Dynamical properties and chaos synchronization of improved Colpitts oscillators

In this paper, the dynamics and synchronization of improved Colpitts oscillators designed to operate in ultrahigh frequency range are considered. The model is described by a continuous time four-dimensional autonomous system with an exponential nonlinearity. The system is integrated numerically and various bifurcation diagrams and corresponding graphs of largest 1D Lyapunov exponent are plotted to summarize different scenarios leading to chaos. It is found that the oscillator moves from the state of fixed point motion to chaos via the usual paths of period-doubling, intermittency and interior crisis routes when monitoring the bias (i.e. power supply) in tiny ranges. In order to promote chaos-based synchronization designs of this type of oscillators, a synchronization strategy based upon the design of a nonlinear state observer is successfully adapted. The suggested approach enables synchronization to be achieved via a scalar transmitted signal which represents a suitable feature for communication applications. Numerical simulations are performed to demonstrate the effectiveness and feasibility of the proposed technique.     

Synchronization in nonlinear oscillators with conjugate coupling

In this work, we investigate the synchronization in oscillators with conjugate coupling in which oscillators interact via dissimilar variables. The synchronous dynamics and its stability are investigated theoretically and numerically. We find that the synchronous dynamics and its stability are dependent on both coupling scheme and the coupling constant. We also find that the synchronization may be independent of the number of oscillators. Numerical demonstrations with Lorenz oscillators are provided.     

Synchronization for two coupled oscillators with inhibitory connection

In this paper we present an oscillatory neural network composed of two coupled neural oscillators with inhibitory connections. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neurons. Regarding time delays τ as the bifurcation parameter, we not only obtain the existence of Hopf bifurcations but also investigate the bifurcation direction and stability of bifurcated periodic solutions by employing normal form theory and center manifold reduction. Finally, numerical simulations are provided to illustrate the theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Synchronization of two coupled multimode oscillators with time-delayed feedback

Effects of synchronization in a system of two coupled oscillators with time-delayed feedback are investigated. Phase space of a system with time delay is infinite-dimensional. Thus, the picture of synchronization in such systems acquires many new features not inherent to finite-dimensional ones. A picture of oscillation modes in cases of identical and non-identical coupled oscillators is studied in detail. Periodical structure of amplitude death and “broadband synchronization” zones is investigated. Such a behavior occurs due to the resonances between different modes of the infinite-dimensional system with time delay.     

Synchronization of time-delay coupled pulse oscillators

We present a detailed study of the dynamics of pulse oscillators with time-delayed coupling. We get the return maps, obtain strict solutions and analyze their stability. For the case of two oscillators, a periodical structure of synchronization regions is found in parameter space, and the regions corresponding to in-phase and antiphase regimes alternate with growth of time delay. Two types of switching between in-phase and antiphase regimes are studied. We also show that for different parameters coupling delay may have synchronizing or desynchronizing effect. Another novel result is that phase locked regimes exist for arbitrary large values. The specificity of system dynamics with large delay is studied.     

Synchronization of hyperchaotic oscillators via single unidirectional chaotic-coupling

In this paper, synchronization of two hyperchaotic oscillators via a single variable’s unidirectional coupling is studied. First, the synchronizability of the coupled hyperchaotic oscillators is proved mathematically. Then, the convergence speed of this synchronization scheme is analyzed. In order to speed up the response with a relatively large coupling strength, two kinds of chaotic coupling synchronization schemes are proposed. In terms of numerical simulations and the numerical calculation of the largest conditional Lyapunov exponent, it is shown that in a given range of coupling strengths, chaotic-coupling synchronization is quicker than the typical continuous-coupling synchronization. Furthermore, A circuit realization based on the chaotic synchronization scheme is designed and Pspice circuit simulation validates the simulated hyperchaos synchronization mechanism.     

Isospectral orbifolds with different maximal isotropy orders

We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which also implies isospectrality on p-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds are quotients of Euclidean and are shown to be isospectral on functions using dimension formulas for the eigenspaces developed in [12]. In the latter type of examples the orbifolds are not isospectral on 1-forms. Along the way we also give several additional examples of isospectral orbifolds which do not have maximal isotropy groups of different size but other interesting properties. All three authors were partially supported by DFG Sonderforschungsbereich 647.     

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.     

General linear methods with external stages of different orders

The approach introduced recently by Albrecht to derive order conditions for Runge-Kutta formulas based on the theory of A-methods is also very powerful for the general linear methods. In this paper, using Albrecht's approach, we formulate the general theory of order conditions for a class of general linear methods where the components of the propagating vector of approximations to the solution have different orders. Using this theory we derive a class of diagonally implicit multistage integration methods (DIMSIMs) for which the global order is equal to the local order. We also derive a class of general linear methods with two nodal approximations of different orders which facilitate local error estimation. Our theory also applies to the class of two-step Runge-Kutta introduced recently by Jackiewicz and Tracogna.The work of the first author was supported by the National Science Foundation under grant NSF DMS-9208048. The work of the second author was supported by the Italian Consiglio Nazionale delle Richerche.     

Projective synchronization of different fractional-order chaotic systems with non-identical orders

This paper investigates the projective synchronization (PS) of different fractional order chaotic systems while the derivative orders of the states in drive and response systems are unequal. Based on some essential properties on fractional calculus and the stability theorems of fractional-order systems, we propose a general method to achieve the PS in such cases. The fractional operators are introduced into the controller to transform the problem into synchronization problem between chaotic systems with identical orders, and the nonlinear feedback controller is proposed based on the concept of active control technique. The method is both theoretically rigorous and practically feasible. We present two examples that illustrate the effectiveness and applications of the method, which include the PS between two 3-D commensurate fractional-order chaotic systems and the PS between two 4-D fractional-order hyperchaotic systems with incommensurate and commensurate orders, respectively. Abundant numerical simulations are given which agree well with the analytical results. Our investigations show that PS can also be achieved between different chaotic systems with non-identical orders. We have further reviewed and compared some relevant methods on this topic reported in several recent papers. A discussion on the physical implementation of the proposed method is also presented in this paper.     

Synchronization transition in degenerate optical parametric oscillators induced by nonlinear coupling

In this paper, the synchronization of certain degenerate optical parametric oscillators is investigated in detail. Complete and/or partial synchronization can be reached when linear controller, constructed by the real part or imaginary part of the subharmonic mode, is imposed on the chaotic degenerate optical parametric oscillators with appropriate coupling intensity. The Lyapunov exponents under different coupling coefficients are calculated to find the critical condition for complete synchronization. Transition from complete synchronization to partial synchronization is observed when nonlinear coupling is introduced into the two chaotic oscillators. It is found that synchronization of chaotic oscillators keeps robust when the intensity of the nonlinear coupling is less than the intensity of the linear coupling; the complete synchronization state is destructed and transient period for partial synchronization is in certain delay when the intensity of the nonlinear coupling is beyond the intensity of the linear coupling.     

On the synchronization of different chaotic oscillators

The synchronization of two different chaotic oscillators is studied, based on an open-loop control – the entrainment control. We consider two types of synchronization: complete synchronization and effectively complete synchronization. The sufficient conditions that two different systems can be synchronized by this method is discussed. Furthermore, a hierarchical idea to synchronize multiple chaotic subsystems is proposed.     

Multiple positive solutions of singularly perturbed differential systems with different orders

In this paper, we consider the multiplicity of positive solutions for a class of singular higher-order perturbed differential systems with different orders. By employing a well-known fixed point theorem, some new existence results are given under the case where nonlinearity can be sign changing.     

Inequality between moduli of smoothness of different orders in different metrics

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 153–155, August, 1991.     

Stochastic resonance with different periodic forces in overdamped two coupled anharmonic oscillators

We study the stochastic resonance phenomenon in the overdamped two coupled anharmonic oscillators with Gaussian noise and driven by different external periodic forces. We consider (i) sine, (ii) square, (iii) symmetric saw-tooth, (iv) asymmetric saw-tooth, (v) modulus of sine and (vi) rectified sinusoidal forces. The external periodic forces and Gaussian noise term are added to one of the two state variables of the system. The effect of each force is studied separately. In the absence of noise term, when the amplitude f of the applied periodic force is varied cross-well motion is realized above a critical value (f_c) of f. This is found for all the forces except the modulus of sine and rectified sinusoidal forces. For fixed values of angular frequency ω of the periodic forces, f_c is minimum for square wave and maximum for asymmetric saw-tooth wave. f_c is found to scale as Ae^0.75ω + B where A and B are constants. Stochastic resonance is observed in the presence of noise and periodic forces. The effect of different forces is compared. The stochastic resonance behaviour is quantized using power spectrum, signal-to-noise ratio, mean residence time and distribution of normalized residence times. The logarithmic plot of mean residence time τ_MR against 1/(D − D_c) where D is the intensity of the noise and D_c is the value of D at which cross-well motion is initiated shows a sharp knee-like structure for all the forces. Signal-to-noise ratio is found to be maximum at the noise intensity D = D_max at which mean residence time is half of the period of the driving force for the forces such as sine, square, symmetric saw-tooth and asymmetric saw-tooth waves. With modulus of sine wave and rectified sine wave, the SNR peaks at a value of D for which sum of τ_MR in two wells of the potential of the system is half of the period of the driving force. For the chosen values of f and ω, signal-to-noise ratio is found to be maximum for square wave while it is minimum for modulus of sine and rectified sinusoidal waves. The values of D_c at which cross-well behaviour is initiated and D_max are found to depend on the shape of the periodic forces.     

On the relations between infinitesimal bendings of different orders

It is established that if a surface has only one nontrivial independent field of first order infinitesimal bendings, then rigidity of order n 3 implies rigidity of any order m n, in particular, nondeformability that is analytic in parameter. All the considerations are carried out in class C ¹.In passing, a new approach to the determination of infinitesimal bendings of higher order is given. Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 118–124, 1992.     

Synchronization of pendula rotating in different directions

We study the synchronization of a number of pendula mounted on a horizontal beam which can roll on the parallel surface. Under the driving moment, the pendula rotate in different directions: one of them rotates counterclockwise, the rest rotate clockwise. It has been shown that after a transient different types of phase synchronization between pendula can be observed, despite opposite directions of rotations.