Symmetric Liapunov center theorem

A Floquet-Liapunov theorem in Fréchet spaces

Based on [4], we prove a variation of the theorem in title, for equations with periodic coefficients, in Fréchet spaces. The main result gives equivalent conditions ensuring the reduction of such an equation to one with constant coefficient. In the particular case of C ∞ , we obtain the exact analogue of the classical theorem. Our approach essentially uses the fact that a Fréchet space is the l...

The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems

and Applied Analysis 3 Here, we assume that the equivariant symmetry S acts antisymplectically and S2 I. Now, we also consider the symmetric property of periodic solutions. This property was not studied for Hamiltonian vector fields without the other structure previously. 2. Main Results Theorem 2.1. Consider an equilibrium 0 of a C∞ equivariant Hamiltonian vector field f , with the equivariant...

Symmetric Versions of Laman's Theorem

Recent work has shown that if an isostatic bar and joint framework possesses non-trivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are ‘fixed’ by various symmetry operations of the framework. For the group C3 which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed in [4] that these restrictions ...

A Converse Liapunov Theorem for Uniformly Locally Exponentially Stable Systems Admitting Carathéodory Solutions ?

This paper provides a converse Liapunov theorem for uniformly locally exponentially stable, locally Lipschitz, non-linear, time-varying, possibly non-smooth systems that admit Carathéodory solutions. The main result proves that a critical point of such a system is uniformly locally exponentially stable if and only if the system admits a local (possibly non-smooth, timevarying) Liapunov function.

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials