International Journal of Bifurcation and Chaos

Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labeling. Addressing these problems corresponds to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, which cover a large variety of known (and less known) examples of chaos. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map, …).

By treating extensively the Rössler and the Lorenz attractors, we extended the description of branched manifold — the highest known taxonomic rank for classifying chaotic attractor — by a linking matrix (or linker) to multicomponent attractors (bounded by a torus whose genus $g\ge 3$).

Hidden attractors have been reported in a significant number of dynamical systems, according to their definition related to the existence of saddle points. Since it has now become common knowledge that there are different types of saddle invariant sets besides a fixed saddle point, it is of great interest to investigate the subject. In this paper, a sudden appearance of a hidden chaotic attractor is observed with the variation of a control parameter in some two-dimensional mapping systems. In order to reveal the mechanism behind the sudden appearance of hidden attractors, the generalized cell mapping method with GPU parallel computing is employed in order to obtain very accurate invariant sets. It is found that hidden attractors are generated from an existing chaotic saddle through a boundary crisis, where a hidden chaotic attractor is touching a periodic or chaotic saddle on its basin boundary. Three examples of two-dimensional mapping systems are given to demonstrate the birth of hidden attractors through a boundary crisis.

This paper investigates the nonlinear dynamical behaviors of a time-delayed supercavitating vehicle system varying with the system parameters, and proposes control strategies to stabilize the trajectory. A simplified dynamical model of the supercavitating vehicle is firstly constructed and then analyzed on its local and global stability in terms of different parameter values and initial conditions. Due to the forces resulting from the fins and the afterbody plane causing one of the system parameters to become a piecewise constant function of position, the system is described as a linear switched one with time delay. Both theoretical analysis and numerical simulations are adopted to describe the system behaviors. Several special phenomena such as bifurcations, deviation behaviors and period-doubling processes emerging from the system for a given set of parameters are discussed in detail in this paper. Besides, the equilibrium and chaotic solutions are accurately located by plotting the Lyapunov exponent map. Chaotic attractors present different strange forms with the variation of the parameters. Finally, based on the system’s global property, the control and anti-control strategies are proposed to stabilize the system or generate desirable chaos and it is shown that the system can be controlled by selecting appropriate control parameters.

These studies focus on the forced vibrations of rectangular plates incorporating oscillating inclusions uniformly distributed in the carrier elastic medium. Plate’s equations of motion consist of the Lamé equation for a carrier material and the equation for oscillators representing the inclusions. The oscillators are acted upon by the nonlinear friction force, described by Coulomb’s modified law, and a cubically nonlinear elastic force. The simply supported forced plate is considered. Using the Galerkin approximation, the problem of plate vibrations is reduced to the study of a system of ODEs. According to numerical and qualitative analyses, the application of harmonic forcing causes the appearance of periodic, quasiperiodic, and chaotic regimes in the system. In particular, the analytical expression is derived for the periodic regimes of the forcing frequency. The scenarios of vibration’s development is identified at the variation of forcing amplitude. The quasiperiodic and chaotic modes are studied by means of Poincaré section technique, and Fourier and Lyapunov spectra analyses. The statistical properties of the sequences of temporal intervals between maxima of system’s solutions are considered in more detail. The sequence extracted from the chaotic trajectory is shown to possess long-term correlations and approximately obeys the Weibull distribution.

This study applies the Kosambi–Cartan–Chern (KCC) theory to the Brusselator model to derive differential geometric quantities related to bifurcation phenomena. Based on these geometric quantities, the KCC stability of the Brusselator model is analyzed in linear and nonlinear cases to determine the extent to which nonequilibrium affects bifurcation and stability. The geometric quantities of the Brusselator model have a constant value in the linear case, and are functions of spatial variables with parameter dependence in the nonlinear case. Therefore, the KCC stability of the nonlinear case shows various distribution patterns, depending on the distance from the equilibrium point (EQP), as follows: in the regions near or far enough from the EQP, the distribution of KCC stability is uniform and regular; and in the intermediate nonequilibrium region, the distribution varies and shows complex patterns with parameter dependence. These results indicate that stability in the intermediate nonequilibrium region plays an important role in the dynamic complex patterns in the Brusselator model.

To investigate the joint effects of diffusion and multiple delays, a toxin producing phytoplankton–zooplankton system with cross-diffusion is studied and two-parameter bifurcation analysis is carried out. Firstly, for the delay-free system, the stability of equilibria, existence of solutions and priori bound are given. The globally asymptotic stability of equilibrium is investigated for the two situations with or without the diffusion terms. The linear stability analyses show that the cross-diffusion is a key mechanism for Turing instability to occur and Turing instability interval becomes larger with increasing cross diffusion coefficient. Secondly, the focus is on the effects of delays on the system including persistence, stability and Hopf bifurcation. The sufficient conditions for the diffusion system with delays to remain persistent are given. Under certain conditions, the boundary equilibrium is globally asymptotically stable. By analyzing the distribution of roots of the characteristic equation, using the crossing curve methods, the stable changes of positive equilibrium in two-delay plane are obtained. Explicit algorithm is derived for the properties of the bifurcating periodic solutions by normal form theory and the center manifold theorem for partial functional differential equations. Finally, numerical simulations confirm the correctness of theoretical analyses and find that double Hopf bifurcation may occur.

The emergence of many fascinating dynamic behaviors is affected by more than one interaction among the elements or cells in a network. In fact, the concurrence and competition of different types of effects among subsystems show a strong connection to the dynamic transition process between oscillation patterns. Here, a network of generic oscillators with mixed attractive-repulsive couplings is introduced to demonstrate the transition from oscillatory states to stationary equilibria, specifically for van der Pol oscillators and Lorenz oscillators. Through the observation of the normalized amplitude changing with the coupling strength, the sudden and irreversible transition appears in both systems, which has a close relation to the mutual repulsion on coupled oscillators. Whereas, for coupled van der Pol oscillators, three typical transition scenarios are found by varying the weight ratio of these two couplings, while the Lorenz system shows only one transition mode no matter how the weight ratio changes. Besides, in the cases of explosive transitions, the coexistence areas of oscillatory and death states also reveal a distinct manifestation for periodic and chaotic systems. The details of theoretical critical transition points on the first-order phase transition are also obtained. Our results pave a new way to control the explosive phenomenon, which is crucial to explain the sudden oscillation quenching and the coexistence of oscillatory and stationary states in biological as well as chemical systems.

The multichannel sales mode is employed by more and more manufacturers to maximize their profits or gain much more than market share. This paper develops a dynamic pricing game model of multichannel supply chain, and focuses on the impacts of channel competition on the complex characteristics of the evolutionary dynamic game, using analyses based on stability domain, bifurcation diagram, the largest Lyapunov exponent, attractor domain and time series. The results indicate that the system will gradually enter chaotic state through flip bifurcation from stable state with the increase of price adjustment speed, and the two-dimensional stability domain will shrink as the competition coefficient increases. Interestingly, although an increase in commission rate will lead to a decrease in profits, it will widen the stability range of the whole system. In the end, for further analysis, the chaotic behavior of the multichannel supply chain is controlled by using the feedback control method. The manufacturer is suggested to adopt a cautious pricing strategy in the stability region, and minimize the direct competition between channels in the sales process and diversify the productions in different channels.

The close relationship between chaos and dynamical systems leads to naturally consider the iteration processes that are related to dynamical systems of fixed point theory. From this natural relationship, the control of chaos that occurs in fixed point iteration dynamics will be the main focus of the article. To achieve this goal, analytical solutions are obtained and used to control chaos that occurs at unstable fixed points of multistep iteration process. Later, we show an effective regime for the parameters of multistep iteration. To illustrate this claim, well-known special cases of multistep iteration process by Noor, Ishikawa, Mann, Krasnoselskij, Picard iteration processes are introduced. In particular, among these iterations, the Noor iteration process is studied in detail in terms of controlled chaos. The Lyapunov exponent is used to estimate the stability and unstability of fixed points and periods that generate chaos in iteration processes. Finally, with the help of MATLAB program, all these results are shown on logistic and cubic equations with chaotic properties.

In this paper, we investigate the transition to a chaotic regime of thermal convection in a five-dimensional model with low Prandtl number in a porous medium. The mathematical formulation of the model includes the heat equation coupled with the equations of motion under the Boussinesq–Darcy approximation. A system of five ordinary differential equations is derived using a spectral method. This system is solved numerically by using the fourth-order Runge–Kutta method. The results show that from a subcritical value of the Rayleigh number, a transition from steady convection to chaos via a Hopf bifurcation produces a limit cycle which can be associated with a homoclinic explosion. Furthermore, we find that for certain values of Rayleigh number and shape parameter which measures the ratio between the dimensions of the computational domain, the transition from periodic oscillatory convection to chaotic convection can occur via a period-doubling.

The paper reports a new generalized Hamiltonian chaotic system with transient quasi-periodic flows and intermittent chaos by investigating the Kolmogorov-type transformation for a 3D chaotic system. The new system meets Hamiltonian energy conservation when $a=b$, and it is always conservative in volume because of a zero divergence. In addition, its Hamiltonian energy is found to be only related to the initial points. Subsequently, some dynamics analyses are presented to investigate its conservative characteristics and coexisting flows including periodic flows, quasi-periodic flows, and chaotic flows. Besides, both transient quasi-periodic flows and intermittent chaos are also found to occur, which further shows complex dynamics of the new generalized Hamiltonian chaotic system. Finally, an FPGA circuit is designed to implement the new generalized Hamiltonian chaotic system, and the results from the circuit experiments are consistent with those from numerical analyses. Complex dynamic characteristics existing in the new generalized Hamiltonian chaotic system seem more appropriate to be used in image encryption and secure communication. And the FPGA circuit not only shows the new generalized Hamiltonian chaotic system from a physical viewpoint, but also provides a new pseudo-random signal generator for engineering applications.

This paper constructs an e-commerce supply chain (ECSC) that includes one e-commerce platform and one low-carbon manufacturer. Based on bounded rationality, the complexity of pricing, service level, and emission reduction effort decisions in an ECSC is investigated in two scenarios: horizontal Nash (HN) game and long-term platform Stackelberg (LPS) game. Finally, the effects of different adjustment speeds and critical parameters on decision variables, profits, and system stability are analyzed through various numerical simulations. The study revealed that the price adjustment parameters have a more noticeable impact on the stability and profit of ECSC in both scenarios. It is interesting that the chaos caused by the adjustment speed of the platform’s decisions leads to a decrease in the platform’s profits. However, it causes continuous fluctuations in low-carbon manufacturers’ profits. In the HN game, when the increase of critical parameters is beneficial to low-carbon manufacturers, the system’s stability will decrease. In addition, the system’s stability in the LPS game is only affected by the adjustment speed of the platform’s decisions. By comparison, we find that the adjustment of the platform’s price plays a vital role in the ECSC under different power structures. Finally, the adaptive adjustment mechanism is used to control the chaos caused by the price of the platform in the ECSC system.

In this paper, we study the bifurcations of a class of planar Hamiltonian systems having a global nilpotent center under the perturbations of polynomials of degree $n\in N$ with a nonlinear switching curve. We obtain the exact bound of the number of limit cycles bifurcating from the period annulus if the first order Melnikov function is not identically zero. We also give some examples to illustrate our results.

In this paper, we investigate the dynamics of three-vortex systems and give a global phase diagram analysis with different types of parameters on the plane. By using the invariants to construct canonical transformations, the Hamiltonian of the three-vortex system is reduced to a desired two-dimensional Hamiltonian. Then, we analyze its phase diagrams by considering all possible situations. In each case, the stabilities of the equilibrium points and singular points are established by using numerical methods, and we obtain the distribution of trajectories in the phase space.

A reaction–diffusion system with two delays, which describes viral infection spreading in the lymphoid tissue, is investigated. The delays promote very complex dynamics of the model. We get the Hopf bifurcation curves and the stability region for the coexisting steady state on a two-parameter plane by finding the stability switching curves and the subsets of stable region. When two delays cross the boundary of the stable region, the system will undergo stability switches. It is shown that two types of bistability are possible: the coexistence of the stable virus-free steady state and the stable coexisting steady state; the coexistence of the stable virus-free steady state and a stable periodic solution. Numerical simulations suggest that delays can also induce tristability including a steady state and two stable periodic solutions.

In this paper, a nitrogen fixation game system with two time delays under nitrogen limitation is investigated. Firstly, we discuss the existence and local stability of the equilibrium points for the nondelay system. Then the nitrogen fixation delay and strategy-dependent delay are used as bifurcation parameters to analyze the local stability and the Hopf bifurcation. In addition, we obtain the direction of Hopf bifurcation, the change of the period for periodic solution and the stability of periodic solution via center manifold theory and normal form method. Finally, numerical simulation is employed to visualize the theoretical analysis results and we find that the nitrogen fixation strategy is the dominant strategy when the values of the two time delays are large enough. This study promotes the investigation of the effects of two time delays on the dynamics of evolutionary games with environmental feedback, especially on stability and bifurcation.

Extreme multistability usually emerges in a mem-element’s circuit or system that possesses a line or plane equilibrium set closely associated with the internal initial state of the mem-element. To extend the investigation of extreme multistability, this paper proposes a nonautonomous memcapacitive oscillator, discovering a new type of extreme multistability due to the infinitely many discrete equilibrium points therein. This memcapacitive oscillator is constructed by connecting a simple memcapacitor-resistor circuit with a sinusoidal voltage. With its normalized model, the infinitely many discrete equilibrium points are computed and the infinitely many necklace-shaped coexisting attractors that were not yet reported are disclosed by numerical methods. Since the number and stability of the equilibrium points vary with time, the attraction basins with complex ripple structures are formed in the memcapacitive oscillator, resulting in the appearance of a special type of extreme multistability. Furthermore, PSIM circuit simulations and microcontroller-based hardware experiments are performed to verify the numerical results.

This note follows up on the study in the paper [Zepeda Ramírez et al., 2021], where we investigate the nonlinear stability of equilibrium points for the planar restricted equilateral four-body problem for the specific values of masses $m_1=0.01$ and $0.99$ of the primary, fixed on the horizontal axis. We give an improvement of the previous results.