In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert’s Sixteenth Problem in this special case of polynomial systems is suggested. Using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper, we present a proof of our earlier conjecture that...

We estimate the expected number of limit cycles situated in a neighbourhood of the origin of a planar polynomial vector field. Our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic functions depending analytically on a parameter. We obtain this inequality by methods of Pluripotential Theory. This inequality also implies versions of a str...

Limit cycles are common in hybrid systems. However the non-smooth dynamics of such systems makes stability analysis difficult. This paper uses recent extensions of trajectory sensitivity analysis to obtain the characteristic multipliers of non-smooth limit cycles. The stability of a limit cycle is determined by its characteristic multipliers. The concepts are illustrated using a compass gait bi...

We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic polynomial differential systems. The family considered is the unique family of weight–homogeneous polynomial differential systems of weight–degree 2 with a center.

Many ecological processes exhibit trajectories which can be suitably represented by stable equilibria or smooth limit cycles. However, a third kind of ecological process involves intermittent, abrupt, and drastic changes in densities, here termed 'outbreak dynamics', which require different modelling frameworks. One such framework, the cusp catastrophe, is used here in a modelling study of a pa...

the aij and bij being real or complex. This vector field is a deformation of the vector field x∂y −y∂x whose trajectories are concentric circles around 0. We prove in this paper a precise version of the following assertion: for any compact K in the space of the (aij ,bij ), there exist a number p(q) and a neighborhood U(q,K) of 0 such that for (a,b) ∈ K , either 0 is again a center of Wa,b (i.e...

In this work, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the limit cycle problem for a general Liénard polynomial systemwith an arbitrary (but finite) number of singular points. © 2012 Elsevier Ltd. All rights reserved.

This paper deals with the question of uniqueness of limit cycles in predator-prey systems of Gause type. By utilizing several transformations, these systems are reduced to a generalized Lienard system as discussed by Cherkas and Zhilevich and by Zhang. As a consequence, criteria for the uniqueness of limit cycles are derived, which include results of Cheng and is related to results in Liou and ...

On the basis of the ideas of Goldman and Beja(1980) and Chiarella(1992), we construct a stochastic model of speculative dynamics that has a stochastic equilibrium. In this model, different expectations for the price of fundamentalists and chartists are introduced into the model. The system is analyzed using concepts and techniques from the theory of Random dynamical systems. The model exhibits ...

A multiscale method for computing the effective slow behavior of a system of weakly coupled general oscillators is presented. The oscillators may be either in the form of a periodic solution or a stable limit cycle. Furthermore, the oscillators may be in resonance with one another and thereby generate some hidden slow dynamics. The proposed method relies on correctly tracking a set of slow vari...