The Focus Case of a Nonsmooth Rayleigh-Duffing Oscillator

In this paper, we study the global dynamics of a nonsmooth Rayleigh-Duffing equation x¨ + ax˙ + bx˙|x˙| + cx + dx3 = 0 for the case d > 0, i.e., the focus case. The global dynamics of this nonsmooth Rayleigh-Duffing oscillator for the case d < 0, i.e., the saddle case, has been studied completely in the companion volume [Int. J. Non-Linear Mech., 129 (2021) 103657]. The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent occurs. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

Microbial insecticide model and homoclinic bifurcation of impulsive control system

A new microbial insecticide mathematical model with density dependent for pest is proposed in this paper. First, the system without impulsive state feedback control is considered. The existence and stability of equilibria are investigated and the properties of equilibria under different conditions are verified by using numerical simulation. Since the system without pulse has two positive equilibria under some additional assumptions, the system is not globally asymptotically stable. Based on the stability analysis of equilibria, limit cycle, outer boundary line and Sotomayor’s theorem, the existence of saddle-node bifurcation and global dynamics of the system are obtained. Second, we consider homoclinic bifurcation of the system with impulsive state feedback control. The existence of order-1 homoclinic orbit of the system is studied. When the impulsive function is slightly disturbed, the homoclinic orbit breaks and bifurcates order-1 periodic solution. The existence and stability of order-1 periodic solution are obtained by means of theory of semi-continuous dynamic system.

Dynamics of a discrete-time pioneer–climax model

We present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation, and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.

Transformations of Closed Invariant Curves and Closed-Invariant-Curve-Like Chaotic Attractors in Piecewise Smooth Systems

The paper describes some aspects of sudden transformations of closed invariant curves in a 2D piecewise smooth map. In particular, using detailed numerically calculated phase portraits, we discuss transitions from smooth to piecewise smooth closed invariant curves. We show that such transitions may occur not only when a closed invariant curve collides with a border but also via a homoclinic bifurcation. Furthermore, we describe an unusual transformation from a closed invariant curve to a large amplitude chaotic attractor and demonstrate that this transition occurs in two steps, involving a small amplitude closed-invariant-curve-like chaotic attractor.

Dynamics of a Discrete-time Pioneer–climax Model

We present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population density growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and exclusion of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.

On the Organization of Homoclinic Bifurcation Curves in Systems with Shilnikov Spiral Attractors

We study spiral chaos in the classical Rössler and Arneodo – Coullet – Tresser systems. Special attention is paid to the analysis of bifurcation curves that correspond to the appearance of Shilnikov homoclinic loop of saddle-focus equilibrium states and, as a result, spiral chaos. To visualize the results, we use numerical methods for constructing charts of the maximal Lyapunov exponent and bifurcation diagrams obtained using the MatCont package.