Qualitative behavior of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres

In this article, the bounds of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres have been studied. Based on Lagrange multiplier method, the function extremum theory and the generalized positive definite and radially unbound Lyapunov functions with respect to the parameters of the system, we derive the ultimate bound and the globally exponentially attractive set for this system. The results that obtained in this article provides theory basis for chaotic synchronization, chaotic control, Hausdorff dimension and the Lyapunov dimension of chaotic attractors. © 2016 Wiley Periodicals, Inc. Complexity 21: 67–72, 2016     

Dynamical analysis and numerical simulation of a new Lorenz-type chaotic system

In this paper, a new 3D autonomous Lorenz-type chaotic system is modelled based on the condition that the system may generate chaos whereas it has only stable or non-hyperbolic equilibrium points. This system also includes some well-known Lorenz-like systems as its special cases, such as the diffusionless Lorenz system, the Burke-Shaw system and some other systems found. Although the new chaotic system is similar to other Lorenz-type systems in algebraic structure, they are topologically non-equivalent. This interesting fact motivates one to further investigate its dynamical behaviours, such as the number and the stability of equilibrium points, Hopf bifurcation and its direction, Poincaré maps, Lyapunov exponents and dissipativity, etc. Given numerical simulations not only verify the corresponding theoretically analytical results, but also demonstrate that this system possesses abundant and complex dynamical properties, which need further attention.     

Hopf bifurcation and topological horseshoe of a novel finance chaotic system

This paper deals with the existence of both Hopf bifurcation and topological horseshoe for a novel finance chaotic system. First, through rigorous mathematical analysis, we show that a Hopf bifurcation occurs at systems’ three equilibriums S_0,1,2 and Hopf bifurcation at equilibrium S₀ is non-degenerate and supercritical. Second, the computer-assisted verifications for horseshoe chaos in the system are given. Simulation results are presented to support the analysis.     

Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci

In order to further understand a complex 3D dynamical system showing strange chaotic attractors with two stable node-foci near Hopf bifurcation point, we propose nonlinear control scheme to the system and the controlled system, depending on five parameters, can exhibit codimension one, two, and three Hopf bifurcations in a much larger parameter regain. The control strategy used keeps the equilibrium structure of the chaotic system and can be applied to degenerate Hopf bifurcation at the desired location with preferred stability.     

Complex dynamics of a simple 3D autonomous chaotic system with four-wing

The present paper revisits a three dimensional (3D) autonomous chaotic system with four-wing occurring in the known literature [Nonlinear Dyn (2010) 60(3): 443--457] with the entitle ``A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems'' and is devoted to discussing its complex dynamical behaviors, mainly for its non-isolated equilibria, Hopf bifurcation, heteroclinic orbit and singularly degenerate heteroclinic cycles, etc. Firstly, the detailed distribution of its equilibrium points is formulated. Secondly, the local behaviors of its equilibria, especially the Hopf bifurcation, are studied. Thirdly, its such singular orbits as the heteroclinic orbits and singularly degenerate heteroclinic cycles are exploited. In particular, numerical simulations demonstrate that this system not only has four heteroclinic orbits to the origin and other four symmetry equilibria, but also two different kinds of infinitely many singularly degenerate heteroclinic cycles with the corresponding two-wing and four-wing chaotic attractors nearby.     

Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419], our new results fill up the gap of the estimate for the cases of 0<a<1 and 0<b<2 [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419].     

统一混沌系统的全局指数吸引集的新结果

研究了参数α∈[1/29,14/173)时,统一混沌系统的全局指数吸引集问题.通过线性变换和广义Lyapunov函数方法,给出了系统最终上界的精确估计.所得结果发展和丰富了现有混沌系统吸引集的结果,并将在混沌控制和同步中得到广泛应用.     

New estimate the bounds for the generalized Lorenz system

The bound of a chaotic system is important for chaos control, chaos synchronization, and other applications. In the present paper, the bounds of the generalized Lorenz system are studied, based on the Lyapunov function theory and the Lagrange multiplier method. We obtain a precise bound for the generalized Lorenz system. The rate of the trajectories is also obtained. Furthermore, we perform the numerical simulations. Numerical simulations are presented to show the effectiveness of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamical analysis of the hyperchaos Lorenz system

This article deals with the ultimate bound on the trajectories of the hyperchaos Lorenz system based on Lyapunov stability theory. The innovation of this article lies in that the method of constructing Lyapunov functions applied to the former chaotic systems is not applicable to this hyperchaos system, and moreover, one Lyapunov function can not estimate the bounds of this hyperchaos Lorenz system. We successfully estimate the bounds of this hyperchaos system by constructing three generalized Lyapunov functions step by step. Some computer simulations are also given to show the effectiveness of the proposed scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 440–445, 2016     

Global chaos synchronization of new chaotic system using linear active control

Chaos synchronization is a procedure where one chaotic oscillator is forced to adjust the properties of another chaotic oscillator for all future states. This research paper studies and investigates the global chaos synchronization problem of two identical chaotic systems and two non‐identical chaotic systems using the linear active control technique. Based on the Lyapunov stability theory and using the linear active control technique, the stabilizing controllers are designed for asymptotically global stability of the closed‐loop system for both identical and non‐identical synchronization. Numerical simulations and graphs are imparted to justify the efficiency and effectiveness of the proposed scheme. All simulations have been done by using mathematica 9. © 2014 Wiley Periodicals, Inc. Complexity 21: 379–386, 2015     

Dynamical analysis and numerical simulation of bloch chaotic system

In this article, some dynamics of Bloch chaotic system have been studied. Based on Lagrange multiplier method, optimization theory, and the generalized positively definite and radially unbound Lyapunov functions with respect to the parameters of the system, we derive the ultimate bound and a family of mathematical expressions of globally exponentially attractive sets for this system with respect to the parameters of system. The results obtained in this article provides theory basis for chaotic synchronization, chaotic control, Hausdorff dimension, and Lyapunov dimension of chaotic attractors of Bloch chaotic system. © 2016 Wiley Periodicals, Inc. Complexity 21: 201–206, 2016     

Zero‐zero‐Hopf bifurcation and ultimate bound estimation of a generalized Lorenz–Stenflo hyperchaotic system

This paper is devoted to the analysis of complex dynamics of a generalized Lorenz–Stenflo hyperchaotic system. First, on the local dynamics, the bifurcation of periodic solutions at the zero‐zero‐Hopf equilibrium (that is, an isolated equilibrium with double zero eigenvalues and a pair of purely imaginary eigenvalues) of this hyperchaotic system is investigated, and the sufficient conditions, which insure that two periodic solutions will bifurcate from the bifurcation point, are obtained. Furthermore, on the global dynamics, the explicit ultimate bound sets of this hyperchaotic system are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamics and bifurcation study on an extended Lorenz system

In this paper, we study dynamics and bifurcation of limit cycles in a recently developed new chaotic system, called extended Lorenz system. A complete analysis is provided for the existence of limit cycles bifurcating from Hopf critical points. The system has three equilibrium solutions: a zero one at the origin and two non-zero ones at two symmetric points. It is shown that the system can either have one limit cycle around the origin, or three limit cycles enclosing each of the two symmetric equilibria, giving a total six limit cycles. It is not possible for the system to have limit cycles simultaneously bifurcating from all the three equilibria. Simulations are given to verify the analytical predictions.     

Modelling and analysis of haemoglobin catalytic reaction kinetic system

ABSTRACT

In order to study whether haemoglobin (Hb) can replace peroxidase and has good catalytic properties. The key to exploring the characteristics of Hb peroxidase is to establish a suitable kinetic model, which is studied in this paper. First, according to the Hb catalytic reaction, a nonlinear system is established and improved. It is proved that the established system is in line with the practical significance. The stability of the original system is judged by analysing the stability of the simplified system. Then, considering the effect of time delay on Hb catalytic reaction, a nonlinear time-delay catalytic reaction system is obtained. For convenient application, the system is linearized using Taylor’s formula, and the dynamic characteristics of Hopf bifurcation are analysed. The response diagrams of three system are plotted by setting perturbation parameters, and their variations are observed to analyse the differences among them. The results show that the nonlinear time-delay system can better describe the characteristics of the catalytic reaction.     

DYNAMICAL BEHAVIORS FOR A THREE－DIMENSIONAL DIFFERENTIAL EQUATION IN CHEMICAL SYSTEM

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Chaos and chaos synchronization for a non-autonomous rotational machine systems

Chaos and chaos synchronization of the centrifugal flywheel governor system are studied in this paper. By mechanics analyzing, the dynamical equation of the centrifugal flywheel governor system is established. Because of the non-linear terms of the system, the system exhibits both regular and chaotic motions. The characteristic of chaotic attractors of the system is presented by the phase portraits and power spectra. The evolution from Hopf bifurcation to chaos is shown by the bifurcation diagrams and a series of Poincaré sections under different sets of system parameters, and the bifurcation diagrams are verified by the related Lyapunov exponent spectra. This letter addresses control for the chaos synchronization of feedback control laws in two coupled non-autonomous chaotic systems with three different coupling terms, which is demonstrated and verified by Lyapunov exponent spectra and phase portraits. Finally, numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

Hopf bifurcation analysis for a delayed predator-prey system with a prey refuge and selective harvesting

In this paper, a delayed predator-prey system with Holling type III functional response incorporating a prey refuge and selective harvesting is considered. By analyzing the corresponding characteristic equations, the conditions for the local stability and existence of Hopf bifurcation for the system are obtained, respectively. By utilizing normal form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations supporting the theoretical analysis are given.     

Steady-state and Hopf bifurcations in the Langford ODE and PDE systems

This paper is concerned with the Langford ODE and PDE systems. For the Langford ODE system, the existence of steady-state solutions is firstly obtained by Lyapunov–Schmidt method, and the stability and bifurcation direction of periodic solutions are established. Then for the Langford PDE system, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalue. Finally, by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions for the PDE system are investigated.     

A hyperchaotic system from a chaotic system with one saddle and two stable node-foci

This paper presents a 4D new hyperchaotic system which is constructed by a linear controller to a 3D new chaotic system with one saddle and two stable node-foci. Some complex dynamical behaviors such as ultimate boundedness, chaos and hyperchaos of the simple 4D autonomous system are investigated and analyzed. The corresponding bounded hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation path, analysis of power spectrum and Poincaré projections. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are rigorous derived and studied.