Self-synchronization of the integrate-and-fire pacemaker model with continuous couplings

The integrate-and-fire cardiac pacemaker model of the pulse coupled oscillators was introduced by C. Peskin. Due to the function of the pacemaker, two famous synchronization conjectures for identical and not identical oscillators were formulated. There are still many issues related to the nature and types of couplings. The couplings may be impulsive, continuous, delayed or advanced, and oscillators may be locally or globally connected. Consequently, it is reasonable to consider various ways of synchronization, if one wants the biological and mathematical analyses to interact productively. We investigate the integrate-and-fire model in both cases-one with identical, and another with not quite identical oscillators. A combination of continuous and pulse couplings that sustain the firing in unison is carefully constructed. Moreover, we obtain conditions on the parameters of continuous couplings that make possible a rigorous mathematical investigation of the problem. The technique developed for differential equations with discontinuities at non-fixed moments and a special continuous map lies on the basis of the analysis. This is the first analytically derived synchronization result for a model with continuous couplings. Illustrative examples are provided.     

Two-cluster bifurcations in systems of globally pulse-coupled oscillators

For a system of globally pulse-coupled phase-oscillators, we derive conditions for stability of the completely synchronous state and all stationary two-cluster states and explain how the different states are naturally connected via bifurcations. The coupling is modeled using the phase-response-curve (PRC), which measures the sensitivity of each oscillator’s phase to perturbations. For large systems with a PRC, which is zero at the spiking threshold, we are able to find the parameter regions where multiple stable two-cluster states coexist and illustrate this by an example. In addition, we explain how a locally unstable one-cluster state may form an attractor together with its homoclinic connections. This leads to the phenomenon of intermittent, asymptotic synchronization with abating beats away from the perfect synchrony.     

Conservative, unconditionally stable discretization methods for Hamiltonian equations, applied to wave motion in lattice equations modeling protein molecules

A new approach is described for generating exactly energy-momentum conserving time discretizations for a wide class of Hamiltonian systems of DEs with quadratic momenta, including mechanical systems with central forces; it is well-suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the Davydov System modeling energetic pulse propagation in protein molecules. The method is unconditionally stable, making it well-suited to equations of broadly “Discrete NLS form”, including many arising in nonlinear optics.Key features of the resulting discretizations are exact conservation of both the Hamiltonian and quadratic conserved quantities related to continuous linear symmetries, preservation of time reversal symmetry, unconditional stability, and respecting the linearity of certain terms. The last feature allows a simple, efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability, avoiding the need for full Newton-type solvers. One distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations, based on a “discrete gradient calculus with product rule” that mimics the essential properties of partial derivatives.This numerical method is then used to study the Davydov system, revealing that previously conjectured continuum limit approximations by NLS do not hold, but that sech-like pulses related to NLS solitons can nevertheless sometimes arise.     

模糊关系几何规划

提出一类目标函数为正项式,约束是模糊关系方程的优化问题。阐述模糊关系方程解集的结构以及求解的方法,基于目标函数中每个单项式的指数取值情况讨论最优解,并且给出解决此类优化问题的一个程序,为了说明该方法的有效性给出具体例子。     

Virtual Excitations and Entanglement Dynamics and Polygamy in Three Ultra-Strongly Coupled Systems

The Milburn dynamics of three non resonant ultra-strongly coupled oscillators are resolved by using symplectic geometry. The Milburn dynamics of virtual excitations and how they affect the pairwise entanglement are looked at. It is found that the dynamics of excitations and entanglement experience similar profiles against time, physical parameters, and decoherence rate. Furthermore, it is shown that the extinction of excitations entails separability, which demonstrates the hierarchy between entanglement and virtual excitations. Additionally, the effects of physical parameters on the redistribution of virtual excitations among the three bi-partitions are analyzed. As a result, the violation of the monogamy of excitations is shown as in quantum discord. This implies that excitations can be considered as signatures of quantum correlations beyond entanglement. Besides, it is emphasized that the treatment can be used to model coupled quantum circuits in real situations (with decoherence).