Normal form theory for volume preserving maps

We discuss the theory of normal forms for volume preserving maps in the non-resonant case. The concept of normal forms is introduced by using the commutative properties with respect to some symmetry groups related to the resonances of the linear eigenvalues. The existence of formal solutions to the conjugation equation, which reduces a map to normal form, is discussed. We also analyze the asymptotic character of the normal forms series and we prove a general theorem which shows that the error between the normal form dynamics and the true dynamics can be exponentially small as a function of the radius of the chosen polydisk centered at the fixed point. As a consequence it is possible to construct analytic manifolds which are approximately invariant under the action of the initial map up to an error which is exponentially small. The connection with a possible KAM theory for volume-preserving maps is suggested. We also show that Noether's theorem can be generalized to volume preserving maps.     

Multiple equilibria,periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Consider the Dirichlet problem for the parabolic equation in , where $\Omega$ is a bounded domain in and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.     

Persistence of equilibria as periodic solutions of forced systems

Given an autonomous system with an isolated equilibrium, we consider general periodic perturbations. We say that the equilibrium persists if it can be continued as a periodic solution. The question of persistence is very classical and we find that the search of sharp conditions is linked with Topology. Besides the topological degree, the notion of diffeotopy and Hopf?s Theorem of homotopy classes play a role. For dimension two we find a complete characterization of persistence.     

Normal form theory for reversible equivariant vector fields

We give a method to obtain formal normal forms of reversible equivariant vector fields. The procedurewe present is based on the classical method of normal forms combined with tools from invariant theory. Normal forms of two classes of resonant cases are presented, both with linearization having a 2-dimensional nilpotent part and a semisimple part with purely imaginary eigenvalues.     

Morse-type index theory for flows and periodic solutions for Hamiltonian Equations

An index theory for flows is presented which extends the classical Morse theory for gradient flows on compact manifolds. The theory is used to prove a Morse-type existence statement for periodic solutions of a time-dependent (periodic in time) and asymptotically linear Hamiltonian equation.     

Nontrivial periodic solutions for delay differential systems via Morse theory

The existence of the nontrivial periodic solutions to the system of delay differential equations

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A new existence theory for periodic solutions to evolution equations

ABSTRACT

This paper deals with a new existence theory for periodic solutions to a broad class of evolution equations. We first establish new fixed point theorems for affine maps in locally convex spaces and ordered Banach spaces. Our new fixed point results extend, encompass and complement a number of well-known theorems in the literature, including the famous Chow and Hale fixed point theorem. With these obtained fixed point results, we investigate the existence of periodic solutions for some class of nonhomogeneous linear systems in Banach spaces with lack of compactness. Some illustrative examples are also given.     

Normal form for linear systems with respect to its vector relative degree

For multi-input multi-output (MIMO) linear systems with existing vector relative degree a normal form is constructed. This normal form is not only structural simple but allows to characterize the system’s zero dynamics for the design of feedback controllers. A characterization of the zero dynamics in terms of the normal form is given.     

Morse theory for periodic solutions of hamiltonian systems and the maslov index

In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over π₂ (M). The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a nondegenerate critical point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every 1-periodic solution has at least one Floquet multiplier which is not equal to 1.     

New exact kink solutions, solitons and periodic form solutions for a combined KdV and Schwarzian KdV equation

In this short letter, new exact solutions including kink solutions, soliton-like solutions and periodic form solutions for a combined version of the potential KdV equation and the Schwarzian KdV equation are obtained using the generalized Riccati equation mapping method.     

New explicit periodic wave solutions and their limits for modified form of Camassa-holm equation

In this paper, we consider the following nonlinear equation ut＋2kux-uxxt＋au^2ux=2uxuxx＋uuxxx,which is a modified form of the Camassa-Holm equation. We construct four new explicit periodic wave solutions by bifurcation method of dynamical systems. We also obtain two explicit solitary wave solutions via the limits of the explicit periodic wave solutions. One of the two solitary wave solutions is new.     

Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations

Using the final value theorem of Laplace transform, it is firstly shown that nonhomogeneous fractional Cauchy problem does not have nonzero periodic solution. Secondly, two basic existence and uniqueness results for asymptotically periodic solution of semilinear fractional Cauchy problem in an asymptotically periodic functions space. Furthermore, existence and uniqueness results are extended to a closed, nonempty and convex set which is a subset of a Fréchet space. Some examples are given to illustrate the results.     

Spatially periodic fundamental solutions of the theory of oscillations

Spatially periodic fundamental solutions of the theory of oscillations are constructed, applicable to anisotropic elastic media with a general form of anisotropy. The results are compared with the isotropic case.     

Stability theory for periodic pulse train solutions of the nonlinear Schrodinger equation

The problem of the stability of periodic and quasiperiodic trainsof soliton pulses in the nonlinear Schrödinger equationis examined using linearized perturbation theory. When the quasiperiodicsoliton pulse train is subjected to perturbations of positionor phase, there are both stable and unstable regions of theparameter space. The stability exponents of these perturbationsare determined in the asymptotic case of large separation betweenthe solitons.     

Useful equilibria for game theory

We suggest two new notions of equilibria for arbitrary game problems, both static and dynamic ones, whose definitions contain no artificial norms for the behavior of the players. In examples of static and differential games, we demonstrate the possibilities of these equilibria and a technique for finding them.     

Instability of periodic BGK equilibria

A collisionless plasma is described by the Vlasov-Poisson equations. The BGK equilibria were proposed in 1957 as the simplest spatially-dependent equilibria. Since that time the question of their instability has been an important open problem. We prove that certain periodic BGK equilibria are unstable. ©1995 John Wiley & Sons, Inc.     

Multiplicity results for periodic solutions to delay differential equations via critical point theory

By the critical point theory, we study the existence and multiplicity of periodic solutions to the following system of delay differential equations:

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