order equations:…”