A General View on Double Limits in Differential Equations
arXiv:2106.01160 · doi:10.1016/j.physd.2021.133105
Abstract
In this paper, we review several results from singularly perturbed differential equations with multiple small parameters. In addition, we develop a general conceptual framework to compare and contrast the different results by proposing a three-step process. First, one specifies the setting and restrictions of the differential equation problem to be studied and identifies the relevant small parameters. Second, one defines a notion of equivalence via a property/observable for partitioning the parameter space into suitable regions near the singular limit. Third, one studies the possible asymptotic singular limit problems as well as perturbation results to complete the diagrammatic subdivision process. We illustrate this approach for two simple problems from algebra and analysis. Then we proceed to the review of several modern double-limit problems including multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each example, we illustrate the previously mentioned three-step process and show that already double-limit parametric diagrams provide an excellent unifying theme. After this review, we compare and contrast the common features among the different examples. We conclude with a brief outlook, how our methodology can help to systematize the field better, and how it can be transferred to a wide variety of other classes of differential equations.
References in corpus (58)
- Networks beyond pairwise interactions: structure and dynamics
- Epidemic dynamics on an adaptive network
- Adaptive Coevolutionary Networks: A Review
- What are higher-order networks?
- Mathematical frameworks for oscillatory network dynamics in neuroscience
- Phase reduction approach to synchronization of nonlinear oscillators
- Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review
- Plasticity and learning in a network of coupled phase oscillators
- Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching
- Chaos in generically coupled phase oscillator networks with nonpairwise interactions
- Exponential ergodicity for Markov processes with random switching
- Phase reduction beyond the first order: the case of the mean-field complex Ginzburg-Landau equation
- Van der Pol and the history of relaxation oscillations: toward the emergence of a concept
- Hopf normal form with symmetry and reduction to systems of nonlinearly coupled phase oscillators
- Phase-amplitude descriptions of neural oscillator models
- Stochastic chaos in a turbulent swirling flow
- Multi-clusters in networks of adaptively coupled phase oscillators networks
- Geometric singular perturbation theory for stochastic differential equations
- Quantitative ergodicity for some switched dynamical systems
- Chaos in Symmetric Phase Oscillator Networks
- Invariant densities for dynamical systems with random switching
- A geometric analysis of the SIR, SIRS and SIRWS epidemiological models
- Shear-Induced Chaos
- Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model
- The Morris-Lecar neuron model embeds a leaky integrate-and-fire model
- Pathwise description of dynamic pitchfork bifurcations with additive noise
- Lotka Volterra in fluctuating environment or "how switching between beneficial environments can make survival harder"
- Hunting French Ducks in a Noisy Environment
- Stochastic switching in infinite dimensions with applications to random parabolic PDEs
- Lyapunov exponents for random perturbations of some area-preserving maps including the standard map
- Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System
- Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes and applications to molecular motors
- Multiscale Geometry of the Olsen Model and Non-Classical Relaxation Oscillations
- Noise and dissipation on coadjoint orbits
- Sensitivity to switching rates in stochastically switched ODEs
- Hopf bifurcation with additive noise
- Stochastic calculus: application to dynamic bifurcations and threshold crossings
- Normal Hyperbolicity and Unbounded Critical Manifolds
- Singularly Perturbed Boundary-Equilibrium Bifurcations
- From limit cycles to strange attractors
- A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential
- From random Poincaré maps to stochastic mixed-mode-oscillation patterns
- Regularity of invariant densities for 1D-systems with random switching
- Itinerant chimeras in an adaptive network of pulse-coupled oscillators
- Characterizing mixed mode oscillations shaped by noise and bifurcation structure
- Bifurcation Analysis of a Stochastically Driven Limit Cycle
- Conditioned Lyapunov exponents for random dynamical systems
- On fast-slow consensus networks with a dynamic weight
- Weak topologies for Carathéodory differential equations. Continuous dependence, exponential Dichotomy and attractors
- Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems
- Discretized Fast-Slow Systems near Transcritical Singularities
- Homogenization of Fully-Coupled Chaotic Fast-Slow Systems via Intermediate Stochastic Regularization
- State-dependent effective interactions in oscillator networks through coupling functions with dead zones
- The Curse of Instability
- Interactions between different predator-prey states. A method for the derivation of the functional and numerical response
- Randomly switched vector fields sharing a zero on a common invariant face
- Lotka-Volterra with randomly fluctuating environments: a full description
- Singularities of invariant densities for random switching between two linear ODEs in 2D