Narrow Results

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold–Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold–heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems.

Bazykin proposed a Lotka–Volterra-type ecological model that accounts for simplified territoriality, which neither depends on territory size nor on food availability. In this study, we describe the global dynamics of the Bazykin model using analytical and numerical methods. We specifically focus on the effects of mutual predator interference and the prey carrying capacity since the variability of each could have especially dramatic ecological repercussions. The model displays a broad array of complex dynamics in space and time; for instance, we find the coexistence of a limit cycle and a steady state, and bistability of steady states. We also characterize super- and subcritical Poincaré–Andronov–Hopf bifurcations and a Bogdanov–Takens bifurcation. To illustrate the system's ability to naturally shift from stable to unstable dynamics, we construct bursting solutions, which depend on the slow dynamics of the carrying capacity. We also consider the stabilizing effect of the intraspecies interaction parameter, without which the system only shows either a stable steady state or oscillatory solutions with large amplitudes. We argue that this large amplitude behavior is the source of chaotic behavior reported in systems that use the MacArthur–Rosenzweig model to describe food-chain dynamics. Finally, we find the sufficient conditions in parameter space for Turing patterns and obtain the so-called "back-eye" pattern and localized structures.

We report the existence of bursting oscillations and mixed-mode oscillations in a Liénard system when it is driven externally by a sinusoidal force. The bursting oscillations transit from a periodic phase to spiking trains through chaotic windows, as the control parameter is varied. The mixed-mode oscillations appear via an alternate sequence of periodic and chaotic states, as well as Farey sequences. The primary and their associated secondary mixed-mode oscillations are detected for the suitable choices of system parameters. Additionally, the system is also found to possess multistability nature. Our investigations involve numerical simulations as well as real time hardware experiments using a simple analog electronic circuit. The experimental observations are in conformation with numerical results.

The main purpose of the paper is to investigate the effect of multiple scales in frequency domain on the complicated oscillations of Filippov system with discontinuous right-hand side. A relatively simple model based on the Chua’s circuit with periodic excitation is introduced as an example. When the exciting frequency is far less than the natural frequency, implying that an order gap between the exciting frequency and the natural frequency exists, the whole exciting term can be considered as a slow-varying parameter, based on which the bifurcations of the two subsystems in different regions divided by the nonsmooth boundary are presented. Two typical cases are considered, which correspond to different distributions of equilibrium branches as well as the related bifurcations. In the first case, periodic symmetric Hopf/Hopf-fold-sliding bursting oscillations can be obtained, in which Hopf bifurcations may cause the alternations between the quiescent states and the spiking states, while fold bifurcations connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. While the second case is the periodic symmetric fold/fold-fold-sliding bursting, where the fold bifurcations not only lead to the alternations between the quiescent states and the spiking states, but also connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. It is pointed out that, different from the bursting oscillations in smooth dynamical systems in which the bifurcations may cause the alternations between quiescent states and spiking states, in the nonsmooth system, bifurcations may not only lead to the alternations, but also connect different forms of quiescent states. Furthermore, in the Filippov system, sliding movement along the nonsmooth boundary can be observed, the mechanism of which is presented based on the analysis of the two subsystems in different regions.

The main purpose of the paper is to explore the influence of the coupling of two scales on the dynamics of a nonsmooth dynamical system. Based on a typical Chua’s circuit, by introducing a nonlinear resistor with piecewise characteristics as well as a harmonically changed electric source, a modified nonsmooth model is established, in which the coupling of two scales in frequency domain exists. Different types of bursting oscillations, appearing in the combination of large-amplitude oscillations, called spiking oscillations ($SP$), and small-amplitude oscillations or at rest, denoted by quiescent states ($QS$), can be observed with the variation of the exciting amplitude. When the exciting frequency is relatively small, by regarding the whole exciting term as a slow-varying parameter, the original system can be transformed into a generalized autonomous system. The phase space can be divided into three regions by the nonsmooth boundaries, in which the trajectory is governed by three different subsystems, respectively. Based on the analysis of the three subsystems as well as the behaviors on the nonsmooth boundaries, all the equilibrium branches and their bifurcations can be obtained, which can be employed to investigate the mechanism of the bursting oscillations. It is found that, for relatively small exciting amplitude, since no bifurcation on the equilibrium branches can be realized with the variation of the slow-varying parameter, the system behaves in periodic movement, which may evolve to bursting oscillations when a pair of fold bifurcations occurs with the increase of the exciting amplitude. Further increase of the exciting amplitude may lead to more complicated bursting oscillations, which may bifurcate into two coexisted asymmetric bursting attractors via symmetric breaking. Interaction between the two attractors may result in an enlarged symmetric bursting attractor, in which more forms of bifurcations at the transitions between the quiescent states and repetitive spiking states can be observed.

This paper is devoted to exploring the effect of the coexisting attractors in the fast subsystem of a full nonlinear system on the complex dynamics. As an example, a modified 3D van der Pol–Duffing circuit is discussed, in which the coexistence of five attractors including a hidden one was reported in 2020. By introducing a parametric excitation, when the exciting frequency is far less than the natural frequency, a slow–fast model with the two-scale coupling is established in the frequency domain. Regarding the exciting term as a slow-varying parameter, all the attractors as well as the bifurcations of the fast subsystem are derived. The coexistence conditions as well as the coexisting attractors, including the equilibrium points and limit cycles, are presented. With the variation of the exciting amplitude, different types of bursting oscillations are observed, the mechanism of which is revealed using a modified slow–fast analysis method, by combining the transformed phase portraits and the attractors as well as the bifurcations. It is found that the coexisting attractors of the fast subsystem may lead to coexisting bursting solutions in the full system, in which a trajectory may visit one of the attracting basins of the attractors, or result in a merged bursting motion, which may visit the attracting basins in turn. The influence of some attractors and the associated bifurcations may vanish in the case that the trajectory passes directly across the parameter interval corresponding to the attractors because of the inertia of the motion; this may also occur in the case that the trajectory does not visit the associated attracting basins, but it starts initially from the basins. It should be pointed out that, when the exciting frequency is small enough, the disappeared effect of stable attractors may reoccur. Furthermore, if the parameter interval between two types of codimension-1 bifurcation points is short enough, the trajectory may behave in a combination of two bifurcations, which is somewhat similar to that caused by a codimension-2 bifurcation.