International Journal of Bifurcation and Chaos

Networks of coupled oscillators have been used to model various real-world self-organizing systems. However, the dynamics, especially chaos and bifurcation, of complex-valued networks are rarely investigated. In this paper, a ring network of interacting complex-valued van der Pol oscillators is studied to model the formation of ring dynamics. Although there are only stable limit cycles in a complex-valued van der Pol oscillator, chaos, hyperchaos, and coexisting chaotic attractors are observed from the ring network, which are analyzed by using the Lyapunov exponent spectrum, bifurcation diagram and 0–1 test. In addition, complexity analysis on nonlinear coefficients and coupling strengths illustrates that the range of parameters within the chaotic (hyperchaotic) region has positive correlation with the number of oscillators. It is shown that the chaotic bifurcation path is highly robust against the size variation of the ring network, which always evolves to chaos directly from period-1 and quasi-periodic states, respectively. Moreover, it is demonstrated that complete synchronization and phase synchronization of oscillations are stable in a large-scale ring network, while chaotic phase synchronization is unstable in a small-scale network.

In this work, we propose to use the distance correlation ($DC$) and the Euclidian distance ($ED$) as powerful quantifiers to describe correlation and synchronization in spatially extended systems. As examples, we use the coupled Kuramoto oscillator model and coupled map lattices to study chimera states. Results for $DC$ and $ED$ are compared and discussed in terms of recurrence plots, local synchronization order parameter, local recurrence rate, and local $DC$ rate. The existence of correlations between desynchronized states, not visible in usual recurrence plots, becomes evident when using $DC$ and $ED$. Since correlation is a broader concept than synchronization, the proposed quantifiers amplify the characterization, description and possible application of spatially extended systems.

This paper presents a detailed investigation of stochastic stability and complex dynamics of a centrifugal governor system with approximately uniform color noise. The centrifugal governor system excited by noise is transformed into Itö equation using polar coordinate transformation and stochastic average method. According to the boundary conditions of attraction and repulsion, the stochastic stability is ensured. In addition, analyses concerning the influence of parameter variation and validity are carried out by employing numerical method. The results manifest that the effects of noise intensity and correlation time on stationary probability density are opposite. The amplitudes of probability density finally tend to a limit value, and the only limit cycle appears, which shows that when the bifurcation occurs, the trivial solution of the system converges to a limit cycle with a higher probability. Finally, the two-dimensional parameter bifurcation analysis of the centrifugal governor system subject to color noise excitation is studied. An interesting distribution characteristic is found that the periodic region is organized according to the sequence of Stern–Brocot trees, and this typical characteristic is a universal characteristic of the system on the two parameter planes. Furthermore, it is concluded that based on the largest Lyapunov exponent diagram and bifurcation diagram in two-dimensional parameter plane, the effects of noise intensity and correlation time on the periodic oscillation state are opposite, but both of them can transform the quasi-periodic oscillation into periodic oscillation. It should be emphasized that with the increase of noise intensity, the coexisting oscillation behavior of the centrifugal governor system will change, which is manifested by the destruction of coexisting attractors and the generation of chaotic attractors.

With people’s increasing awareness of the dangers of climate warming, the cap-and-trade scheme in supply chain has been gradually implemented worldwide. However, many companies have not engaged in carbon emission reduction, and the market is under asymmetric competition. In this context, this article focuses on the asymmetric competition between manufacturers under the cap-and-trade regulation and compares different retail modes. At the same time, four models are constructed from two different perspectives: the short-term game and the long-term repeated game. The results show that the manufacturer’s optimal price increases with the commission coefficient when the carbon gap is significant in the short-term game, while the carbon emission reduction decreases with the coefficient. And the profits of the manufacturer who contributes to carbon emission reduction are more sensitive to changes in the commission coefficient. Under the reselling mode, the retailer’s participation in pricing will intensify supply chain competition. When the carbon gap is significant, carbon emission reduction will decrease with the increasing carbon trading price. By comparison, the carbon emission reduction level under the reselling mode is higher when the carbon gap is significant; when the commission coefficient is moderate, the retailer in the agency mode has higher profits. In the long-term game, the system is more likely to be in chaos when the manufacturer’s adjustment speed is too high.

Nonlinear dynamical systems with hidden attractors belong to a recent and hot area of research. Such systems can exist in different forms, such as without equilibrium or with a stable equilibrium point. This paper focuses on the dynamics of a new 4D chaotic hyper-jerk system with a unique equilibrium point. It is shown that the new hyper-jerk system effectively exhibits different hidden behaviors, which are hidden point attractor, hidden periodic attractor, and hidden chaotic state. Collective behaviors of the system are studied in terms of the equilibrium point, bifurcation diagrams, phase portraits, frequency spectra, and two-parameter Lyapunov exponents. Some remarkable and exciting properties are found in the new snap system, such as period-doubling transition, asymmetric bubbles, and coexisting bifurcations. Also, we demonstrate that it is possible to generate different varieties of two, three, four, or five coexisting hidden and self-excited attractors in the introduced model. In addition, the amplitude and offset of the hidden chaotic attractors are perfectly controlled for possible application in engineering. Furthermore, a circuit design has been implemented to support the physical feasibility of the proposed model.

In this paper, the evolution of three types of border collision bifurcations are investigated thoroughly for a high-order single-inductor double-output DC-DC converter with one-cycle control (OCC). Multi-piecewise inductor current with more than one border and cross-coupling effect between different output ports are prominent features of the system, resulting in unexpected period incrementing cascade and transition from periodicity to chaos as a parameter is varied. A high-order nonlinear iterative map model is proposed, which reveals the underlying mechanisms of those complicated nonstandard and unsmooth bifurcations in the multiple-output system. Additionally, theoretical analysis shows that those nonstandard bifurcations arise from the boundary collisions, and more specifically, a fixed point suddenly appears in the borderline between various regions. Besides eigenvalue loci changes abruptly at the bifurcation point, the characteristic of border collision bifurcation is the topology sequence changes. Furthermore, the parameter boundaries are given to provide useful guidance for design. Finally, PSpice circuit experiments are performed to verify the above theoretical and numerical results.

We classify the global dynamics of the five-parameter family of planar Kolmogorov systems

Hopfield neural network attracts particular attention as it serves as a relatively simple mathematical model that describes some properties of the brain function. We investigate analog Hopfield neural networks with two time delays. It is shown that the neural network with all inhibitory connections demonstrates growing oscillations after exceeding the threshold, and oscillations become synchronous after a relatively short period of time of the order of the larger time delay. The oscillation amplitude of the neural network as function of the time delay in one subnetwork demonstrates resonance-like phenomena with multiple peaks. The oscillation period of the neural network with two time delays shows discrete structure and changes within relatively narrow intervals in contrast to the oscillation period of the neural network with one time delay, which shows continuous changes of oscillation amplitude and period with time delay. Multiresonance behavior is sustained for both inhibitory and excitatory connections of the second subnetwork, while the first subnetwork possesses only inhibitory connections. The oscillation period of the subnetwork with smaller time delay and equal coupling strengths dominates the whole neural network activity. Turning on and off smaller time delay in one subnetwork allows for the control of the oscillation period and demonstrates memory-like behavior. Mechanisms of the simulated phenomena are disclosed and their similarity to brain function is discussed.

Obtaining the master stability function is a well-known approach to study the synchronization in networks of chaotic oscillators. This method considers a normalized coupling parameter which allows for a separation of network topology and local dynamics of the nodes. The present study aims to understand how the dynamics of oscillators affect the master stability function. In order to examine the effect of various characteristics of oscillators, a flexible oscillator with adjustable amplitude, Lyapunov exponent, and Kaplan–Yorke dimension is used. Not surprisingly, it is demonstrated that the amplitude of the oscillations has no substantial effect on the master stability function, i.e. the coupling strength needed for the complete synchronization is not changed. However, the flexible oscillators with larger maximal Lyapunov exponent synchronize with larger coupling strength. Interestingly, it is shown that there is no linear connection between the Kaplan–Yorke dimension and coupling strength needed for complete synchronization.

In this paper, a fractional-order susceptible-infected-recovered-susceptible (SIRS) model is studied, focusing on delay effects and high-order interactions. Two types of time delays are considered to describe latent period and healing cycle, respectively. From the ecological point of view, we found that the increasing delays caused by either the latent period or the healing cycle lead to the periodic outbreak of disease. The finding provided us with an important implication to preventing periodic outbreaks of disease by reducing the time delay, like accelerating the healing process with effective medication and medical intervention. Specifically, taking the time delays as bifurcation parameters, the stability of endemic equilibria and the existence of Hopf bifurcation are studied by analyzing the characteristic equation of the SIRS model. From a general point of view, based on the establishment of a fractional-order SIRS model, we found that the order of the fractional order is critical for describing the dynamic behavior of the model. Typically, the decrease of the order appears to bring about the disappearance of the periodic phenomenon (i.e. the periodic oscillation) of the originally stable system.

Vegetation pattern generated by aeolian sand movements is a typical type of vegetation pattern in arid and semi-arid areas. Considering the effects of sand movements on vegetation diffusion as a cross-diffusion term, an improved vegetation-sand is studied to explore the effects of cross-diffusion on vegetation pattern formation and transition. Turing bifurcation analysis and amplitude analysis are performed. Theoretical analysis shows that three types of vegetation patterns can be formed: stripes, hexagons and mixture of stripes and hexagons. Numerical simulations on the vegetation pattern formation are consistent with theoretical results, which also show hysteresis in pattern transition process. Moreover, when striped vegetation is formed, with the increase of cross-diffusion coefficient, vegetation stripes break into spots. And when spotted vegetation is formed, with the decrease of cross-diffusion coefficient, spots persist but the statistics of vegetation pattern vary.

The saturated function series have been successfully used to generate multiscroll chaotic attractors. In this paper, we revisit multiscroll chaotic attractors via saturated function series. We find that with a small constant drift acting on the saturated function, the number of scrolls will greatly decrease. This phenomenon brings extra difficulty in pinning synchronization of networked multiscroll chaotic attractors. A new output-feedback pinning controller is designed based on the unknown-input distributed observer to estimate the synchronization error, which has the advantage of saving the communication cost as the transmission of the observer information is not needed.

In this paper, we investigate the dynamics of a modified Leslie–Gower predator–prey model with Allee effect on prey. And the Holling type II functional response is considered in this model. When prey reproduces much faster than predator, by combining the normal form theory of slow–fast systems and the geometric singular perturbation theory, we observe much richer new dynamical phenomena than the existing ones. In the case of strong Allee effect, we prove the existence of canard cycles, homoclinic orbits and heteroclinic orbits. Furthermore, we focus on the case of weak Allee effect. In addition to the similar dynamics of that exhibited by the system with strong Allee effect, we also demonstrate the occurrence of relaxation oscillations created by entry-exit function. Moreover, the existence of canard explosion is further explained analytically and numerically with the help of sophisticated slow–fast techniques.

In this paper, we study the number of limit cycles for a class of cubic systems with general quadratic polynomial perturbations. By using the Melnikov function theory we obtain that five limit cycles can be bifurcated from a period annulus. We also study the Hopf bifurcation at the center surrounded by the annulus.

In this paper, we consider a class of switching systems perturbed by cubic homogeneous polynomials. This class of systems is separated by a straight line: $y=0$, and has three equilibria: $(0,0)$ and $(\pm 1,0)$ which are in the separation line. A new version of the Gasull–Torregrosa method based on Poincaré return maps is presented, and used to compute the Lyapunov constants. Based on this method, a complete classification on the center conditions is obtained for the studied class of systems. Furthermore, by perturbing the cubic switching integral system with cubic homogeneous polynomials, we show that at least ten small-amplitude limit cycles are obtained around one of the centres. This is a new lower bound for the number of limit cycles bifurcating from a center in such switching systems with cubic homogeneous nonlinearities.

We study the characteristics of chaos evolution of initially localized energy excitations in the one-dimensional nonlinear disordered Klein–Gordon lattice of anharmonic oscillators, by computing the time variation of the fundamental frequencies of the motion of each oscillator. We focus our attention on the dynamics of the so-called “weak chaos” and “strong chaos” spreading regimes [Laptyeva et al., 2010], for which Anderson localization is destroyed, as the initially restricted excitation at the central region of the lattice propagates in time to more lattice sites. Based on the fact that large variations of the fundamental frequencies denote strong chaotic behavior, we show that in both regimes chaos is more intense at the central regions of the wave packet, where also the energy content is higher. On the other hand, the oscillators at the wave packet’s edges, through which the energy propagation happens, exhibit regular motion up until the time they gain enough energy to become part of the highly excited portion of the wave packet. Eventually, the percentage of chaotic oscillators remains practically constant, despite the fact that the number of excited sites grows as the wave packet spreads, but the portion of highly chaotic sites decreases in time. Thus, albeit the number of chaotic oscillators is constantly growing the strength of their chaotic behavior decreases, indicating that although chaos persists it is becoming weaker in time. We show that the extent of the zones of regular motion at the edges of the wave packet in the strong chaos regime is much smaller than in the weak chaos case. Furthermore, we find that in the strong chaos regime the chaotic component of the wave packet is not only more extended than in the weak chaos one, but in addition the fraction of strongly chaotic oscillators is much higher. Another important difference between the weak and strong chaos regimes is that in the latter case a significantly larger number of frequencies is excited, even from the first stages of the evolution. Moreover, our computations confirmed the shifting of fundamental frequencies outside the normal mode frequency band of the linear system in the case of the so-called “selftrapping” regime where a large part of the wave packet remains localized.

When chaotic systems are applied to stream ciphers, chaotic real-valued sequences generally need to be converted into binary sequences with the purpose of encrypting data. However, the performance of binary sequences will be degraded under the joint influence of round-off and quantization errors. In this case, the randomness of some chaotic binary sequences may be weakened in a local range. Taking advantage of parallel computing, a fast period detection algorithm is designed to locate all local “periodicities” contained in chaotic binary sequences quickly and accurately. This algorithm evaluates the randomness of a chaotic binary sequence from a new perspective of periodicity which enriches the randomness test methods for binary sequences. Different logistic binary sequences are analyzed to demonstrate the effectiveness and practicability of the proposed algorithm.

For the modified Camassa–Holm equation, by using the methodologies of dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to its corresponding traveling wave system, under different parameter conditions, all possible exact explicit bounded solutions (solitary wave solutions, periodic wave solutions, peakon, periodic peakons, as well as compactons) are obtained. More than 23 explicit exact parametric representations of the above-mentioned traveling wave system are presented.