相关论文: Analytic and algebraic conditions for bifurcations…

The existence and bifurcation of homoclinic orbits in planar piecewise linear homogeneous systems with two regions separated by a discontinuity boundary are investigated in this paper. In addition, existence of periodic orbits and stability…

In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the Invariant Manifolds Theorem, construct finite Markov partitions and use them to prove the existence…

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in $\mathbb{R}^4$ unfolds generically a bifurcation…

In this paper, we study the existence for the homoclinic orbits for the second order Hamiltonian systems. Under suitable conditions on the potential $V$, we apply the direct method of variations and the Fourier analysis to prove the…

The nonlinear Schr\"odinger/Gross-Pitaevskii (NLS/GP) equation is considered in the presence of three equally-spaced potentials. The problem is reduced to a finite-dimensional Hamiltonian system by a Galerkin truncation. Families of…

We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter $\varepsilon>0$ and give several necessary conditions for their persistence. Here we…

This paper introduces techniques of symplectic topology to the study of homoclinic orbits in Hamiltonian systems. The main result is a strong generalization of homoclinic existence results due to Sere and to Coti-Zelati, Ekeland and Sere,…

We study the linear instabilities and bifurcations in the Selkov model for glycolysis with diffusion. We show that this model has a zero wave-vector, finite frequency Hopf bifurcation to a growing oscillatory but spatially homogeneous state…

In this paper we present a computer-assisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vector fields. The idea is to compute high-order Taylor approximations of…

We study the dynamics of a pair of parametrically-driven coupled nonlinear mechanical resonators of the kind that is typically encountered in applications involving microelectromechanical and nanoelectromechanical systems (MEMS & NEMS). We…

Steady states of the Swift--Hohenberg equation are studied. For the associated four--dimensional ODE we prove that on the energy level $E=0$ two smooth branches of even periodic solutions are created through the saddle-node bifurcation. We…

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle…

We consider a $\mathbb{Z}_{2}$-equivariant 4-dimensional system of ODEs with a smooth first integral $H$ and a saddle equilibrium state $O$. We assume that there exists a transverse homoclinic orbit $\Gamma$ to $O$ that approaches $O$ along…

We discuss the bifurcation structure of homoclinic orbits in bimodal one dimensional maps. The universal structure of these bifurcations with singular bifurcation points and the web of bifurcation lines through the parameter space are…

We examine the Melnikov criterion for transition to chaos in case of a single degree of freedom nonlinear oscillator with the Ueda well potential and an external periodic excitation term. Using effective Hamiltonian we have examined…

This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are…

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales-Ramis…

Bifurcations of periodic orbits as an external parameter is varied are a characteristic feature of generic Hamiltonian systems. Meyer's classification of normal forms provides a powerful tool to understand the structure of phase space…

We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point, and implement the method for the saddle-focus libration points of the planar equilateral restricted four body problem. We…

We consider the scalar delay differential equation $$ \dot{x}(t)=-x(t)+f_{K}(x(t-1)) $$ with a nondecreasing feedback function $f_{K}$ depending on a parameter $K$, and we verify that a saddle-node bifurcation of periodic orbits takes place…