Bursting multistability induced by double-Hopf bifurcation.

We study the slow-fast dynamics of a system with a double-Hopf bifurcation and a slowly varying parameter. The model consists of coupled Bonhöffer-van der Pol oscillators excited by a periodic slow-varying AC source. We consider two cases where the slowly varying parameter passes by or crosses the double-Hopf bifurcation, respectively. Due to the system's multistability, two bursting solutions are observed in each case: single-mode bursting and two-mode bursting. Further investigation reveals that the double-Hopf bifurcation causes a stable coexistence of these two bursting solutions. The mechanism of such coexistence is explained using the slowly changing phase portraits of the fast subsystem. We also show the robustness of the observed effect in the vicinity of the double-Hopf bifurcation.

Route to bursting via pulse-shaped explosion.

This Rapid Communication reports on the discovery of a route to bursting, called a pulse-shaped explosion (PSE), for a paradigmatic class of nonlinear oscillators. We find that both an equilibrium and a limit cycle can exhibit pulse-shaped sharp quantitative changes in relation to the variation of system parameters, which are interesting explosive behaviors, the PSE. It leads to large-amplitude oscillations in the rest phase (i.e., small-amplitude oscillations) of bursting, giving rise to additional active phases alternating with the rest phase, and finally determines compound bursting structures. This way, the route to complex bursting dynamics by PSE is explained and its robustness is shown. PSE opens different ways for the control dynamics of complex systems.

Obtaining amplitude-modulated bursting by multiple-frequency slow parametric modulation.

Amplitude-modulated bursting (AMB), characterized by oscillations appearing in the envelope of the active phase of bursting, is a novel class of bursting rhythms reported recently. The present paper aims to report a simple and effective method, i.e., the multiple-frequency slow parametric modulation (MFSPM) method, for obtaining such a bursting pattern. We show that the MFSPM can be well controlled so that it may exhibit multiple continuous ups and downs in the active area. Then, the amplitude of the traced active state alternates between increases and decreases accordingly, which leads to oscillations in the envelope of the active phase, and AMB is thus created. Based on this, the route to AMB by the MFSPM is presented. The validity of the approach is demonstrated by several examples. The proposed approach does not depend on specific systems or bifurcations and thus is a general method.

Approximation to Hadamard Derivative via the Finite Part Integral.

In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green's function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique.

Two novel bursting patterns in the Duffing system with multiple-frequency slow parametric excitations.

This paper aims to report two novel bursting patterns, the turnover-of-pitchfork-hysteresis-induced bursting and the compound pitchfork-hysteresis bursting, demonstrated for the Duffing system with multiple-frequency parametric excitations. Typically, a hysteresis behavior between the origin and non-zero equilibria of the fast subsystem can be observed due to delayed pitchfork bifurcation. Based on numerical analysis, we show that the stable equilibrium branches, related to the non-zero equilibria resulted from the pitchfork bifurcation, may become the ones with twists and turns. Then, the novel bursting pattern turnover-of-pitchfork-hysteresis-induced bursting is revealed accordingly. In particular, we show that additional pitchfork bifurcation points may appear in the fast subsystem under certain parameter conditions. This creates multiple delay-induced hysteresis behavior and helps us to reveal the other novel bursting pattern, the compound pitchfork-hysteresis bursting. Besides, effects of parameters on the bursting patterns are studied to explore the relation of these two novel bursting patterns.

Inverse period-doubling bifurcations determine complex structure of bursting in a one-dimensional non-autonomous map.

We propose a simple one-dimensional non-autonomous map, in which some novel bursting patterns (e.g., "fold/double inverse flip" bursting, "fold/multiple inverse flip" bursting, and "fold/a cascade of inverse flip" bursting) can be observed. Typically, these bursting patterns exhibit complex structures containing a chain of inverse period-doubling bifurcations. The active states related to these bursting can be period-2(n) (n = 1, 2, 3,…) attractors or chaotic attractors, which may evolve to quiescence by a chain of inverse period-doubling bifurcations when the slow excitation decreases through period-doubling bifurcation points of the map. This accounts for the complex inverse period-doubling bifurcation structures observed in bursting patterns. Our findings enrich the possible routes to bursting as well as the underlying mechanisms of bursting.

Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies.

We present a general method for analyzing mixed-mode oscillations (MMOs) in parametrically and externally excited systems with two low excitation frequencies (PEESTLEFs) for the case of arbitrary m:n relation between the slow frequencies of excitations. The validity of the approach has been demonstrated using the equations of Duffing and van der Pol, separately. Our study shows that, by introducing a slow variable and finding the relation between the slow variable and the slow excitations, PEESTLEFs can be transformed into a fast-slow form with a single slow variable and therefore MMOs observed in PEESTLEFs can be understood by the classical machinery of fast subsystem analysis of the transformed fast-slow system.

Multiple-mode wave solutions to display superpositions and collisions in nonlinear evolution equations.

We present a general method for obtaining multiple-mode waves (MMWs), which is introduced as a concept expressed in the form of nonlinear superpositions of single-mode waves (SMWs) with different wave speeds, for nonlinear evolution equations. The validity of the approach has been demonstrated using two wave equations. It is shown that MMWs may combine different types of SMWs such as periodic waves, kink waves, compactons, solitary waves, etc., to form more general solutions, which can be used to display the whole evolution process of interactions between different types of waves, especially to reveal the dynamic details of the wave patterns.