Lagrangian structures, integrability and chaos for 3D dynamical equations

Lagrangian structures, integrability and chaos for 3D dynamical equations

In this paper we consider the general setting for constructing Action Principles for three–dimensional first order autonomous equations. We present the results for some integrable and non–integrable cases of the Lotka–Volterra equation, and we show Lagrangian descriptions which are valid for systems satisfying Shil’nikov criteria on the existence of strange attractors, though chaotic behavior o...

Lagrangian Structures for the Stokes, Navier-stokes and Euler Equations

— We prove that the Navier-Stokes, the Euler and the Stokes equations admit a Lagrangian structure using the stochastic embedding of Lagrangian systems. These equations coincide with extremals of an explicit stochastic Lagrangian functional, i.e. they are stochastic Lagrangian systems in the sense of [6].

Meromorphic Non-Integrability of Several 3D Dynamical Systems

In this paper, we apply the differential Galoisian approach to investigate the meromorphic non-integrability of a class of 3D equations in mathematical physics, including Nosé–Hoover equations, the Lü system, the Rikitake-like system and Rucklidge equations, which are well known in the fields of molecular dynamics, chaotic theory and fluid mechanics, respectively. Our main results show that all...

Dynamical complexity and symplectic integrability

We introduce two numerical conjugacy invariants for dynamical systems – the complexity and weak complexity indices – which are well-suited for the study of “completely integrable” Hamiltonian systems. These invariants can be seen as “slow entropies”, they describe the polynomial growth rate of the number of balls (for the usual “dynamical” distances) of coverings of the ambient space. We then d...

Integrability and Chaos – algebraic and geometric approach