平面亏一质量作用系统中的振荡

An algorithm for finding weakly reversible deficiency zero realizations of polynomial dynamical systems

Systems of differential equations with polynomial right-hand sides are very common in applications. On the other hand, their mathematical analysis is very challenging in general, due to the possibility of complex dynamics: multiple basins of attraction, oscillations, and even chaotic dynamics. Even if we restrict our attention to mass-action systems, all of these complex dynamical behaviours are still possible. On the other hand, if a polynomial dynamical system has a weakly reversible deficiency zero ($WR_0$) realization, then its dynamics is known to be remarkably simple: oscillations and ch…

Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems

A reaction network together with a choice of rate constants uniquely gives rise to a system of differential equations, according to the law of mass-action kinetics. On the other hand, different networks can generate the same dynamical system under mass-action kinetics. Therefore, the problem of identifying "the" underlying network of a dynamical system is not well-posed, in general. Here we show that the problem of identifying an underlying weakly reversible deficiency zero network is well-posed, in the sense that the solution is unique whenever it exists. This can be very useful in applicatio…

Delay-induced blow-up in a planar oscillation model

In this paper we study a system of delay differential equations from the viewpoint of a finite time blow-up of the solution. We prove that the system admits a blow-up solution, no matter how small the length of the delay is. In the non-delay system every solution approaches to a stable unit circle in the plane, thus time delay induces blow-up of solutions, which we call "delay-induced blow-up" phenomenon. Furthermore, it is shown that the system has a family of infinitely many periodic solutions, while the non-delay system has only one stable limit cycle. The system studied in this paper is an…

Canard explosion and relaxation oscillation in planar, piecewise-smooth, continuous systems

Classical canard explosion results in smooth systems require the vector field to be at least $C^3$, since canard cycles are created as the result of a Hopf bifurcation. The work on canards in nonsmooth, planar systems is recent and has thus far been restricted to piecewise-linear or piecewise-smooth Van der Pol systems, where an extremum of the critical manifold arises from the nonsmoothness. In both of these cases, a canard (or canard-like) explosion is created through a nonsmooth bifurcation as the slow nullcline passes through a corner of the critical manifold. Additionally, it is possible …

Double canard cycles in singularly perturbed planar systems with two canard points

We consider double canard cycles including two canards in singularly perturbed planar systems with two canard points. Previous work studied the complex oscillations including relaxation oscillations and canard cycles in singularly perturbed planar systems with one-parameter layer equations, which have precisely one canard point, two jump points or one canard point and one jump point. Based on the normal form theory, blow-up technique and Melnikov theory, we investigate double canard cycles induced by two Hopf breaking mechanisms at two non-degenerate canard points. Finally, we apply the obtain…

用平均法分析耦合平面振子的同步

Synchronization analysis of coupled planar oscillators by averaging

利用平均技术研究了耦合Lienard型振子同步的充分条件。这里考虑的耦合是成对的、单向的，并且由相对距离（被耦合的状态对之间的）矢量的某个投影的非线性函数（其图形驻留在第一和第三象限中）来描述。在互连拓扑定义了一个连通图的假设下，证明了在振荡频率足够大且振荡器的初相位都位于一个开周期内的条件下，可以使振荡器的解彼此任意接近地收敛，而初始解彼此任意远离地收敛.它还表明，（几乎）同步振荡总是发生在一些固定的幅度独立的初始条件。对于非线性耦合谐振子也得到了类似的结果。