Periodic and subharmonic solutions of second-order Hamiltonian systems

Some solvability conditions of periodic solutions and subharmonic solutions are obtained for a class of new superquadratic non-autonomous second-order Hamiltonian systems by the minimax methods in critical point theory.     

Brake subharmonic solutions of first order Hamiltonian systems

In this paper, we mainly use the Galerkin approximation method and the iteration inequalities of the L-Maslov type index theory to study the properties of brake subharmonic solutions for the first order non-autonomous Hamiltonian systems. We prove that when the positive integers j and k satisfy the certain conditions, there exists a jT-periodic nonconstant brake solution z _j such that z _j and z _kj are distinct.     

Multiplicity of solutions for second-order Hamiltonian systems with impulses

In this paper, we study the existence of infinitely many solutions for second-order Hamiltonian systems with impulses. By using an infinitely many critical points theorem and Fountain theorem, we obtain some new criteria for guaranteeing that the impulsive Hamiltonian systems have infinitely many solutions. No symmetric condition on the nonlinear term is assumed. Some examples are also given in this paper to illustrate our main results.     

Periodic and subharmonic solutions of a class of superquadratic Hamiltonian systems

Periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic Hamiltonian systems by using the minimax methods in critical point theory.     

Periodic and subharmonic solutions for a class of non-autonomous Hamiltonian systems

Some existence theorems are obtained for periodic and subharmonic solutions of a class of non-autonomous Hamiltonian systems. Our technical approach is based on a version of the Local Linking Theorem, and the Generalized Mountain Pass Theorem.     

Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems

Some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second-order Hamiltonian systems by the minimax methods in critical point theory.     

Multiplicity of periodic solutions for second-order discrete Hamiltonian systems with a small forcing term

Some sufficient conditions for the existence of three periodic solutions are obtained for the second-order discrete Hamiltonian systems with a small forcing term by using a three-critical-point theorem.     

Bifurcation of subharmonic solutions in time-reversible systems

We study bifurcation of 2q-periodic solutions in one-parameter families of 2-periodic time-reversible systems. We obtain generically satisfied conditions which imply the bifurcation of 2q branches of such subharmonic solutions. Whenq5 the solutions alongq of these branches are unstable, while the solutions along the otherq branches are stable in a weak sense. Special results hold forq=3 andq=4. We also describe a situation in which there is secundary bifurcation and give a brief discussion of what happens under a perturbation which breaks the time-reversibility.

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Zusammenfassung Wir untersuchen die Verzweigung von 2q-periodischen Lösungen in einer einparametrigen Familie von 2-periodischen reversiblen Differentialgleichungssystemen. Wir erhalten Bedingungen, welche die Verzweigung von 2q solcher subharmonischer Lösungen garantieren und die generisch erfüllt sind. Fürq5 sindq Lösungen instabil undq Lösungen (linear) stabil. Spezielle Resultate gelten fürq=3 undq=4. Wir untersuchen auch einen Fall in dem sekundäre Verzweigung eintritt und diskutieren kurz die Wirkung einer Störung, die die Reversibilität zerstört.

     

Subharmonics near an equilibrium point for hamitonian systems

This work is devoted to the study of subharmonic solutions near an equilibrium for certain Hamiltonian systems. We impose a weak condition on the Floquet exponents of the linearized system, and a superquadratic condition on the higher order term. This last condition is reduced to the center manifold. This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-87-0202.     

On subharmonic solutions of systems of difference equations with periodic perturbations Part II: Multiplicity and stability

We consider the multiplicity and stability of subharmonic solutions of discrete dynamic systems with periodic perturbations, whose existence was established in the first part of this series. Under the hypotheses that the perturbation is locally Lipschitz continuous, we determine the precise number of subharmonic solutions, and we also show that these subharmonic solutions are stable (unstable) provide the periodic orbit of the unperturbed system is stable (unstable).     

Periodic and subharmonic solutions of discrete p-Laplacian systems

Some solvability conditions of periodic and subharmonic solutions are obtained for a class of discrete p-Laplacian systems by the minimax methods in critical point theory.     

Bifurcation and forced symmetry breaking in Hamiltonian systems

We consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian system on a symplectic manifold. In particular we study the persistence of an initial relative equilibrium subjected to this forced symmetry breaking. We see that, under certain nondegeneracy conditions, an estimate can be made on the number of bifurcating relative equilibria. To cite this article: F. Grabsi et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Multiplicity of solutions for resonant elliptic systems

We establish the existence and multiplicity of solutions for some resonant elliptic systems. The results are proved by applying minimax arguments and Morse theory.     

Hyperbolic-like solutions for singular Hamiltonian systems

We study the existence of unbounded solutions of singular Hamiltonian systems: where is a potential with a singularity. For a class of singular potentials with a strong force , we show the existence of at least one hyperbolic-like solutions. More precisely, for given and , we find a solution q(t) of (*) satisfying Received October 1998     

Infinitely many solutions for Hamiltonian systems

We consider two classes of the second-order Hamiltonian systems with symmetry. If the systems are asymptotically linear with resonance, we obtain infinitely many small-energy solutions by minimax technique. If the systems possess sign-changing potential, we also establish an existence theorem of infinitely many solutions by Morse theory.