International Journal of Bifurcation and Chaos最新文献

The impact of power-law memory on the dynamics of the Caputo standard fractional map is investigated in this paper. The definition of a complexity measure for the Caputo standard fractional map is introduced. This measure evaluates both the average algebraic complexity of a trajectory and the distribution of the trajectories of different types in the phase space of the system. The interplay between the small-scale spatial chaos and the large-scale spatial behavior is observed and measured during the transition of the Caputo standard fractional map from the circle map to the classical standard map. It is demonstrated that the impact of the fractional derivative on the complexity of the fractional system is not straightforward and is predetermined by the physical properties governing the dynamics of that system.

The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator’s food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles.

In this work, we present a method for the Global Asymptotic Stabilization (GAS) of an affine control chaotic Lorenz system, via admissible (bounded and regular) feedback controls, where the control bounds are given by a class of (convex) polytopes. The proposed control design method is based on the control Lyapunov function (CLF) theory introduced in [Artstein, 1983; Sontag, 1998]. Hence, we first recall, with parameters including those in [Lorenz, 1963], that these equations are point-dissipative, i.e. there is an explicit absorbing ball${ℬ}$ given by the level set of a certain Lyapunov function, $V_{\infty }(x)$. However, since the minimum point of $V_{\infty }(x)$ does not coincide with any rest point of Lorenz system, we apply a modified solution to the “uniting CLF problem” (to unify local (possibly optimal) controls with global ones, proposed in [Andrieu & Prieur, 2010]) in order to obtain a CLF $V(x)$ for the affine system with minimum at a desired equilibrium point. Finally, we achieve the GAS of “any” rest point of this system via bounded and regular feedback controls by using the proposed CLF method, which also contains the following controllers: (i) damping controls outside ${ℬ}$, so they collaborate with the beneficial stable free dynamics, and (ii) (possibly optimal) feedback controls inside ${ℬ}$ that stabilize the control system at “any” desired rest point of the (unforced) Lorenz system.

We study canard cycles and their cyclicity of a fast–slow Leslie–Gower predator–prey system with Allee effect. More specifically, we find necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its location (or their locations), and then we further completely determine its (or their) dynamics under explicit conditions. Besides, by geometric singular perturbation theory and the slow–fast normal form, we find explicit sufficient conditions to characterize singular Hopf bifurcation and canard explosion of the system. Additionally, the cyclicity of canard cycles is completely solved, and of particular interest is that we show the existence and uniqueness of a canard cycle, whose cyclicity is at most two, under corresponding precise explicit conditions.

For the quadratic reversible systems of genus one, all of their periodic orbits are higher-order algebraic curves. When they are perturbed by polynomials of degree $n$, the numbers of zeros of their Abelian integrals will change and we study the upper bounds of these numbers by using the methods of Riccati equation and Picard–Fuchs equation. We consider both the highest and lowest degrees of polynomials, and more importantly, we consider the law of polynomials and the range of values for their variables. Consequently, some laws of the polynomials are discovered and many upper bounds are obtained, and these upper bounds are sharper than the results obtained by other techniques.

In this paper, the qualitative properties of a physically extended six-dimensional Lorenz system, with additional physical terms describing rotation and density, which was proposed in [Moon et al., 2019] have been investigated. The dissipation, invariance, Lyapunov exponents, Kaplan–Yorke dimension, ultimate boundedness and global attractivity of this six-dimensional Lorenz system have been discussed in detail according to the chaotic systems theory. We find that this system exhibits chaos phenomena for a new set of parameters. It is well known that the general method for studying the bounds of a chaotic system is to construct a suitable Lyapunov-like function (or the generalized positive definite and radically unbounded Lyapunov function). However, the higher the dimension of a chaotic system, the more difficult it is to construct the Lyapunov-like function. The innovation of this paper is that we first construct the suitable Lyapunov-like function for this six-dimensional Lorenz system, and then we prove that this system is not only globally bounded for varying parameters, but it also gives a collection of global absorbing sets for this system with respect to all parameters of this system according to Lyapunov’s direct method and the optimization method. Furthermore, we obtain the rate of the trajectories going from the exterior to the global absorbing set. Some numerical simulations are presented to validate our research results. Finally, we give a direct application of the results obtained in this paper. According to the results of this paper, we can conclude that the equilibrium point $O(0,0,0,0,0,0)$ of this system is globally exponentially stable.

In this paper, we study the dynamic behaviors and chaos synchronization of the Chen system described by Caputo–Hadamard fractional derivative. First, the existence and uniqueness of a solution to the Chen system with Caputo–Hadamard derivative are proved by qualitative analysis. Further, the stability of equilibria of the considered system is analyzed with the aid of Routh–Hurwitz criteria. Meanwhile, the bifurcation condition of the Caputo–Hadamard Chen system is compared with the integer-order Chen system, where the differences between the two systems are demonstrated numerically. In the study of chaos synchronization of the drive–response Chen systems with Caputo–Hadamard derivative, two control schemes are developed: three nonlinear controllers and single linear controller. The feasibility of two control schemes is verified, and the synchronization performances of these two schemes are compared by numerical simulations. Based on this, the influence of the fractional-order on chaos synchronization performance is illustrated as well.

In this paper, we present a concise four-dimensional (4D) conservative chaotic system with a wide parameter range. Since there are no terms higher than first order, the circuit does not contain multipliers, resulting in a simple circuit implementation. The nonlinear dynamic characteristics, such as phase diagrams, equilibrium points, divergence, Poincaré cross-sections, Lyapunov exponents, bifurcation diagrams, and Lyapunov dimension, are analyzed in detail, which illustrates the conservativity. Besides, the system exhibits different offset boosting behaviors. Through offset boosting, the system can propagate along a line, convert signal polarity, control variable amplitude, generate coexisting attractors, and even induce changes in its state. Specially, we realize the transition from a single-vortex attractor to a multivortex one by some changes in the initial values. Furthermore, the Pearson correlation coefficient is used to demonstrate the higher initial value sensitivity of the system. Finally, the system is implemented through Multisim simulation and analog circuit separately, and their consistency validates the system effectively.

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.