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Concept# Stable manifold

Summary

In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set.
Physical example
The gravitational tidal forces acting on the rings of Saturn provide an easy-to-visualize physical example. The tidal forces flatten the ring into the equatorial plane, even as they stretch it out in the radial direction. Imagining the rings to be sand or gravel particles ("dust") in orbit around Saturn, the tidal forces are such that any perturbations that push particles above or below the equatorial plane results in that particle feeling a restoring force, pushing it back into the plane. Particles effectively oscillate in a harmonic well, damped by collisions. The stable direction is perpendicular to the ring. The unstab

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MATH-325: Dynamics and bifurcation

Introduction to local and global behavior of
nonlinear dynamical systems arising from maps and ordinary differential equations. Theoretical and computational aspects studied.

PHYS-460: Nonlinear dynamics, chaos and complex systems

The course provides students with the tools to approach the study of nonlinear systems and chaotic dynamics. Emphasis is given to concrete examples and numerical applications are carried out during the exercise sessions.

MICRO-311(a): Signals and systems II (for MT)

Ce cours aborde la théorie des systèmes linéaires discrets invariants par décalage (LID). Leurs propriétés et caractéristiques fondamentales y sont discutées, ainsi que les outils fondamentaux permettant de les étudier (transformée de Fourier et transformée en Z).

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I will try to explain, without going into too much detail, how one can
consider a non-linear wave equation as a dynamical system and what it brings to the study of its
solutions. We begin by considering our model case, the non-linear Klein-Gordon equation and state
its basic properties. We will then see what happens for solutions with energies below that of the
ground state. After that, we place ourselves energetically around the ground state and we show the
apparition of the so-called invariant manifolds. Finally, we consider the critical "pure" (without
the mass term) wave equation and describe some of its interesting solutions. The last part will be
concerned with an attempt to rely what we have learn so far with the critical case.

Joachim Krieger, Daniel Tataru

We consider finite-energy equivariant solutions for the wave map problem from ℝ2+1 to S2 which are close to the soliton family. We prove asymptotic orbital stability for a codimension-two class of initial data which is small with respect to a stronger topology than the energy.

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In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get