A geometric characterization of unstable blow-up solutions with computer-assisted proof

03/23/2021
by   Jean-Philippe Lessard, et al.
0

In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial conditions are studied. Combining dynamical systems machinery (e.g. phase space compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g. rigorous integrators, parameterization method for invariant manifolds), these unstable blow-up solutions are obtained as trajectories on stable manifolds of hyperbolic (saddle) equilibria at infinity. In this process, important features are obtained: smooth dependence of blow-up times on initial points near blow-up, level set distribution of blow-up times, singular behavior of blow-up times on unstable blow-up solutions, organization of the phase space via separatrices (stable manifolds). In particular, we show that unstable blow-up solutions themselves, and solutions defined globally in time connected by those blow-up solutions can separate initial conditions into two regions where solution trajectories are either globally bounded or blow-up, no matter how the large initial points are.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/09/2023

Gentlest ascent dynamics on manifolds defined by adaptively sampled point-clouds

Finding saddle points of dynamical systems is an important problem in pr...
research
02/27/2022

Computer-assisted proofs of Hopf bubbles and degenerate Hopf bifurcations

We present a computer-assisted approach to prove the existence of Hopf b...
research
09/05/2022

Numerical dynamics of integrodifference equations: Periodic solutions and invariant manifolds in C^α(Ω)

Integrodifference equations are versatile models in theoretical ecology ...
research
04/21/2022

Staying the course: Locating equilibria of dynamical systems on Riemannian manifolds defined by point-clouds

We introduce a method to successively locate equilibria (steady states) ...
research
12/13/2021

Computing complete hyperbolic structures on cusped 3-manifolds

A fundamental way to study 3-manifolds is through the geometric lens, on...
research
03/27/2023

A Compositional Approach to Certifying the Almost Global Asymptotic Stability of Cascade Systems

In this work, we give sufficient conditions for the almost global asympt...

Please sign up or login with your details

Forgot password? Click here to reset