International Journal of Bifurcation and Chaos

We analyze benchmark models for reaction dynamics associated with a time-dependent index-2 saddle point. The influence of index-2 saddle points on chemical reaction dynamics has received a great deal of attention in recent years and we extend this work in a new and important direction. Our model allows us to incorporate time dependence of a general form, subject to an exponential growth restriction. Under these conditions, we analytically compute the time-dependent normally hyperbolic invariant manifold and its time-dependent stable and unstable manifolds. These time-dependent phase space structures allow us to quantitatively determine the effect of the time dependence on reacting and nonreacting trajectories.

We analyze the effects of extreme multistability in a mechanical system which describes the movement of an idealized ball over a surface like a Mexican hat. The mathematical model is given by a system of autonomous ordinary differential equations with parameters. In particular cases of rotational symmetry and cosymmetry, the system has a curve of asymptotically stable equilibria. The symmetry gives a circle of equilibria with identical stability spectra, whereas the cosymmetry produces an ellipse of equilibria with nonidentical properties. The destruction of both symmetry and cosymmetry leads to a finite number of equilibria (multistability). We study the dynamics for conservative (without dissipation) and dissipative (linear damping) cases using analytical methods and computer simulation. We found interesting effects caused by extreme multistability: nontrivial selection of equilibria of the family, high sensitivity to initial data because of memory about conservative chaos, and essential difference in dynamics in rotational symmetry and cosymmetry cases.

This paper introduces a new statistical tool: dispersion Havrda–Charvat entropy plane, which is used to analyze the complexity characteristics of time series. The Havrda–Charvat entropy with one parameter can provide flexibility in applications and provide more information about time series. The dispersion entropy algorithm is a fast and powerful algorithm for evaluating time series, which has been proposed in recent years. The statistical complexity measure defined by Jensen–Shannon divergence reflects the chaotic degrees of complex systems. The dispersion Havrda–Charvat entropy plane is constructed using the above conceptions. The performance of the dispersion entropy plane is evaluated by simulated chaotic processes and fractional Brownian motions, and then we apply the method to stock data. This demonstrates that dispersion Havrda–Charvat entropy plane can distinguish the intensive properties of time series well and is a powerful method to classify stock markets. In addition, the multiscale measure is experimented, the results show that it can eliminate the noise contained in the data and effectively extract the information contained in time series with different time scales.

Even if a system has only one stable equilibrium point, its dynamics can still be complicated. In the case of the equilibrium point with a complicated boundary of the attraction basin, it is probably difficult to predict the long-term behavior of the trajectory starting from a point outside but near the boundary. The exploration of such systems is helpful for deepening our understanding of the dynamics of complex systems. This paper studies a class of three-dimensional polynomial systems with a nonelementary singularity. With parameters satisfying some conditions, the asymptotic stability of the origin is proved and the complexity of the attraction basin is investigated. It is demonstrated that the boundary of the attraction basin of the origin has a fractal structure in the following sense: An invariant set homeomorphic to the well-known Lorenz strange attractor is contained in the basin boundary. Based on the non-negative function we constructed, how fast a trajectory of the system tends to the asymptotically stable nonelementary singularity is measured.

The dynamic behavior of a new financial system with diffusion effect and two delays is studied. The local stability of equilibrium is studied by analyzing the corresponding characteristic equation and developing the crossing curve method. Hopf bifurcation curves are obtained by three procedures and the critical value of delay of Hopf bifurcation is obtained. By using the central manifold theorem and the normal form theory of partial differential equations, the normal form of Hopf bifurcation with two delays is derived, and the bifurcation direction and the stability of periodic solution are determined. The numerical simulations not only verify these theoretical results, but also show that double Hopf bifurcation may occur and the two delays play a key role in chaos suppression. This study has important theoretical value of innovation and practical value in the macro-economic system.

In the paper, the main parameters affecting the stability of a rotating pretwisted cylindrical shell model with a presetting angle are found by the extended singularity theory. Based on the nonlinear second-order ordinary differential equations of rotating pretwisted cylindrical shell, the universal unfoldings of these equations are given, such that the derivative terms of these equations are treated as periodic perturbation parameters. Analysis results show that these equations have: one form of the universal unfolding codimension 4; one form of the universal unfolding codimension 2; one form of the universal unfolding codimension 2. Calculating the transition set of the universal unfolding codimension 2, the transition set covers all parameters which affect the stability of the rotating pretwisted cylindrical shell. The numerical results show that the amplitudes of periodic excitation and part of the functions of the geometric and physical parameters have strong effect on bifurcation and hysteresis of the rotating pretwisted cylindrical shell subjected to small perturbations near the singularity; however, small external disturbances have weak effect on bifurcation and hysteresis. Due to the periodicity of perturbation parameters, settled in different persistent regions by change in other parameters, the governing equations produce chaotic solutions. The present results are validated by comparison with those in the existing literature.

In this paper, we study the global dynamics of the 5D structural leukemia model with 14 parameters as developed by Clapp et al. [2015]. This model describes the interaction between leukemic cell populations and the immune system. Our analysis is based on the localization method of compact invariant sets. We develop this method by introducing the notion of a partitioning of the parameter space and the notion of a localization set corresponding to this partitioning as its parameters change. Further, we obtain ultimate upper and lower bounds for all variables of a state vector without imposing additional restrictions. Local asymptotic stability conditions with respect to the leukemia-free equilibrium point (EP) are given. We deduce formulas describing inner EPs expressed in terms of positive roots of one 7D equation. Based on this equation, it is shown that the number of inner EPs cannot exceed 3 and one case of a global bifurcation of EPs is detected. Next, we prove the existence of the attracting set. Further, in two theorems, the global eradication/extinction leukemia theorems are described. The impact of using parametrically variable localization sets for a qualitative analysis of the ultimate behavior of leukemic cell populations is discussed.

In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results.

The so-called coupled smooth and discontinuous (SD) oscillator whose stiffness term leads to a transcendental function is a simple mass-spring system constrained to a straight line by two parameters, which are the dimensionless distances to the fixed point. This paper studies the homoclinic–heteroclinic chaos in a coupled SD oscillator subjected to Gaussian colored noise. In order to investigate the chaos thresholds analytically, the piecewise linearization approximation is used to fit the transcendental function. Stochastic nonsmooth Melnikov method with homoclinic–heteroclinic orbits is developed to study chaos thresholds of oscillators with tri-stable potential. Based on stochastic Melnikov process, the mean square criterion and the rate of phase-space flux function theory are used to study the chaotic motions of a coupled SD oscillator under weak noise and strong noise, respectively. The obtained results show that it is effective to use the piecewise linear approximation to analyze chaos in the coupled SD oscillator subjected to Gaussian colored noise. It also lays the foundation for chaos research of other nonsmooth mechanical vibration systems under random excitation.

In this paper, we investigate the dynamical behavior of a one-dimensional piecewise map based on the logistic map, where generalized multistability can be observed. The proposed system has the unique property that the function is symmetric with respect to the origin but not its behavior, furthermore this system can display three types of multistability, and chaos for both, monostable and bistable behaviors. The stability analysis of the proposed system is presented. We describe the structure of bistable regions in the bifurcation diagram. Particular attention is paid to the chaotic regions. Corresponding to coexisting attractors, three scenarios of coexisting attractors, namely fixed points, periodic orbits, and chaotic attractors, can be found, which are unreported behaviors in discrete chaotic systems. The mechanism that leads to multistability phenomenon including pitchfork bifurcation, period-halving bifurcations, and the coexisting invariant sets is demonstrated. Furthermore, the Lyapunov exponent is analyzed with the type of multistability distinguished for a given set of parameters.

In this paper, a delayed rumor propagation model with a discontinuous control mechanism is investigated. First, we investigate the model’s existence and boundedness. Then, conditions for the existence of the rumor-endemic equilibrium are derived utilizing differential inclusions. We examine the local asymptotic stability and Hopf bifurcation of the rumor-endemic equilibrium when the discontinuous control function is differential at the rumor-endemic equilibrium, considering the delay as the bifurcating parameter. When the discontinuous control function is not differential at the rumor-endemic equilibrium, we investigate the global asymptotically stable rumor-free and rumor-endemic equilibria using the Lyapunov functional technique. We conclude by providing two examples to validate our theoretical predictions. It is demonstrated that delay is a critical system parameter that can result in both Hopf bifurcation and chaos.

In ecosystems, almost all organisms suffer from external influences, which are assumed to be random perturbations. The aim of this paper is to investigate the effects of random perturbations on interacting populations. To solve this problem, we propose a discrete-time plant quality and larch budmoth interaction model with biological control. First of all, the model is theoretically analyzed, and the dynamic behavior of the population is qualitatively analyzed by numerical simulation. In addition, random perturbations are added to the model to observe the effects of random perturbations on the dynamic behavior of the budmoth population. The main results show that a random perturbation changes the number of blurred orbits of the system compared to the single population model with the same perturbation. Undeniably, it enhances the stability of the system. The model results, after adding random perturbations, are closer to reality, and in practice, we can also use random perturbations for resource management and pest control.

Reliable neuron models play an essential role in identifying the electrical activities, global bifurcation patterns, and dynamic mechanisms of neurons in electromagnetic environments. Considering that memristive autapse can characterize the self-induced effect of neurons, a five-dimensional Hindmarsh–Rose (HR) neuron model involving electric and magnetic fields is established. The detailed existence and stability analyses for equilibrium points are performed, and the complex time-varying stability, saddle-node bifurcation, and Hopf bifurcation behaviors are demonstrated. Interestingly, the bistable structures consisting of quiescent state and periodic bursting modes near the subcritical Hopf bifurcation and counterintuitive dynamic phenomena can be induced via appropriately adjusting the memristive current. Accordingly, the mechanism of positive feedback autaptic current decreases its firing frequency, while negative feedback autaptic current promotes its excitability and is revealed by the fast–slow dynamic analysis. Generally, the system possesses period-adding bifurcation patterns and comb-shaped chaotic structures as demonstrated by the numerical results. Importantly, it can be confirmed that the electrical activities and multistability of the system can be accurately predicted by analyzing the global dynamic behaviors of the Hamilton energy. Furthermore, it is verified that the unidirectional coupling controller involving energy is far more efficient and consumes lower energy than electrical synaptic coupling in achieving complete synchronization with mismatched parameters. These results provide potential guidance and help for further research in computational neuroscience and the design and control of intelligent sensors.

We provide an upper bound for the maximum number of limit cycles for the class of discontinuous piecewise differential systems formed by two differential systems separated by the straight line $x=0$, one of which is a linear rigid center while the other is a rigid center formed of a linear part plus a homogeneous polynomial of odd degree. We solve the extended 16th Hilbert problem for this class of discontinuous piecewise differential systems.

This paper proves that some shadowing properties are sufficient conditions for being transitive or point-transitive for a nonautonomous discrete dynamical system. Moreover, considering weak mixing property and transitivity via Furstenberg family, this paper reveals the relationship for transitivity and mixing between $p$-periodic systems and their induced autonomous discrete dynamical systems.

The existence, direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady state of a reaction–diffusion–advection model with distributed delay and Dirichlet boundary condition are considered in this paper. By analyzing the characteristic equation and taking the bound of time delay as the bifurcation parameter, we show that Hopf bifurcation will occur when the shape parameter $n\ge 1$, and the positive steady state is always stable when $n=0$. The properties of Hopf bifurcations are then obtained by center manifold reduction and norm form methods. The effects of the distributed delay, shape parameter and advection on Hopf bifurcation values are also investigated. Finally, numerical simulations illustrate the effectiveness of the theoretical results.

In this paper, we introduce Allee effect and predator competition in the Bazykin’s model with Holling I functional response. Theoretically, we analyze the existence and stability of equilibria, and derive the existence conditions of saddle-node bifurcation and Hopf bifurcation. In addition, in order to determine the stability of limit cycles, we explicitly calculate the first Lyapunov coefficient and prove that the positive equilibrium is not a center, but a weak focus with a multiplicity of at least two. Therefore, the system has Hopf bifurcation and Bautin bifurcation with two limit cycles. Our results indicate that the Allee effect and predator competition lead to a series of complex dynamic phenomena. Finally, numerical simulation verifies the effectiveness of the theoretical results.

In a previous publication, we established a new paradigmatic model of laser with feedback including the minimal nonlinearity leading to chaos. In this paper, the jerk dynamics of this minimal universal model of laser is presented. It is proved that two equivalent forms of the model in jerk dynamics can be derived. The electronic circuit of the simpler dynamics is designed and implemented. The link between the minimal universal model of laser and the search for simple jerk circuits is established.