Dynamical behaviors of a system with switches between the Rssler oscillator and Chua circuits

The behaviors of a system that alternates between the R¨ossler oscillator and Chua’s circuit is investigated to explore the influence of the switches on the dynamical evolution.Switches related to the state variables are introduced,upon which a typical switching dynamical model is established.Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points,which divide the parameters into several regions corresponding to different types of attractors.The dynamics behave typically in period orbits with the variation of the parameters.The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement.The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches.Furthermore,period-decreasing sequences have been obtained,which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.     

Dynamical behaviors of a system with switches between the Rssler oscillator and Chua circuits

The behaviors of a system that alternates between the R¨ossler oscillator and Chua＇s circuit is investigated to explore the influence of the switches on the dynamical evolution.Switches related to the state variables are introduced,upon which a typical switching dynamical model is established.Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points,which divide the parameters into several regions corresponding to different types of attractors.The dynamics behave typically in period orbits with the variation of the parameters.The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement.The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches.Furthermore,period-decreasing sequences have been obtained,which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.     

Complete synchronization of double-delayed Rssler systems with uncertain parameters

In this paper,we investigate complete synchronization of double-delayed Rssler systems with uncertain parameters as the master system is in chaotic synchronization.The uncertain parameters can be nonlinearly expressed in the system.The analysis and proof are given by means of the Lyapunov stability theorem.Based on theoretical analysis,some sufficient conditions of complete synchronization are proved.In order to validate the proposed scheme,numerical simulations are performed and the numerical results show that our scheme is very effective.     

Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems

We consider the energy dependent Schrödinger operator , which we have previously shown to be associated with multi-Hamiltonian structures [2]. In this paper we use an unusual form of the Lax approach to derive by asingle construction the time evolutions of the eigenfunctions of , the associated Hamiltonian operators and the Hamiltonian functionals. We then generalise the well known factorisation of standard Lax operators to the case of energy-dependent operators. The simple product of linear factors is replaced by a -dependent quadratic form. We thus generalise the resulting construction of Miura maps and modified equations. We show that for some of our systems there exists a sequence ofN such modifications, ther ^th modification possessing (N–r+1) Hamiltonian structures.On leave of absence from Institute of Theoretical Physics, Warsaw University, Hoza 69, PL-00-681 Warsaw, Poland (present address)     

Establishing generalized synchronization in Rössler oscillator networks

The processes of establishing of generalized chaotic synchronization in a network of mutually coupled continuous-time systems are investigated. The nature of interaction between the network elements in transitioning from synchronous to asynchronous behavior while increasing the communication parameter is studied. A synchronization regime, the nearest neighbors method, and calculations of the spectrum of Lyapunov exponents are used to clarify features of the interaction between network elements and the occurrence of a generalized chaotic.     

Mittag–Leffler synchronization of fractional-order uncertain chaotic systems

This paper deals with the synchronization of fractional-order chaotic systems with unknown parameters and unknown disturbances.An adaptive control scheme combined with fractional-order update laws is proposed.The asymptotic stability of the error system is proved in the sense of generalized Mittag–Leffler stability.The two fractional-order chaotic systems can be synchronized in the presence of model uncertainties and additive disturbances.Finally these new developments are illustrated in examples and numerical simulations are provided to demonstrate the effectiveness of the proposed control scheme.     

Stochastic period-doubling bifurcation analysis of a R?ssler system with a bounded random parameter

This paper aims to study the stochastic period-doubling bifurcation of the three-dimensional R?ssler system with an arch-like bounded random parameter. First, we transform the stochastic R?ssler system into its equivalent deterministic one in the sense of minimal residual error by the Chebyshev polynomial approximation method. Then, we explore the dynamical behaviour of the stochastic R?ssler system through its equivalent deterministic system by numerical simulations. The numerical results show that some stochastic period-doubling bifurcation, akin to the conventional one in the deterministic case, may also appear in the stochastic R?ssler system. In addition, we also examine the influence of the random parameter intensity on bifurcation phenomena in the stochastic R?ssler system.     

Stochastic period-doubling bifurcation analysis of a Rssler system with a bounded random parameter

This paper aims to study the stochastic period-doubling bifurcation of the three-dimensional Rssler system with an arch-like bounded random parameter. First, we transform the stochastic Rssler system into its equivalent deterministic one in the sense of minimal residual error by the Chebyshev polynomial approximation method. Then, we explore the dynamical behaviour of the stochastic Rssler system through its equivalent deterministic system by numerical simulations. The numerical results show that some stochastic period-doubling bifurcation, akin to the conventional one in the deterministic case, may also appear in the stochastic Rssler system. In addition, we also examine the influence of the random parameter intensity on bifurcation phenomena in the stochastic Rssler system.     

Function projective synchronization of identical and non-identical modified finance and Shimizu–Morioka systems

In this paper, function projective synchronizations (FPS) of identical and non-identical modified finance systems (MFS) and Shimizu?CMorioka system (S-MS) are studied via active control technique. The technique is applied to construct a response system which synchronizes with a given drive system for a desired function relation between identical MFS, identical S-MS and between MFS and S-MS. The results are validated via numerical simulations.     

Modeling and dynamics analysis of the fractional-order Buck—Boost converter in continuous conduction mode

In this paper,the fractional-order mathematical model and the fractional-order state-space averaging model of the Buck-Boost converter in continuous conduction mode(CCM) are established based on the fractional calculus and the Adomian decomposition method.Some dynamical properties of the current-mode controlled fractional-order Buck-Boost converter are analysed.The simulation is accomplished by using SIMULINK.Numerical simulations are presented to verify the analytical results and we find that bifurcation points will be moved backward as α and β vary.At the same time,the simulation results show that the converter goes through different routes to chaos.     

A new kind of nonlinear fractional-order chaotic phenomenon in coupled systems： coexistence of anti-phase and complete synchronization

In this paper, we have found a kind of interesting nonlinear phenomenon hybrid synchronization in linearly coupled fractional-order chaotic systems. This new synchronization mechanism, i.e., part of state variables are anti- phase synchronized and part completely synchronized, can be achieved using a single linear controller with only one drive variable. Based on the stability theory of the fractional-order system, we investigated the possible existence of this new synchronization mechanism. Moreover, a helpful theorem, serving as a determinant for the gain of the controller, is also presented. Solutions of coupled systems are obtained numerically by an improved Adams Bashforth-Moulton algorithm. To support our theoretical analysis, simulation results are given.     

Orthogonal transforms of the Lorenz- and Rössler-equation

It is shown that none of the proper or improper orthogonal transformations transforms the Lorenz-equation into a kinetic equation, i.e. into an equation representing reasonable chemistry. It is also shown that none of the proper orthogonal transformations transforms a model by Rössler into a kinetic model either. The importance of the presence of negative cross-effects is hereby emphasized.     

Mössbauer and X-ray phase analysis of carburized steel

⁵⁷Fe backscattering Mössbauer phase analysis of the ASTM A295-70 type chromium bearing steel gas was performed for samples with varying phase composition achieved by gas carburization up to 3 wt% C followed by standard hardening, i.e. austenitizing and quenching procedures. Variations of contents of principal metallographic phases (alloyed martensite, austenite, magnetic and non-magnetic carbides) were determined and compared with reported X-ray data. Results are discussed in connection with the composition and heat treatment. Good mutual agreement of results was found, the Mössbauer phase analysis being superior for distinguishing magnetic and non-magnetic carbides, for finding the dependence on composition and also for detecting alloying element segregation. On the other hand, the Mössbauer method of determination of a small content of alloyed retained austenite suffers from the small difference between its satellites and non-magnetic carbides.     

Stability analysis of two coupled Lorenz lasers and the coupling-induced periodic →chaotic transition

We have performed a linear stability analysis of two Lorenz lasers coupled by their electric fields and have shown that the bad cavity condition becomes a function of coupling and that a good cavity instability may occur if the injected fields are inverted before injection. In addition, we show that the symmetrically coupled Lorenz system is isomorphic to the original Lorenz system with new parameters. The stability analysis also predicts a lowering of the second laser threshold with coupling for both the chaotic and self-pulsing regimes. Numerical integration of the equations is in agreement with these predictions and has revealed a coupling induced transition from self-pulsing to chaotic behavior. The classification of the behavior of the coupled system in the parameter space of the coupling constants has been investigated and shows that the results of symmetric coupling allow enough of a margin for an experimental test of the theory. This would allow experimentalists to observe the actual Lorenz instability at excitations as low as 4–5 times above threshold.     

Statistical energy analysis of similarly coupled systems

1 IntroductionStatistical Energy Analysis (SEA)I'] is a powerful method fOr dealing with dynamic prolylems of a complex structu-ral or acoustical system. The development of SEA for non--con-ser\u-tively coupled systems1'], as well as SEA of coupled systems under correlative excitationsI3],makes the method more appllcable and practical in the analysis and control of structuralnoise/vibration. In a corn-mon sense, the consertutive coup1ing and the non--consertutive oneare the only two coup…     

General existence conditions of homoclinic trajectories in dissipative systems. Lorenz,Shimizu–Morioka,Lu and Chen systems

The general method for proving the existence of homoclinic trajectories in dissipative systems is developed. The applications of this method to Lorenz-like systems: Lorenz, Shimizu–Morioka, Lu and Chen systems are demonstrated. A criterion for the existence of a homoclinic trajectory within a given family of differential equations (Fishing principle) is presented. New numerical algorithm for the approximation of a homoclinic point in parameters space is constructed. The comparison with Kaplan–Yorke and Shilnikov results is made.