Nonlinear analysis in a Lorenz-like system

In this paper we study the nonlinear dynamics of a Lorenz-like system. More precisely, we study the stability and bifurcations which occur in a new three parameter quadratic chaotic system. We also study the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters. As a consequence we show the existence of chaotic attractors when these cycles disappear.     

A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system

This paper introduces a new 3-D quadratic autonomous system, which can generate two coexisting single-wing chaotic attractors and a pair of diagonal double-wing chaotic attractors. More importantly, the system can generate a four-wing chaotic attractor with very complicated topological structures over a large range of parameters. Some basic dynamical behaviors and the compound structure of the new 3-D system are investigated. Detailed bifurcation analysis illustrates the evolution processes of the system among two coexisting sinks, two coexisting periodic orbits, two coexisting single-wing chaotic attractors, major and minor diagonal double-wing chaotic attractors, and a four-wing chaotic attractor. Poincaré-map analysis shows that the system has extremely rich dynamics. The physical existence of the four-wing chaotic attractor is verified by an electronic circuit. Finally, spectral analysis shows that the system has an extremely broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

分数阶双指数混沌系统的自适应滑模同步

研究了分数阶双指数混沌系统的自适应滑模同步问题.通过设计滑模函数和控制器,构造了平方Lyapunov函数进行稳定性分析.利用Barbalat引理证明了同步误差渐近趋于零,获得了系统取得自适应滑模同步的充分条件.数值仿真结果表明:选取适当的控制器及与滑模函数,分数阶双指数混沌系统取得自适应滑模同步.     

The reducibility of the equations of free motion of a complex mechanical system

A mechanical system consists of an unchangeable rigid body (a carrier) and a subsystem whose configuration and composition may vary with time (the motion of its elements relative to the carrier is given). The free motion of the system in a uniform gravitational field is investigated, on the assumption that there is no dynamic symmetry. Necessary and sufficient conditions are derived for the existence of two integrals, each quadratic in the components of the absolute angular velocity of the carrier. Lt is shown that the initial dynamical system can be reduced to an autonomous gyrostat system if and only if the motion has these two quadratic integrals; the explicit form of a linear transformation to the autonomous system is indicated. The explicit form of the integrals and conditions for their existence are obtained. Examples of motion with two quadratic integrals are considered.     

On quadratic differential systems with equal reflecting functions

For a two-dimensional quadratic system, we obtain necessary conditions for the existence of a triangular quadratic system with the same Mironenko reflecting function as the original system. We suggest an algorithm that permits establishing the coincidence of the reflecting functions of a quadratic nonstationary system and some stationary system.     

Complex dynamics in a linear impulsive system

The dynamical behavior of a linear impulsive system is discussed by means of both theoretical and numerical ways. This paper investigates the existence and stability of the equilibrium and period-one solution, the discontinuous jumps of eigenvalues, and the conditions for system possessing infinite period-two, period-three, and period-six solutions. By using discrete maps, the conditions of existence for Neimark–Sacker bifurcation are derived. In particular, chaotic behavior in the sense of Marotto’s definition of chaos is rigorously proven. Moreover, some detailed numerical results of the phase portraits, the periodic solutions, the bifurcation diagram, and the chaotic attractors, which are illustrated by some interesting examples, are in good agreement with the theoretical analysis.     

On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations

A system of linear differential equations with pulse action at fixed times is considered. We obtain sufficient conditions for the existence of a positive-definite quadratic form whose derivative along the solutions of differential equations and whose variation at the points of pulse action are negative-definite quadratic forms regardless of the times of pulse action.     

Stability,bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response

A discrete predator-prey system with square root functional response is presented. We study the existence and local stability analysis of the system. The conditions of existence of flip and Niemark-Sacker bifurcations in the system are derived. Furthermore, the chaotic behavior of the system in the sense of Marotto is proved. Numerical simulations are performed to show the consistence with analytical results and also to exhibit the complexity of the system. Finally, chaos control in the system is achieved via OGY feedback control method.     

二次多项式微分系统的反射函数

本文给出了二次多项式微分系统的反射函数的第一分量不依赖第二分量的充要条件，以及此时第二分量的表示式，及该系统存在周期解的条件。     

A chaotic system with only one stable equilibrium

If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is actually chaotic? Although chaos theory for three-dimensional autonomous systems has been intensively and extensively studied since the time of Lorenz in the 1960s, and the theory has become quite mature today, it seems that no one would anticipate a possibility of finding a three-dimensional autonomous quadratic chaotic system with only one stable equilibrium. The discovery of the new system, to be reported in this Letter, is indeed striking because for a three-dimensional autonomous quadratic system with a single stable node-focus equilibrium, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Although the equilibrium is changed from an unstable saddle-focus to a stable node-focus, therefore the familiar Ši’lnikov homoclinic criterion is not applicable, it is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous broad frequency spectrum, and a period-doubling route to chaos.     

Superstability and optimal multiparameter suppression of chaotic dynamics for a class of autonomous systems with quadratic nonlinearities

We obtain superstability conditions for a class of three-dimensional autonomous chaotic systems with quadratic nonlinearities and with singular points. On the basis of correction of the entries of the Jacobian matrix of the system, we develop a technique of optimal multiparameter correction, which permits one to study the problem of optimally reaching the superstable type of behavior.     

Complex dynamics of a discrete-time predator-prey system with Holling IV functional response

In this paper, a discrete-time predator-prey system with Holling-IV functional response is studied. We first classify the existence of the fixed points of the system, and further investigate their local stabilities. Then the local bifurcation theory for maps is applied to explore the variety of dynamics of the system. Sufficient conditions for the flip bifurcation and Neimark–Sacker bifurcation are provided. Numerical results demonstrate that the system may have more complex dynamical behaviors including multiple periodic orbits, quasi-periodic orbits and chaotic behavior. The maximum Lyapunov exponent and sensitivity analysis also confirm the chaotic dynamical behaviors of the system.     

Quadratic control problems

The present paper is concerned with the study of quadratic control problems on linear spaces. In particular, we are concerned with the conditions under which a quadratic criterion function is positive on certain linear spaces. This involves the elementary theory of conjugate and focal points, the existence of a conjugate system with a nonvanishing determinant, and the existence of extremal fields. The results given are in part a translation into control language of known theory for the problem of Bolza. The method used is based on the Hilbert space techniques developed earlier by the author.     

Parametric Perturbations and Dynamic System Control

The article analyzes dynamical systems with externally applied periodic perturbations in a general setting. We provide a rigorous justification of an approach that reduces such systems to autonomous systems and thus simplifies the analysis. The behavior of families of quadratic one-dimensional maps and circle maps in the presence of parametric perturbations is studied in detail. We prove the existence of periodic perturbations acting strictly on a chaotic subset that stabilize the dynamics and induce the emergence of stable cycles in initially chaotic maps. The analytical results are supplemented with numerical data. It is shown that chaos may be suppressed by a sufficiently complex periodic perturbation.     

Generation of multi-wing chaotic attractor in fractional order system

In this paper, a novel approach is proposed for generating multi-wing chaotic attractors from the fractional linear differential system via nonlinear state feedback controller equipped with a duality-symmetric multi-segment quadratic function. The main idea is to design a proper nonlinear state feedback controller by using four construction criterions from a fundamental fractional differential nominal linear system, so that the controlled fractional differential system can generate multi-wing chaotic attractors. It is the first time in the literature to report the multi-wing chaotic attractors from an uncoupled fractional differential system. Furthermore, some basic dynamical analysis and numerical simulations are also given, confirming the effectiveness of the proposed method.     

Relationships between limit cycles and algebraic invariant curves for quadratic systems

Algebraic limit cycles for quadratic systems started to be studied in 1958. Up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2, 4, 5 and 6. Here we present some new results on the limit cycles and algebraic limit cycles of quadratic systems. These results provide sometimes necessary conditions and other times sufficient conditions on the cofactor of the invariant algebraic curve for the existence or nonexistence of limit cycles or algebraic limit cycles. In particular, it follows from them that for all known examples of algebraic limit cycles for quadratic systems those cycles are unique limit cycles of the system.     

周期扰动下卫星运动系统的动力学性质

本文运用Melnikov方法对平面卫星运动系统在周期扰动下所表现出来的动力学性质进行了探讨.首先运用次谐Melnikov方法给出了卫星轨道在周期扰动下存在次谐周期轨道的条件,并进一步运用同宿.Melnikov方法证实了该系统存在Smale马蹄意义下的混沌性质.     

Existence of periodic solutions for Hamiltonian systems with super-linear and sign-changing nonlinearities

In this paper, we consider the existence of periodic solutions for the super quadratic second order Hamiltonian system, and primitive functions of nonlinearities are allowed to be sign-changing. By using some weaker conditions, our result extends and improves some existed results in the literature.     

Centers and isochronous centers of a class of quasi-analytic switching systems

In this paper, we study the integrability and linearization of a class of quadratic quasi-analytic switching systems. We improve an existing method to compute the focus values and periodic constants of quasianalytic switching systems. In particular, with our method, we demonstrate that the dynamical behaviors of quasi-analytic switching systems are more complex than those of continuous quasi-analytic systems, by showing the existence of six and seven limit cycles in the neighborhood of the origin and infinity, respectively, in a quadratic quasi-analytic switching system. Moreover, explicit conditions are obtained for classifying the centers and isochronous centers of the system.     

Decidability of Chaos for Some Families of Dynamical Systems

We show that the existence of positive Lyapounov exponents and/or SRB measures are undecidable (in the algorithmic sense) properties within some parametrized families of interesting dynamical systems: the quadratic family and Hénon maps. Because the existence of positive exponents (or SRB measures) is, in a natural way, a manifestation of chaos, these results may be understood as saying that the chaotic character of a dynamical system is undecidable. Our investigation is directly motivated by questions asked by Carleson and Smale in this direction.