International Journal of Bifurcation and Chaos

Self-organization of hexagonal distributions of cities of various sizes in Southern Germany is envisaged in central place theory. Yet scientific verification of this theory has been lacking over the years. To scientifically support this theory, we propose a group-theoretic double Fourier spectrum analysis procedure that can detect self-organizing patterns bifurcating from a uniform state. This procedure is reinforced by the optimization of the choice of an analysis domain and the spatial eigen-decomposition to capture agglomeration effects. Strong power spectra for hexagonal networks of cities were detected by the procedure for population data in Southern Germany. Moreover, a gigantic hexagonal distribution of cities in Eastern USA was found as an assemblage of a series of hexagonal networks. The amazing geometrical regularity observed in this distribution manifests the existence of such distribution in the real world, thereby underpinning central place theory.

In this paper, analytical solutions of periodic evolutions of the Brusselator with a harmonic diffusion are obtained through the generalized harmonic balance method. The stability and bifurcation of the periodic evolutions are determined. The bifurcation tree of period-1 to period-8 evolutions of the Brusselator is presented through frequency-amplitude characteristics. To illustrate the accuracy of the analytical periodic evolutions of the Brusselator, numerical simulations of the stable period-1 to period-8 evolutions are completed. The harmonic amplitude spectrums are presented for the accuracy of the analytical periodic evolution, and each harmonics contribution on the specific periodic evolution can be achieved. This study gives a better understanding of periodic evolutions to chaos in the slowly varying Brusselator system, and the bifurcation tree of period-1 evolution to chaos are clearly demonstrated, which can help one understand a route of periodic evolution to chaos in chemical reaction oscillators. From this study, the generalized harmonic balance method is a good method for slowly varying systems, and such a method provides very accurate solutions of periodic motions in such nonlinear systems.

Mixed-mode oscillation-incrementing bifurcations (MMOIBs) are a universal phenomenon appearing in a large class of mixed-mode oscillation (MMO)-generating dynamics that appear to be governed by asymmetric Farey arithmetic. This report presents experimental and numerical observations of the simplest MMOIBs generated by an extended Bonhoeffer–van der Pol circuit. The results indicate that maintaining a constant temperature during the circuit experiments is required to yield stable MMOs and MMOIBs. We also clarify how the firing number variations related to MMOIB phenomena follow asymmetric Farey arithmetic.

Tooth surface contact deformation is part of the main causes for gear system vibration. The safety region of the single-stage spur system vibration is established based on the tooth surface contact deformation. Safe basin and its erosion are calculated numerically according to the safety region as the system control parameters are varied. The basin of attraction in the safe basin is computed by combining the simple cell-to-cell method. Vibration safety and global dynamics of the gear system are investigated. Some bifurcations and mechanisms of the erosion of safe basin are studied by using phase portraits, Poincaré maps, bifurcation diagrams under multi-initial values and top Lyapunov exponents (TLE). The sensitivity of system behavior and bifurcation to initial values are discussed as well. Hidden bifurcating points and attractors are revealed. To get a better understanding of the sensitivity of the system behavior to initial values, a bifurcation dendrogram under multi-initial values is designed. It is found that there is the erosion of safe basin existing in the examining area. Both vibration amplitude changing of coexisting attractors and appearing or disappearing of coexisting attractors are the main causes for the erosion of the safe basin. Bifurcation of the system behavior is selective to initial values. With varying frequency, backlash and comprehensive transmission error, the period-1 response gradually bifurcates under multi-initial conditions and evolves into other new periodic responses that coexist with the period-1 response. The new coexisting response has a key effect on the erosion of the safe basin. The study is helpful to the optimization design of the gear system and the control of the system behavior.

Symmetry is usually prevented by the broken balance in polarity. If the offset boosting returns the balance of polarity when some of the variables have their polarity reversed, the corresponding system becomes conditionally symmetric and in turn produces coexisting attractors with that type of symmetry. In this paper, offset boosting in one dimension or in two dimensions in a 3D system is made for producing conditional symmetry, where the symmetric pair of coexisting attractors exist from one-dimensional or two-dimensional offset boosting, which is identified by the basin of attraction. The polarity revision from offset boosting provides a general method for constructing chaotic systems with conditional symmetry. Circuit implementation based on FPGA verifies the coexisting attractors with conditional symmetry.

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for $n$-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the $𝜀$-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.

To the best of authors’ knowledge, little work has been focused on the noisy vibro-impact systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise excitation is investigated. First, the viscoelastic force is approximately replaced by damping force and stiffness force. Thus the original oscillator is converted to an equivalent oscillator without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the discontinuity of the vibro-impact oscillator. Third, the stochastic averaging method is utilized to obtain analytical solutions of which the effectiveness can be verified by numerical solutions. We also find that the viscoelastic parameters, fractional coefficient and fractional derivative order can induce stochastic bifurcation.

Many studies have been done on different aspects of biped robots such as motion, path planning, control and stability. Dynamical properties of biped robot on a sloping surface such as equilibria and their stabilities, bifurcations and basin of attraction are investigated in this paper. Basin of attraction is an important property since it can determine the unseen conditions which affect the attractor of the system with multistabilities. By the help of basin of attractions, the paper claims that the strange attractors of compass-gait robot are hidden.

In this paper, a novel unimodal chaotic system is presented which has stable Lyapunov exponents, a wider range of parameters and a larger phase space. In order to generate evenly distributed chaotic sequences, a general homogenization scheme is proposed with a rigorous mathematical proof. To generate high-quality pseudo-random number sequences, a PRNG scheme based on cross-coupling is designed. Theoretical derivation and experimental analysis fully demonstrate that the proposed PRNG has the ability to produce high-security random numbers with excellent random characteristics.

An epidemic model for pest management with impulsive control over a patchy environment is proposed in this paper. We investigate the dynamical behaviors on extinction and permanence and obtain the threshold value $R_0$ of dynamics for the impulsive system by utilizing a small amplitude perturbation method, matrix spectral analysis and persistence theory. We prove that the periodic pest-eradication solution of the system is globally asymptotically stable if $R_0<1$, while the system is persistent if $R_0>1$. Furthermore, by discussion on the two-patch case, we analyze the effects of the dispersal and impulsive control on dynamical behaviors of the system. Some numerical examples are given to illustrate the effectiveness of the obtained results and to demonstrate the complexity such as chaotic characteristic of the system.

This paper considers the center problem for quasi-homogeneous polynomial systems with a given weight degree. We provide the necessary conditions such that these systems have a center at the origin. Especially, we present the necessary and sufficient conditions on the existence of a center for some class of such systems.

In this paper, we investigate the number of limit cycles for two classes of discontinuous Liénard polynomial perturbed differential systems. By the second-order averaging theorem of discontinuous differential equations, we provide several criteria on the lower upper bounds for the maximum number of limit cycles. The results show that the second-order averaging theorem of discontinuous differential equations can predict more limit cycles than the first-order one.

The one-dimensional Sine map and Chebyshev map are classical chaotic maps, which have clear chaotic characteristics. In this paper, we establish a chaotic framework based on a Sine–Cosine compound function system by analyzing the existing one-dimensional Sine map and Chebyshev map. The sensitive dependence on initial conditions, topological transitivity and periodic-point density of this chaotic framework is proved, showing that the chaotic framework satisfies Devaney’s chaos definition. In order to illustrate the chaotic behavior of the chaotic framework, we propose three examples, called Cosine–Polynomial (C–P) map, Sine–Tangent (S–T) map and Sine–Exponent (S–E) map, respectively. Then, we evaluate the chaotic behavior with Sine map and Chebyshev map by analyzing bifurcation diagrams, Lyapunov exponents, correlation dimensions, Kolmogorov entropy and $C_0$ complexity. Experimental results show that the chaotic framework has better unpredictability and more complex chaotic behaviors than the classical Sine map and Chebyshev map. The results also verify the effectiveness of the theoretical analysis of the proposed chaotic framework.

In this paper, we further study a discontinuous piecewise-linear financial market model established in our previous paper. The model dynamics is driven by a two-dimensional discontinuous area-preserving map. We exploit its complex dynamics from the viewpoint of open conservative systems. We first give the classification of fixed points, and then theoretically present the existence of periodic saddle orbit and a homoclinic chaos for some parameter setting. We finally apply the Conley–Moser conditions to verify the existence of Smale horseshoe-like dynamics and chaos for another parameter setting. The result is helpful for understanding the internal change rule of the finance market.

Many definitions of chaos have appeared in the last decades and there is a concerned question if they are equivalent in some more specific spaces. Our focus will be on distributional chaos, first defined in 1994 and later subdivided into three major types (and even more subtypes). These versions of chaos are equivalent on a closed interval, but distinct in more complicated spaces. Since dendrites have much in common with the interval, we explore whether or not we can distinguish these kinds of chaos already on dendrites. At the end of the paper we will also briefly look at their correlation with other types of chaos.

This paper is concerned with a delayed predator–prey model with nonconstant death rate and constant-rate prey harvesting. We mainly study the impact of the time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively. By choosing time delay $\tau$ as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay passes some critical values. In addition, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are carried out to depict our theoretical results.

In this paper, a delayed fractional-order SIR (susceptible, infected, and removed) epidemic model with saturated incidence and treatment functions is presented. Firstly, the non-negativity and boundedness of solutions of the proposed model are proved. Next, some sufficient conditions are established to ensure the local asymptotic stability of the disease-free equilibrium point $E_0$ and the endemic equilibrium point $E_1$ for any delay. Meanwhile, global asymptotic stability of the endemic equilibrium point $E_1$ is investigated by constructing a suitable Lyapunov function. Some sufficient conditions are established for the global asymptotic stability of this endemic equilibrium point. Finally, some numerical simulations are illustrated to verify the correctness of the theoretical results.

The period-doubling sequence is one of the most well-known aperiodic $0$–$1$ sequences. In this paper, a complete description of its symbolic recurrence plot is given, and formulas for asymptotic values of basic recurrence quantifiers are derived.