Multibump solutions of a class of second-order discrete Hamiltonian systems

For a class of second-order discrete Hamiltonian systems $\Delta^2x(t-1)-L(t)x(t)+V'_x(t,x(t))=0$, we investigate the existence of homoclinic orbits by applying variational method, where $L$ and $V(\cdot,x)$ are periodic functions. Further, we show that there exist either uncountable many homoclinic orbits or multibump solutions under certain conditions

Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential

We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -ε2u′′+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the Poincaré map. We discuss the periodic and the Neumann boundary conditions. The value of the term ε>0, although small, can be explicitly estimated

Periodic solutions and chaotic dynamics in a Duffing equation model of charged particles

The emergence of chaotic behavior in many physical systems has triggered the curiosity of scientists for a long time. Their study has been concentrated in understanding which are the underlying laws that govern such dynamics and eventually aim to suppress such (often) undesired behavior. In layman terms, a system is defined chaotic when two orbits that initially are very near to each other will diverge in exponential time. Clearly, this translates to the fact that a chaotic system can hardly have regular behavior, a property that is also often required even for human-made systems. An example is that of particle accelerators used a lot in the study of experimental physics. The main principle is that of forcing a large number of particles to move periodically in a toroidal space in order to collide with each other. Another example is that of the tokamak, a particular accelerator built to generate plasma, one of the states of the matter. In both cases, it is crucial for the sake of the accelerating process, to have regular periodic behavior of the particles instead of a chaotic one. In this dissertation, we have studied the question of chaos in mathematical models for the motion of magnetically charged particles inside the tokamak in the presence or absence of plasma. We start by a model introduced by Cambon et al., which describes in general mathematical terms, also known as the Duffing modes, the formalism of the above problem. The central core of this work reviews the necessary mathematical tools to tackle this problem, such as the theorem of the Linked Twisted maps and the variational Hamiltonian equations which describe the evolutionary dynamics of the system under consideration. Extensive analytical and numerical tools are required in this thesis work to investigate the presence of chaos, known as chaos indicator. The main ones we have used here are the Poincar \u301e Map, the Maximum Lyapunov Exponent (MLE), and the SALI and GALI methods. Using the techniques mentioned above, we have studied our problem analytically and validated our results numerically for the particular case of the Duffing equation, which applies to the motion of charged particles in the tokamak. In detail, we first discuss the presence of chaotic dynamics of charged particles inside an idealized magnetic field, sug- gested by a tokamak type configuration. Our model is based on a periodically perturbed Hamiltonian system in a half-plane r \ubf 0. We propose a simple mechanism producing complex dynamics, based on the theory of Linked Twist Maps jointly with the method of stretching along the paths. A key step in our argument relies on the monotonicity of the period map associated with the unperturbed planar system. In the second part of our results, we give an analytical proof of the presence of complex dynamics for a model of charged particles in a magnetic field. Our method is based on the theory of topological horseshoes and applied to a periodically perturbed Duffing equation. The existence of chaos is proved for sufficiently large, but explicitly computable, periods. In the latter part, we study the generalized forementioned Duffing equations and study the chaoticity using the Melnikov topological method and verify the results numerically for the models of Wang & You and the tokamak one

A Perturbative Analysis of Modulated Amplitude Waves in Bose-Einstein Condensates

We apply Lindstedt's method and multiple scale perturbation theory to analyze spatio-temporal structures in nonlinear Schr\"odinger equations and thereby study the dynamics of quasi-one-dimensional Bose-Einstein condensates with mean-field interactions. We determine the dependence of the intensity of modulated amplitude waves on their wave number. We also explore the band structure of Bose-Einstein condensates in detail using Hamiltonian perturbation theory and supporting numerical simulations.Comment: 24 pages, 20 figs (numbered to 9), 6 tables, to appear in Chao

Nonlinear oscillations, bifurcations and chaos in ocean mooring systems

Complex nonlinear and chaotic responses have been recently observed in various compliant ocean systems. These systems are characterized by a nonlinear mooring restoring force and a coupled fluid-structure interaction exciting force. A general class of ocean mooring system models is formulated by incorporating a variable mooring configuration and the exact form of the hydrodynamic excitation. The multi-degree of freedom system, subjected to combined parametric and external excitation, is shown to be complex, coupled and strongly nonlinear. Stability analysis by a Liapunov function approach reveals global system attraction which ensures that solutions remain bounded for small excitation. Construction of the system's Poincare map and stability analysis of the map's fixed points correspond to system stability of near resonance periodic orbits. Investigation of nonresonant solutions is done by a local variational approach. Tangent and period doubling bifurcations are identified by both local stability analysis techniques and are further investigated to reveal global bifurcations. Application of Melnikov's method to the perturbed averaged system provides an approximate criterion for the existence of transverse homoclinic orbits resulting in chaotic system dynamics. Further stability analysis of the subharmonic and ultraharmonic solutions reveals a cascade of period doubling which is shown to evolve to a strange attractor. Investigation of the bifurcation criteria obtained reveals a steady state superstructure in the bifurcation set. This superstructure identifies a similar bifurcation pattern of coexisting solutions in the sub, ultra and ultrasubharmonic domains. Within this structure strange attractors appear when a period doubling sequence is infinite and when abrupt changes in the size of an attractor occur near tangent bifurcations. Parametric analysis of system instabilities reveals the influence of the convective inertial force which can not be neglected for large response and the bias induced by the quadratic viscous drag is found to be a controlling mechanism even for moderate sea states. Thus, stability analyses of a nonlinear ocean mooring system by semi-analytical methods reveal the existence of bifurcations identifying complex periodic and aperiodic nonlinear phenomena. The results obtained apply to a variety of nonlinear ocean mooring and towing system configurations. Extensions and applications of this research are discussed

Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems

We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems u¨+atWuu=0, (HS) where -∞<t<+∞, u=u1,u2, …,uN∈ℝNN≥3, a:ℝ→ℝ is a continuous bounded function, and the potential W:ℝN∖{ξ}→ℝ has a singularity at 0≠ξ∈ℝN, and Wuu is the gradient of W at u. The novelty of this paper is that, for the case that N≥3 and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of W. Different from the cases that (HS) is autonomous at≡1 or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and N≥3. Besides the usual conditions on W, we need the assumption that a′t<0 for all t∈ℝ to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved