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Mean-field games (MFGs) provide a statistical physics-inspired modelling framework for decision-making in large populations of strategic, non-cooperative agents. Mathematically, these systems consist of a forwards–backwards in time-system of two coupled nonlinear partial differential equations (PDEs), namely, the Fokker–Plank (FP) and the Hamilton–Jacobi–Bellman (HJB) equations, governing the agent state and control distribution, respectively. In this work, we study a finite-time MFG with a rich global bifurcation structure using a reduced-order model (ROM). The ROM is a four-dimensional (4D) two-point boundary value problem (BVP) obtained by restricting the controlled dynamics to the first two moments of the agent state distribution, i.e. the mean and the variance. Phase space analysis of the ROM reveals that the invariant manifolds of periodic orbits around the so-called ‘ergodic MFG equilibrium’ play a crucial role in determining the bifurcation diagram and imparting a topological signature to various solution branches. We show a qualitative agreement of these results with numerical solutions of the full-order MFG PDE system.
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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.
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- Award ID(s):
- 1722578
- PAR ID:
- 10354189
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 34
- Issue:
- 5
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 2970 to 3000
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6544/abe456
