Journal of Chaos The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. On Matrix Projective Synchronization and Inverse Matrix Projective Synchronization for Different and Identical Dimensional Discrete-Time Chaotic Systems Sun, 10 Jan 2016 09:04:12 +0000 The problem of matrix projective synchronization (MPS) in discrete-time chaotic systems is investigated, and a new type of discrete chaos synchronization called inverse matrix projective synchronization (IMPS) is introduced. Sufficient conditions are derived for achieving MPS and IMPS between chaotic dynamical systems in discrete-time of different and identical dimensions. Based on new control schemes, Lyapunov stability theory, and stability theory of linear dynamical systems in discrete-time, some synchronization criteria are obtained. Numerical examples and simulations are used to illustrate the use of the proposed schemes. Adel Ouannas and Raghib Abu-Saris Copyright © 2016 Adel Ouannas and Raghib Abu-Saris. All rights reserved. Complex Dynamical Behavior of a Two-Stage Colpitts Oscillator with Magnetically Coupled Inductors Tue, 21 Oct 2014 06:27:10 +0000 A five-dimensional (5D) controlled two-stage Colpitts oscillator is introduced and analyzed. This new electronic oscillator is constructed by considering the well-known two-stage Colpitts oscillator with two further elements (coupled inductors and variable resistor). In contrast to current approaches based on piecewise linear (PWL) model, we propose a smooth mathematical model (with exponential nonlinearity) to investigate the dynamics of the oscillator. Several issues, such as the basic dynamical behaviour, bifurcation diagrams, Lyapunov exponents, and frequency spectra of the oscillator, are investigated theoretically and numerically by varying a single control resistor. It is found that the oscillator moves from the state of fixed point motion to chaos via the usual paths of period-doubling and interior crisis routes as the single control resistor is monitored. Furthermore, an experimental study of controlled Colpitts oscillator is carried out. An appropriate electronic circuit is proposed for the investigations of the complex dynamics behaviour of the system. A very good qualitative agreement is obtained between the theoretical/numerical and experimental results. V. Kamdoum Tamba, H. B. Fotsin, J. Kengne, F. Kapche Tagne, and P. K. Talla Copyright © 2014 V. Kamdoum Tamba et al. All rights reserved. Transport Catastrophe Analysis as an Alternative to a Monofractal Description: Theory and Application to Financial Crisis Time Series Sun, 14 Sep 2014 00:00:00 +0000 The goal of this investigation was to overcome limitations of a persistency analysis, introduced by Benoit Mandelbrot for monofractal Brownian processes: nondifferentiability, Brownian nature of process, and a linear memory measure. We have extended a sense of a Hurst factor by consideration of a phase diffusion power law. It was shown that precatastrophic stabilization as an indicator of bifurcation leads to a new minimum of momentary phase diffusion, while bifurcation causes an increase of the momentary transport. An efficiency of a diffusive analysis has been experimentally compared to the Reynolds stability model application. An extended Reynolds parameter has been introduced as an indicator of phase transition. A combination of diffusive and Reynolds analyses has been applied for a description of a time series of Dow Jones Industrial weekly prices for the world financial crisis of 2007–2009. Diffusive and Reynolds parameters showed extreme values in October 2008 when a mortgage crisis was fixed. A combined R/D description allowed distinguishing of market evolution short-memory and long-memory shifts. It was stated that a systematic large scale failure of a financial system has begun in October 2008 and started fading in February 2009. Sergey A. Kamenshchikov Copyright © 2014 Sergey A. Kamenshchikov. All rights reserved. 2D Chaos in the Interaction of Inflation and Unemployment: Moving Averages and the Modeling of High Frequency Macrodynamics Wed, 18 Jun 2014 07:08:25 +0000 The paper argues that applicable macro is high frequency macro and the data generating process is therefore to be modeled in continuous time. It exemplifies this with a misuse of a 2D period model of monetarist type which becomes extremely overshooting, allowing for routes to “chaos,” when iterated at low frequencies. Instead of such low frequency procedures, we augment the model by a Keynesian feedback chain (the real rate of interest channel) to introduce local instability into the model. We also introduce heterogeneous opinion dynamics into it. The implied 4D dynamics are made bounded thereby, but seem to allow only complex limit cycles, with no transition towards strange attractors anymore. Peter Flaschel and Christian Proaño Copyright © 2014 Peter Flaschel and Christian Proaño. All rights reserved. Bifurcation and Feedback Control of an Exploited Prey-Predator System Tue, 22 Apr 2014 00:00:00 +0000 This paper makes an attempt to highlight a differential algebraic model in order to investigate the dynamical behavior of a prey-predator system due to the variation of economic interest of harvesting. In this regard, it is observed that the model exhibits a singularity induced bifurcation when economic profit is zero. For the purpose of stabilizing the proposed model at the positive equilibrium, a state feedback controller is therefore designed. Finally, some numerical simulations are carried out to show the consistency with theoretical analysis and to illustrate the effectiveness of the proposed controller. Uttam Das Copyright © 2014 Uttam Das. All rights reserved. Clustering and Uncertainty in Perfect Chaos Systems Wed, 26 Mar 2014 09:18:32 +0000 The goal of this investigation was to derive strictly new properties of chaotic systems and their mutual relations. The generalized Fokker-Planck equation with a nonstationary diffusion has been derived and used for chaos analysis. An anomalous transport turned out to be natural property of this equation. A nonlinear dispersion of the considered motion allowed us to find a principal consequence: a chaotic system with uniform dynamic properties tends to instable clustering. Small fluctuations of particles density increase by time and form attractors and stochastic islands even if the initial transport properties have uniform distribution. It was shown that an instability of phase trajectories leads to the nonlinear dispersion law and consequently to a space instability. A fixed boundary system was considered, using a standard Fokker-Planck equation. We have derived that such a type of dynamic systems has a discrete diffusive and energy spectra. It was shown that phase space diffusion is the only parameter that defines a dynamic accuracy in this case. The uncertainty relations have been obtained for conjugate phase space variables with account of transport properties. Given results can be used in the area of chaotic systems modelling and turbulence investigation. Sergey A. Kamenshchikov Copyright © 2014 Sergey A. Kamenshchikov. All rights reserved. Adaptive Control for Modified Projective Synchronization-Based Approach for Estimating All Parameters of a Class of Uncertain Systems: Case of Modified Colpitts Oscillators Wed, 12 Mar 2014 08:03:27 +0000 A method of estimation of all parameters of a class of nonlinear uncertain dynamical systems is considered, based on the modified projective synchronization (MPS). The case of modified Colpitts oscillators is investigated. Through a suitable transformation of the dynamical system, sufficient conditions for achieving synchronization are derived based on Lyapunov stability theory. Global stability and asymptotic robust synchronization of the considered system are investigated. The proposed approach offers a systematic design procedure for robust adaptive synchronization of a large class of chaotic systems. The combined effect of both an additive white Gaussian noise (AWGN) and an artificial perturbation is numerically investigated. Results of numerical simulations confirm the effectiveness of the proposed control strategy. Soup Tewa Kammogne and Hilaire Bertrand Fotsin Copyright © 2014 Soup Tewa Kammogne and Hilaire Bertrand Fotsin. All rights reserved. Theoretical Analysis and Adaptive Synchronization of a 4D Hyperchaotic Oscillator Thu, 27 Feb 2014 16:05:10 +0000 We propose a new mathematical model of the TNC oscillator and study its impact on the dynamical properties of the oscillator subjected to an exponential nonlinearity. We establish the existence of hyperchaotic behavior in the system through theoretical analysis and by exploiting electronic circuit experiments. The obtained numerical results are found to be in good agreement with experimental observations. Moreover, the new technique on adaptive control theory is used on our model and it is rigorously proven that the adaptive synchronization can be achieved for hyperchaotic systems with uncertain parameters. T. Fonzin Fozin, J. Kengne, and F. B. Pelap Copyright © 2014 T. Fonzin Fozin et al. All rights reserved. Reduced Order Projective and Hybrid Projective Combination-Combination Synchronization of Four Chaotic Josephson Junctions Thu, 13 Feb 2014 16:10:44 +0000 This paper investigates the reduced order projective and hybrid projective combination-combination synchronization of four chaotic Josephson junctions consisting of two third order Josephson junctions as the drives and two second order chaotic Josephson junctions as the response systems via active backstepping technique. The investigation confirms the achievement of reduced order projective and hybrid projective combination-combination synchronization among four chaotic Josephson junctions via active backstepping technique. Numerical simulations are validated to show the effectiveness of the synchronization scheme. Reduced order combination-combination synchronization scheme has more significant applications to neural encoding and decoding of information in biological systems and to the security of information transmission in communication systems than the usual one drive system and one response system synchronization scheme. K. S. Ojo, A. N. Njah, O. I. Olusola, and M. O. Omeike Copyright © 2014 K. S. Ojo et al. All rights reserved. Synchronization and Stabilization of Chaotic Dynamics in a Quasi-1D Bose-Einstein Condensate Thu, 12 Dec 2013 14:07:06 +0000 A nonlinear control is proposed for the exponential stabilization and synchronization of chaotic behaviour in a model of Bose-Einstein condensate (BEC). The active control technique is designed based on Lyapunov stability theory and Routh-Hurwitz criteria. The control design approach in both cases guarantees the stability of the controlled states. Whereas the synchronization of two identical BEC in their chaotic states can be realized using the scheme; a suitable controller is also capable of driving the otherwise chaotic oscillation to a stable state which could be expected in practice. The effectiveness of this technique is theoretically and numerically demonstrated. B. A. Idowu and U. E. Vincent Copyright © 2013 B. A. Idowu and U. E. Vincent. All rights reserved. Dynamical Properties and Finite-Time Hybrid Projective Synchronization Using Fractional Nonsingular Sliding Mode Surface in Fractional-Order Two-Stage Colpitts Oscillators Wed, 11 Dec 2013 13:54:14 +0000 The dynamics and robust finite-time hybrid projective synchronization of a fractional-order four-dimensional nonlinear system based on a two-stage Colpitts oscillator is investigated. The study of the fractional order stability of the equilibrium states of the system is carried out. The bifurcation diagram confirms the occurrence of Hopf bifurcation in the proposed system when the fractional-order passes a sequence of critical values; the Lyapunov exponent shows the different chaotic sequences of the system. Further, a fractional nonsingular terminal sliding surface and an appropriate robust fractional sliding mode control law are proposed for the finite-time hybrid projective synchronization of a fractional-order chaotic two-stage Colpitts oscillator by taking into account the effects of model uncertainties and the external disturbances. The fractional version of the Lyapunov stability is used to prove the finite-time existence of the sliding motion. Finally, some numerical simulations are presented to demonstrate the effectiveness and applicability of the proposed technique. Romanic Kengne, Robert Tchitnga, Arnaud Nzeusseu Tchikankou, Aurelle Tchagna Kouanou, and Anaclet Fomethe Copyright © 2013 Romanic Kengne et al. All rights reserved. On a New Cournot Duopoly Game Mon, 30 Sep 2013 10:51:38 +0000 This paper presents a new Cournot duopoly game. The main advantage of this game is that the outputs are nonnegative for all times. We investigate the complexity of the corresponding dynamical behaviors of the game such as stability and bifurcations. Computer simulations will be used to confirm our theoretical results. It is found that the chaotic behavior of the game has been stabilized on the Nash equilibrium point by using delay feedback control method. H. N. Agiza, A. A. Elsadany, and M. M. El-Dessoky Copyright © 2013 H. N. Agiza et al. All rights reserved. Synchronization of Uncertain Fractional-Order Hyperchaotic Systems via Unidirectional Linear Error Feedback Coupling Scheme Wed, 18 Sep 2013 11:55:23 +0000 A simple method for synchronization of uncertain fractional-order hyperchaotic systems is proposed in this paper. The method makes use of a unidirectional linear coupling approach due to its simple configuration and ease of implementation. To determine the coupling parameters, the synchronization error dynamics is first formulated as a fractional-order linear interval system. Then, the parameters are obtained by solving a linear matrix inequality (LMI) stability condition for stabilization of fractional-order linear interval systems. Thanks to the existence of an LMI solution, the convergence of the synchronization errors is guaranteed. The effectiveness of the proposed method is numerically illustrated by the uncertain fractional-order hyperchaotic Lorenz system. Suwat Kuntanapreeda Copyright © 2013 Suwat Kuntanapreeda. All rights reserved. Finite-Time Combination-Combination Synchronization for Hyperchaotic Systems Sun, 01 Sep 2013 09:45:22 +0000 A new type of finite-time synchronization with two drive systems and two response systems is presented. Based on the finite-time stability theory, step-by-step control and nonlinear control method, a suitable controller is designed to achieve finite-time combination-combination synchronization among four hyperchaotic systems. Numerical simulations are shown to verify the feasibility and effectiveness of the proposed control technique. Huini Lin, Jianping Cai, and Jincheng Wang Copyright © 2013 Huini Lin et al. All rights reserved. Control of Chaos in Rate-Dependent Friction-Induced Vibration Using Adaptive Sliding Mode Control and Impulse Damper Sun, 21 Jul 2013 10:32:13 +0000 Two different control methods, namely, adaptive sliding mode control and impulse damper, are used to control the chaotic vibration of a block on a belt system due to the rate-dependent friction. In the first method, using the sliding mode control technique and based on the Lyapunov stability theory, a sliding surface is determined, and an adaptive control law is established which stabilizes the chaotic response of the system. In the second control method, the vibration of this system is controlled by an impulse damper. In this method, an impulsive force is applied to the system by expanding and contracting the PZT stack according to efficient control law. Numerical simulations demonstrate the effectiveness of both methods in controlling the chaotic vibration of the system. It is shown that the settling time of the controlled system using impulse damper is less than that one controlled by adaptive sliding mode control; however, it needs more control effort. Ehsan Maani Miandoab, Aghil Yousefi-Koma, and Saeed Hashemnia Copyright © 2013 Ehsan Maani Miandoab et al. All rights reserved. Difficulties in Evaluating Lyapunov Exponents for Lie Governed Dynamics Thu, 11 Jul 2013 13:51:09 +0000 We consider here an environment in which the fact that a semiquantum Hamiltonian obeys SU(2) symmetries poses serious difficulties if one wants to compute Lyapunov exponents. C. M. Sarris, A. Plastino, and A. N. Proto Copyright © 2013 C. M. Sarris et al. All rights reserved. Unstable Manifolds of Continuous Self-Mappings Wed, 05 Jun 2013 18:04:36 +0000 Unstable manifolds of continuous self-mappings on completely densely ordered linear ordered topological spaces (CDOLOTS) are discussed. Let be a continuous self-map. First, the interval with endpoints of two adjacent fixed points is contained in the unilateral unstable manifold of one of the endpoints. Then, by using the above conclusion, we prove that periodic points of not belong to the unstable manifold of their iteration points of (for some ), unless the iteration points are themselves. Tianxiu Lu and Peiyong Zhu Copyright © 2013 Tianxiu Lu and Peiyong Zhu. All rights reserved.