Planar Differential Systems Research Articles

Two kinds of singularities in planar differential systems and the response to external forces.

A Lyapunov function is defined for two-dimensional differential systems. The relationship between the Lyapunov function and the stationary probability distribution of the corresponding Fokker-Planck equations in the weak-noise limit is analyzed. Based on the structure of the Lyapunov function we classify two types of singularities. The classification of different kinds of singularities is shown to be significant for the intrinsical dynamics of the deterministic flow. The bifurcation behavior of periodically forced planar systems that contain distinctive kinds of singularities in the autonomous cases is investigated. It is found that the responses of planar systems to the external force, i.e., the resulting bifurcation sequences, are substantially different if the trajectories circle around different types of singularities.

Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation

We give here a planar quadratic differential system depending on two parameters, λ, δ. There is a curve in the λ-δ space corresponding to a homoclinic loop bifurcation (HLB). The bifurcation is degenerate at one point of the curve and we get a narrow tongue in which we have two limit cycles. This is the first example of such a bifurcation in planar quadratic differential systems. We propose also a model for the bifurcation diagram of a system with two limit cycles appearing at a singular point from a degenerate Hopf bifurcation, and dying in a degenerate HLB. This model shows a deep duality between degenerate Hopf bifurcations and degenerate HLBs. We give a bound for the maximal number of cycles that can appear in certain simultaneous Hopf and homoclinic loop bifurcations. We also give an example of quadratic system depending on three parameters which has at one place a degenerate Hopf bifurcation of order 3, and at another place a Hopf bifurcation of order 2 together with a HLB. We characterize the planar quadratic systems which are integrable in the neighbourhood of a homoclinic loop.

Singular points of planar ordinary differential systems

Singular points of planar ordinary differential systems

Equivalence of planar dynamical and differential systems

where x is a 2-vector, f(x) is homogeneous of degree N 3 1, g(x) is o (11 x 11”) for sufficiently small I] x I), and the usual existence and uniqueness properties hold for the solutions of each system close enough to the origin. The origin is assumed to be an isolated critical point of each system.A familiar problem is that of determining the local behavior of trajectories of system( 1) neartheorigin [2]. In this paper it is shown that under certain assumptions on f this behavior is qualitatively the same as that of the trajectories of (2), the system of first approximation. In particular, it is shown (Theorem 5) that there is a homeomorphism of a neighborhood of the origin onto another neighborhood of the origin which maps arcs of trajectories of system (1) onto arcs of trajectories of system (2). Two such systems are said to be locally trajectory equivalent or locally integral equivalent at the origin. The analysis rests upon two theorems, one by Forster [I], the other by Nemytski [4]. Forster’s theorem, stated in full in Section 3, resembles in spirit our main result in that it also relates the behavior of the trajectories of (1) to that of the trajectories of (2). H owever, trajectory equivalence is not proven-only that for each trajectory of (2) with a specified type of behavior near the origin there is a trajectory of (1) which behaves similarly near the origin.