A new four-dimensional energy resources system and its linear feedback control

This paper reports a new four-dimensional energy resources chaotic system. The system is obtained by adding a new variable to a three-dimensional energy resource demand–supply system established for two regions of China. The dynamics behavior of the system will be analyzed by means of Lyapunov exponents and bifurcation diagrams. Linear feedback control methods are used to suppress chaos to unstable equilibrium or unstable periodic orbits. Numerical simulations are presented to show these results.     

Feedback control and adaptive control of the energy resource chaotic system

In this paper, the problem of control for the energy resource chaotic system is considered. Two different method of control, feedback control (include linear feedback control, non-autonomous feedback control) and adaptive control methods are used to suppress chaos to unstable equilibrium or unstable periodic orbits. The Routh–Hurwitz criteria and Lyapunov direct method are used to study the conditions of the asymptotic stability of the steady states of the controlled system. The designed adaptive controller is robust with respect to certain class of disturbances in the energy resource chaotic system. Numerical simulations are presented to show these results.     

The stability of attractors for non-autonomous perturbations of gradient-like systems

We study the stability of attractors under non-autonomous perturbations that are uniformly small in time. While in general the pullback attractors for the non-autonomous problems converge towards the autonomous attractor only in the Hausdorff semi-distance (upper semicontinuity), the assumption that the autonomous attractor has a ‘gradient-like’ structure (the union of the unstable manifolds of a finite number of hyperbolic equilibria) implies convergence (i.e. also lower semicontinuity) provided that the local unstable manifolds perturb continuously.We go further when the underlying autonomous system is itself gradient-like, and show that all trajectories converge to one of the hyperbolic trajectories as t→∞. In finite-dimensional systems, in which we can reverse time and apply similar arguments to deduce that all bounded orbits converge to a hyperbolic trajectory as t→−∞, this implies that the ‘gradient-like’ structure of the attractor is also preserved under small non-autonomous perturbations: the pullback attractor is given as the union of the unstable manifolds of a finite number of hyperbolic trajectories.     

A new hyperchaotic system and its circuit implementation

In this paper, a new hyperchaotic system is presented by adding a nonlinear controller to the three-dimensional autonomous chaotic system. The generated hyperchaotic system undergoes hyperchaos, chaos, and some different periodic orbits with control parameters changed. The complex dynamic behaviors are verified by means of Lyapunov exponent spectrum, bifurcation analysis, phase portraits and circuit realization. The Multisim results of the hyperchaotic circuit were well agreed with the simulation results.     

Computation of bifurcation manifolds of linearly independent homoclinic orbits

Regarding the small perturbation as a parameter in an appropriate space of functions, we can discuss co-existence of homoclinic orbits for non-autonomous perturbations of an autonomous system in Rn and describe conditions of parameters for such degenerate homoclinic bifurcations with some bifurcation manifolds of infinite dimension. Since those manifolds determine the relation among parameters for such bifurcations, in this paper we give an algorithm to compute approximately those manifolds and concretely obtain their first order approximates.     

Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.     

A hyperchaotic system from the Rabinovich system

This paper presents a new 4D hyperchaotic system which is constructed by a linear controller to the 3D Rabinovich chaotic system. Some complex dynamical behaviors such as boundedness, chaos and hyperchaos of the 4D autonomous system are investigated and analyzed. A theoretical and numerical study indicates that chaos and hyperchaos are produced with the help of a Liénard-like oscillatory motion around a hypersaddle stationary point at the origin. The corresponding bounded hyperchaotic and chaotic attractors are first numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation path and Poincaré projections. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are rigorously derived and studied.     

Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation

A system with more than one positive Lyapunov exponent can be classified as a hyperchaotic system. In this study, a sinusoidal perturbation was designed for generating hyperchaos from the Chen–Lee chaotic system. The hyperchaos was identified by the existence of two positive Lyapunov exponents and bifurcation diagrams. The system is hyperchaotic in several different regions of the parameters c, ε, and ω. It was found that this method not only can enhance or suppress chaotic behavior, but also induces chaos in non-chaotic parameter ranges. In addition, two interesting dynamical behaviors, Hopf bifurcation and intermittency, were also found in this study.     

Suppression and generation of chaos for a three-dimensional autonomous system using parametric perturbations

In this paper, we attempt to suppress or generate chaos in the newly presented Lü system using parametric perturbation. We find that this method not only suppresses chaotic behavior, but also induces chaos in non-chaotic parameter ranges. When we add the small sinusoidal perturbations, the system becomes four-dimension. From the calculation of Lyapunov exponents, we discover hyperchaos in the perturbed system, which has not yet been reported before.     

江苏-西部能源需求-供给模型及其动力学分析

根据江苏省能源需求与西部能源供给及江苏省能源进口量之间相互支持、相互制约的复杂关系建立了江苏-西部能源需求-供给系统．分析了系统的稳态、周期、分叉、混沌等若干动力学行为．通过调节参数可以使系统处于稳定发展或周期震荡区域中．基于Silnikov定理,应用待定系数法,求出了能源系统的同宿轨的表达式．证明了在两个平衡点处均有Silnikov型的同宿轨．根据Silnikov定理确保了能源系统有Smale马蹄和马蹄形混沌．     

A new hyperchaotic system from the Lü system and its control

This paper presents a 4D new hyperchaotic system which is constructed by a linear controller to a 3D Lü system. Some complex dynamical behaviors such as Hopf bifurcation, chaos and hyperchaos of the simple 4D autonomous system are investigated and analyzed. The corresponding hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation path, analysis of power spectrum and Poincaré projections. Furthermore, the design is illustrated with both simulations and experiments. Finally, the control problem of a new hyperchaotic system is investigated using negative feedback control. Ordinary feedback control, dislocated feedback control and speed feedback control are used to suppress hyperchaos to an unstable equilibrium. Numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.     

Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach

We are concerned with non-autonomous radially symmetric systems with a singularity, which are T-periodic in time. By the use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T. Precise estimates are then given in the case of Keplerian-like systems, showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system.     

A reduction principle for asymptotically triangular differential systems

Problems of the stability of non-linear non-autonomous systems of differential equations with a special class of asymptotically vanishing perturbations are considered. The problem of reducing a problem on the stability of the equilibrium of a perturbed system to a problem on stability with respect to a non-linear approximation system which has a triangular form is solved. Applications of the results of the investigations to mechanical systems with a variable mass and time-varying equations of the constraints are presented.     

Stability analysis for a class of functional differential equations and applications

The problem of Lyapunov stability for functional differential equations in Hilbert spaces is studied. The system to be considered is non-autonomous and the delay is time-varying. Known results on this problem are based on the Gronwall inequality yielding relative conservative bounds on nonlinear perturbations. In this paper, using more general Lyapunov-Krasovskii functional, neither model variable transformation nor bounding restriction on nonlinear perturbations is required to obtain improved conditions for the global exponential stability of the system. The conditions given in terms of the solution of standard Riccati differential equations allow to compute simultaneously the two bounds that characterize the stability rate of the solution. The proposed method can be easily applied to some control problems of nonlinear non-autonomous control time-delay systems.     

Bifurcation of periodic orbits in a class of planar Filippov systems

In this paper we discuss the perturbations of a general planar Filippov system with exactly one switching line. When the system has a limit cycle, we give a condition for its persistence; when the system has an annulus of periodic orbits, we give a condition under which limit cycles are bifurcated from the annulus. We also further discuss the stability and bifurcations of a nonhyperbolic limit cycle. When the system has an annulus of periodic orbits, we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching.     

Asymptotic behaviour of reaction–diffusion equations with non-damped impulsive effects

In this paper we consider non-autonomous reaction–diffusion systems with impulsive effects at fixed moments of time from the point of view of the theory of global attractors. For a translation-compact nonlinear term which does not provide the uniqueness of the Cauchy problem, and for different classes of non-damped multivalued impulse perturbations, we construct a multivalued non-autonomous dynamical system and prove for it the existence of a compact global attractor.     

EXISTENCE OF HOMOCLINIC SOLUTIONS TO NONAUTONOMOUS SECOND-ORDER p-LAPLACIAN SYSTEM WITH A COERCIVE POTENTIAL

In this paper, we investigate a non-autonomous second-order p-Laplacian system.Based on critical point theory, we discuss the existence of homoclinic orbits of the system.     

On averaging,reduction, and symmetry in hamiltonian systems

The existence of periodic orbits for Hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged Hamiltonian on an associated “reduced space.” Alternatively, in classical (kinetic plus potential energy) Hamiltonians the existence of such orbits can often be established by elementary geometrical arguments. The present paper unifies the two approaches by exploiting discrete symmetries, including reversing diffeomorphisms, that occur in a given system. The symmetries are used to locate the periodic orbits in the averaged Hamiltonian, and thence in the original Hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries. In applications to the Hénon-Heiles Hamiltonian, it is illustrated how “higher order” averaging can sometimes be used to overcome degeneracies encountered at first order.     

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.