Adaptive control of chaotic systems based on a single layer neural network

This Letter presents an adaptive neural network control method for the chaos control problem. Based on a single layer neural network, the dynamic about the unstable fixed period point of the chaotic system can be adaptively identified without detailed information about the chaotic system. And the controlled chaotic system can be stabilized on the unstable fixed period orbit. Simulation results of Henon map and Lorenz system verify the effectiveness of the proposed control method.     

Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control

A new kind of generalized synchronization of two chaotic systems with uncertain parameters is proposed. Based on a pragmatical asymptotical stability theorem and an assumption of equal probability for ergodic initial conditions, an adaptive control law is derived so that it can be proved strictly that the common null solution of error dynamics and of parameter dynamics is actually asymptotically stable, i.e. these two identical systems are in generalized synchronization and the estimated parameters approach the uncertain values. It is called pragmatical generalized synchronization. Finally, two numerical examples are studied for two Quantum-CNN oscillator chaotic systems to show the effectiveness of the proposed generalized synchronization strategy with a double Duffing chaotic system as a goal system.     

Locking bandwidth and birefringence effects for polarized optical injection in vertical-cavity surface-emitting lasers

The aim of this paper is to investigate the effects of external optical injection taking account of polarization and electron spin properties in vertical-cavity surface-emitting lasers (VCSELs). Using external polarized injection we seek the locked phases and amplitudes of specific polarized fields in terms of injection level and frequency detuning, taking account of two kinds of distinguishable carrier density (spin-up and spin-down). For the conventional form of optical injection without taking account of spin-polarized fields there are three fundamental equations describing the carrier density, field amplitude and phase. However, by using the spin flip model (SFM), the combined effect of polarized fields along two perpendicular crystal axes and electron spin properties results in six equations. We analyse the conditions for stable locking and also the influence of birefringence effects on the stability map of detuning versus optical injection for both cases of injection polarized parallel and perpendicular to the lasing mode of the solitary VCSEL. For given values of pumping and spin relaxation rate there is a minimum birefringence rate for orthogonal injection. Above this value three regions of elliptical polarization are found in the stability map, namely “quasi-stability” (QS), “coupled limit cycle” (CLC) and “coupled chaos” (CC). The three regions of linear polarization, namely chaos, limit cycle and stability, are reduced in area compared to the case of parallel injection. For orthogonal injection it is found that increased birefringence or reduced spin relaxation rate causes the stable locking region to begin at higher injected power and frequency detuning.     

利用单模激光Lorenz系统实现混沌反控制

利用Lyapunov函数方法，对混沌反控制问题进行了研究.以单模激光Lorenz系统和描述心脏搏动的Bonhoeffer-Van der Pol系统为例，设计了一种控制器，成功地使Bonhoeffer-Van der Pol系统混沌化.给出了控制器的具体设计方案以及单模激光Lorenz系统与Bonhoeffer-Van der Pol系统状态之间误差系统的结构.仿真结果表明，在控制器的作用下，Bonhoeffer-Van der Pol系统所有状态变量严格地跟踪了单模激光Lorenz系统的混沌轨迹，对应的相空间中Bonhoeffer-Van der Pol系统的轨迹也由极限环转变为与单模激光Lorenz系统的轨迹完全相同的混沌吸引子，Bonhoeffer-Van der Pol系统严格地跟踪了单模激光Lorenz系统混沌的动态行为.     

Dynamical behaviors of the brusselator system with impulsive input

Responses of dynamic system to pulse perturbations were investigated theoretically and experimentally. The model used in this paper has been proved dissipative by impulsive and dynamic theory. Complex phenomena such as limit cycles, periodic solutions, and chaos were numerically demonstrated.     

Productive public expenditures, expectation formations and nonlinear dynamics

In this paper, we investigate the dynamic properties of an overlapping generations’ model with capital accumulation and publicly funded inventions under three different expectations: perfect foresight, myopic expectations and adaptive expectations. We show that considering productive public expenditures in the model will increase the dimension of the dynamical system. To study the dynamic behavior of a high-dimensional dynamical system, we focus on the case when the elasticity of publicly funded invention to output is small and approximate the system by using a one-dimensional dynamical system. This approximation method provides an efficient way to rigorously prove the existence of chaos in high-dimensional dynamical systems. We show that when agents are perfectly foresighted, there exists a unique, nontrivial steady state which is a global attractor. Cycles or even chaos may occur under myopic and adaptive expectations when the inter-temporal elasticity of substitution of consumption is large enough. Furthermore, we find that the impact of fiscal policy is sensible to the expectation formation.     

Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control

The stability and bifurcation of a van der Pol-Duffing oscillator with the delay feedback are investigated, in which the strength of feedback control is a nonlinear function of delay. A geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and Hopf bifurcations. The Hopf bifurcation curves and multi-stable regions are obtained as two parameters vary. Some weak resonant and non-resonant double Hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the “death island” regions simultaneously. By applying the center manifold theory, the normal forms near the double Hopf bifurcation points, as well as classifications of local dynamics are analyzed. Furthermore, some quasi-periodic and chaotic motions are verified in both theoretical and numerical ways.     

Dynamics of a small vibro-impact pile driver

A mathematical model is developed to study periodic-impact motions and bifurcations in dynamics of a small vibro-impact pile driver. Dynamics of the small vibro-impact pile driver can be analyzed by means of a three-dimensional map, which describes free flight and sticking solutions of the vibro-impact system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the Poincaré map. The piecewise property is caused by the transitions of free flight and sticking motions of the driver and the pile immediately after the impact, and the singularity of map is generated via the grazing contact of the driver and the pile immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularities and parameter variation on the performance of the vibro-impact pile driver is analyzed. The global bifurcation diagrams for the impact velocity of the driver versus the forcing frequency are plotted to predict much of the qualitative behavior of the actual physical system, which enable the practicing engineer to select excitation frequency ranges in which stable period one single-impact response can be expected to occur, and to predict the larger impact velocity and shorter impact period of such response.     

Unknown input observer based chaotic secure communication

Based on the idea of unknown input observers (UIOs), this Letter considers the problem of chaos secure communication within the master-slave configuration. The unknown constant disturbance of parameters in chaotic systems and the unknown input in chaotic signals can be regarded as messages to be recovered in the slave end. If certain conditions are satisfied, we can recover all the unknown messages by designing the UIO based slave. Theoretical analysis and simulation results fully verify the effectiveness of the proposed scheme.     

The trajectory scaling function for period doubling bifurcations in flows

We extend the trajectory scaling function as defined for maps to flows whose dynamics is governed by ordinary differential equations. The results are obtained for the Duffing oscillator and are expected to be the same for other dissipative flows as well.