A Note on Homoclinic Orbits for Second Order Hamiltonian Systems

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 existence of homoclinc orbits

Resonances of the SD oscillator due to the discontinuous phase

Resonance phenomena of a harmonically excited system with mul-tiple potential well play an important role in nonlinear dynamics research.In this paper, we investigate the resonant behaviours of a discontinuous dynamical system with double well potential derived from the SD oscillator to gain better understanding of the transition of resonance mechanism. Firstly,the time dependent Hamiltonian is obtained for a Duffing type discontinuous system modelling snap-through buckling. This system comprises two subsystems connected at x = 0, for which the system is discontinuous. We constructa series of generating functions and canonical transformations to obtain the canonical form of the system to investigate the complex resonant behavioursof the system. Furthermore, we introduce a composed winding number to explore complex resonant phenomena. The formulation for resonant phenomena given in this paper generalizes the formulation of n Omega0 = m Omega used in the regular perturbation theory, where n and m are relative prime integers, Omega 0 and Omega are the natural frequency and external frequencies respectively. Understanding the resonant behaviour of the SD oscillator at the discontinuousphase enables us to further reveal the vibrational energy transfer mechanism between smooth and discontinuous nonlinear dynamical system

Accurate reduction of a model of circadian rhythms by delayed quasi steady state assumptions

Quasi steady state assumptions are often used to simplify complex systems of ordinary differential equations in modelling of biochemical processes. The simplified system is designed to have the same qualitative properties as the original system and to have a small number of variables. This enables to use the stability and bifurcation analysis to reveal a deeper structure in the dynamics of the original system. This contribution shows that introducing delays to quasi steady state assumptions yields a simplified system that accurately agrees with the original system not only qualitatively but also quantitatively. We derive the proper size of the delays for a particular model of circadian rhythms and present numerical results showing the accuracy of this approach.Comment: Presented at Equadiff 2013 conference in Prague. Accepted for publication in Mathematica Bohemic