Invariant critical sets of conserved quantities

Abstract

For a dynamical system we will construct various invariant sets starting from its conserved quantities. We will give conditions under which certain solutions of a nonlinear system are also solutions for a simpler dynamical system, for example when they are solutions for a linear dynamical system. We will apply these results to the example of Toda lattice

The stability problem and special solutions for the 5-components Maxwell–Bloch equations

“…We will look for invariant sets of the system (1.1) using the technique presented in [3]. We have the following vectorial conserved quantity F : R 5 → R 3 , F(p) = (H(p), I(p), C(p)).…”

“…We have the following vectorial conserved quantity F : R 5 → R 3 , F(p) = (H(p), I(p), C(p)). In [3], Theorem 2.3, it has been proved that the set M F (2) = {p ∈ R 5 | rank ∇F(p) = 2} is invariant under the dynamics of the system. By direct computation we obtain that M F (2) = M 1 ∪ M 2 , where…”

Dynamics under Geometric Dissipation

“…We loose exactly that equilibrium points x e for which the vectors ∇F 1 (x e ),...,∇F k (x e ) and ∇G(x e ) are linear independent. The set Inv is invariant under the unperturbed dynamics (1.1) (see Corollary 2.4 in [4]). Next we will prove that the set Inv is also an invariant set for the geometrically dissipated dynamics (1.5).…”

“…The subsets G * and Y are also invariant subsets for the perturbed dynamics. Indeed, G * is invariant under the unperturbed dynamic (see [4], [8]) and if x ∈ G * is an initial condition then x p (t, x) = x un (t, x) ∈ G * for all t ∈ R, which implies that G * is invariant under the perturbed dynamics. Consequently, Y is also an invariant set for the perturbed dynamics.…”

Stability of equilibrium states in the Zhukovski case of heavy gyrostat using algebraic methods

“…This method assumes the existence of a certain invariant set of the dynamics generated by (3). The invariant set can be found in the method presented in [10]. For the following considerations, we suppose that n D 2m, and there exists the C 1 -function s :…”