A simple trick is illustrated, whereby nonlinear evolution equations can be modified so that they feature a lot – or, in some cases, only – periodic solutions. Several examples (ODEs and PDEs) are exhibited.

. In this paper, we develop a quadratic spline collocation method for integrating the nonlinear partial differential equations (PDEs) of a plug flow reactor model. The method is proposed in order to be used for the operation of control design and/or numerical simulations. We first present the Crank-Nicolson method to temporally discretize the state variable. Then, we develop and analyze the pro...

Abstract The synthesis of suboptimal feedback laws for controlling nonlinear dynamics arising from semi-discretized PDEs is studied. An approach based on the State-dependent Riccati Equation (SDRE) presented 2 and ∞ control problems. Depending nonlinearity dimension resulting problem, offline, online, hybrid offline-online alternatives to SDRE are proposed. method reduces sequential solution Ly...

-In this paper a parabolic equation with memory operator is considered. CNN model for such equation is made. Dynamic behavior of the CNN model is studied using describing function method. Traveling wave solutions are proved for the CNN model. An example of one-dimensional wave in medium with memory arising in classical mechanics is presented. Key-Words:Cellular Neural Networks, Partial Differen...

In some cases, solutions to nonlinear PDEs happen to be asymptotically (for large x and/or t) invariant under a group G which is not a symmetry of the equation. After recalling the geometrical meaning of symmetries of differential equations – and solution-preserving maps – we provide a precise definition of asymptotic symmetries of PDEs; we deal in particular, for ease of discussion and physica...

We prove that under certain stability and smoothing properties of the semi-groups generated by the partial differential equations that we consider, manifolds left invariant by these flows persist under C perturbation. In particular, we extend well known finite-dimensional results to the setting of an infinite-dimensional Hilbert manifold with a semi-group that leaves a submanifold invariant. We...

This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking...