1,071 research outputs found
The local cyclicity problem : Melnikov method using Lyapunov constants
In 1991, Chicone and Jacobs showed the equivalence between the computation of the first-order Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, so M(6) ≥ 44. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that Mcp(4) ≥ 43 and Mcp(5) ≥ 65
Reversibility and Non-reversibility in Stochastic Chemical Kinetics
Mathematical problems with mean field and local type interaction related to
stochastic chemical kinetics,are considered. Our main concern various
definitions of reversibility, their corollaries (Boltzmann type equations,
fluctuations, Onsager relations, etc.) and emergence of irreversibility
An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems: the Averaging Method Revisited
This paper introduces an algorithmic approach to the analysis of bifurcation
of limit cycles from the centers of nonlinear continuous differential systems
via the averaging method. We develop three algorithms to implement the
averaging method. The first algorithm allows to transform the considered
differential systems to the normal formal of averaging. Here, we restricted the
unperturbed term of the normal form of averaging to be identically zero. The
second algorithm is used to derive the computational formulae of the averaged
functions at any order. The third algorithm is based on the first two
algorithms that determines the exact expressions of the averaged functions for
the considered differential systems. The proposed approach is implemented in
Maple and its effectiveness is shown by several examples. Moreover, we report
some incorrect results in published papers on the averaging method.Comment: Proc. 44th ISSAC, July 15--18, 2019, Beijing, Chin
Counter-propagating wave patterns in a swarm model with memory
Hyperbolic transport-reaction equations are abundant in the description of movement of motile organisms. Here, we focus on a system of four coupled transport-reaction equations that arises from an age-structuring of a species of turning individuals. By modeling how the behavior depends on the time since the last reversal, we introduce a memory effect. The highlight consists of the explicit construction and characterization of counter-propagating traveling waves, patterns which have been observed in bacterial colonies. Stability analysis reveals conditions for the wave formation as well as pulsating-in-time spatially constant solutions
Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes
We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. {\bf 19}1-24 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multi-well potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov equation), the minimization of the free energy functional and a continuation algorithm for the stationary solutions
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