Journal of Nonlinear Dynamics http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Hyers-Ulam Stability of Third Order Euler’s Differential Equations Tue, 04 Nov 2014 00:00:00 +0000 http://www.hindawi.com/journals/jndy/2014/487257/ We investigate the Hyers-Ulam stability of third order Euler's differential equations of the form on any open interval , or , where , and are complex constants. A. K. Tripathy and A. Satapathy Copyright © 2014 A. K. Tripathy and A. Satapathy. All rights reserved. Experiment on Bifurcation and Chaos in Coupled Anisochronous Self-Excited Systems: Case of Two Coupled van der Pol-Duffing Oscillators Wed, 29 Oct 2014 12:23:48 +0000 http://www.hindawi.com/journals/jndy/2014/815783/ The analog circuit implementation and the experimental bifurcation analysis of coupled anisochronous self-driven systems modelled by two mutually coupled van der Pol-Duffing oscillators are considered. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of their velocities (i.e., dissipative coupling). Interest in this problem does not decrease because of its significance and possible application in the analysis of different, biological, chemical, and electrical systems (e.g., coupled van der Pol-Duffing electrical system). Regions of quenching behavior as well as conditions for the appearance of Hopf bifurcations are analytically defined. The scenarios/routes to chaos are studied with particular emphasis on the effects of cubic nonlinearity (that is responsible for anisochronism of small oscillations). When monitoring the control parameter, various striking dynamic behaviors are found including period-doubling, symmetry-breaking, multistability, and chaos. An appropriate electronic circuit describing the coupled oscillator is designed and used for the investigations. Experimental results that are consistent with results from theoretical analyses are presented and convincingly show quenching phenomenon as well as bifurcation and chaos. J. Kengne, F. Kenmogne, and V. Kamdoum Tamba Copyright © 2014 J. Kengne et al. All rights reserved. On Full-State Hybrid Projective Synchronization of General Discrete Chaotic Systems Tue, 14 Oct 2014 12:59:19 +0000 http://www.hindawi.com/journals/jndy/2014/983293/ The problems of full-state hybrid projective synchronization (FSHPS) and inverse full-state hybrid projective synchronization (IFSHPS) for general discrete chaotic systems are investigated in 2D. Based on nonlinear control method and Lyapunov stability theory, new controllers are designed to study FSHPS and IFSHPS, respectively, for 2D arbitrary chaotic systems in discrete-time. Numerical example and simulations are used to validate the main results of this paper. Adel Ouannas Copyright © 2014 Adel Ouannas. All rights reserved. Catastrophe and Hysteresis by the Emerging of Soliton-Like Solutions in a Nerve Model Tue, 07 Oct 2014 13:56:34 +0000 http://www.hindawi.com/journals/jndy/2014/710152/ The nonlinear problem of traveling nerve pulses showing the unexpected process of hysteresis and catastrophe is studied. The analysis was done for the case of one-dimensional nerve pulse propagation. Of particular interest is the distinctive tendency of the pulse nerve model to conserve its behavior in the absence of the stimulus that generated it. The hysteresis and catastrophe appear in certain parametric region determined by the evolution of bubble and pedestal like solitons. By reformulating the governing equations with a standard boundary conditions method, we derive a system of nonlinear algebraic equations for critical points. Our approach provides opportunities to explore the nonlinear features of wave patterns with hysteresis. Fernando Ongay Larios, Nikolay P. Tretyakov, and Maximo A. Agüero Copyright © 2014 Fernando Ongay Larios et al. All rights reserved. Multimode Analysis of the Dynamics and Integrity of Electrically Actuated MEMS Resonators Thu, 25 Sep 2014 06:22:35 +0000 http://www.hindawi.com/journals/jndy/2014/735712/ We present a theoretical investigation of the dynamic behavior of a microelectromechanical system (in brief, MEMS) device modelled as a clamped-clamped microbeam subjected to electrostatic and electrodynamic actuation. We use the Galerkin projection technique to reduce the partial integro-differential equation governing the dynamics of the microbeam to a system of coupled ordinary differential equations which describe the interactions of the linear mode shapes of the microbeam. Analytical solutions are derived and their stability is studied for the simplest reduced-order model which takes into account only the first linear mode in the Galerkin procedure. We further investigate the influence of the first few higher modes on the Galerkin procedure, and hence its convergence, by analysing the boundaries between pull-in and pull-in-free vibrations domains in the space of actuation parameters. These are determined for the various multimode combinations using direct numerical time integration. Our results show that unsafe domains form V-like shapes for actuation frequencies close to the superharmonic, fundamental, and subharmonic resonances. They also reveal that the single first-mode reduced model usually considered underestimates the left branches and overestimates the right branches of these boundaries. Serge Bruno Yamgoué and Alain Juvenal Tchiegang Copyright © 2014 Serge Bruno Yamgoué and Alain Juvenal Tchiegang. All rights reserved. The Effect of Slow Invariant Manifold and Slow Flow Dynamics on the Energy Transfer and Dissipation of a Singular Damped System with an Essential Nonlinear Attachment Mon, 01 Sep 2014 05:38:54 +0000 http://www.hindawi.com/journals/jndy/2014/208171/ We study the effect of slow flow dynamics and slow invariant manifolds on the energy transfer and dissipation of a dissipative system of two linear oscillators coupled with an essential nonlinear oscillator with a mass much smaller than the masses of the linear oscillators. We calculate the slow flow of the system, the slow invariant manifold, the total energy of the system, and the energy that is stored in the nonlinear oscillator for different sets of the parameters and show that the bifurcations of the SIM and the dynamics of the slow flow play an important role in the energy transfer from the linear to the nonlinear oscillator and the rate of dissipation of the total energy of the initial system. Jamal-Odysseas Maaita and Efthymia Meletlidou Copyright © 2014 Jamal-Odysseas Maaita and Efthymia Meletlidou. All rights reserved. Exact Solitary Wave Solution in the ZK-BBM Equation Sun, 31 Aug 2014 06:52:07 +0000 http://www.hindawi.com/journals/jndy/2014/468392/ The traveling wave solution for the ZK-BBM equation is considered, which is governed by a nonlinear ODE system. The bifurcation structure of fixed points and bifurcation phase portraits with respect to the wave speed c are analyzed by using the dynamical system theory. Furthermore, the exact solutions of the homoclinic orbits for the nonlinear ODE system are obtained which corresponds to the solitary wave solution curve of the ZK-BBM equation. Juan Zhao and Wei Li Copyright © 2014 Juan Zhao and Wei Li. All rights reserved. Optimal Control of a Delayed HIV Infection Model via Fourier Series Tue, 26 Aug 2014 05:34:05 +0000 http://www.hindawi.com/journals/jndy/2014/945158/ We present a delayed optimal control which describes the interaction of the immune system with the human immunodeficiency virus (HIV) and CD4+ T-cells. In order to improve the therapies, treatment and the intracellular delays are incorporated into the model. The optimal control in this model represents the efficiency of drug treatment in preventing viral production and new infections. The optimal pair of control and trajectories of this nonlinear delay system with quadratic cost functional is obtained by Fourier series approximation. The method is based on expanding time varying functions in the nonlinear delay system into their Fourier series with unknown coefficients. Using operational matrices for integration, product, and delay, the problem is reduced to a set of nonlinear algebraic equations. Gh. Ghanbari and M. H. Farahi Copyright © 2014 Gh. Ghanbari and M. H. Farahi. All rights reserved. Increase in Equilibrium Price by Fast Oscillations Tue, 20 May 2014 08:28:24 +0000 http://www.hindawi.com/journals/jndy/2014/593254/ The dynamics of a market can be described by a differential equation. Using the concept of fast oscillation, the system (typical market) can also oscillate around a new equilibrium price, with an increase. Previously that increase was established by applying harmonic force. In present work, harmonic force is replaced by an arbitrary periodic force with zero mean. Hence the increase in equilibrium price can be controlled by varying the external arbitrary periodic force. Babar Ahmad and Khalid Iqbal Mahr Copyright © 2014 Babar Ahmad and Khalid Iqbal Mahr. All rights reserved. Control and Synchronization of Chaotic and Hyperchaotic Lorenz Systems via Extended Backstepping Techniques Tue, 06 May 2014 08:17:19 +0000 http://www.hindawi.com/journals/jndy/2014/861727/ We propose novel controllers for stabilization and tracking of chaotic and hyperchaotic Lorenz systems using extended backstepping techniques. Based on the proposed approach, generalized weighted controllers were designed to control chaotic behaviour as well as to achieve synchronization in chaotic and hyperchaotic Lorenz systems. The effectiveness and feasibility of the proposed weighted controllers were verified numerically and showed their robustness against noise. O. S. Onma, O. I. Olusola, and A. N. Njah Copyright © 2014 O. S. Onma et al. All rights reserved. Parameter Estimation and Hybrid Lag Synchronization in Hyperchaotic Lü Systems Sun, 30 Mar 2014 09:21:22 +0000 http://www.hindawi.com/journals/jndy/2014/842790/ The antiphase and complete lag synchronization of hyperchaotic Lü systems with unknown parameters is investigated. Based on the Lyapunov stability theory, the sufficient conditions for achieving hybrid lag synchronization are derived. The optimized parameter observers are approached analytically via adaptive control approach. Numerical simulation results are presented to verify the effectiveness of the proposed scheme. Qing Wei and Zuolei Wang Copyright © 2014 Qing Wei and Zuolei Wang. All rights reserved. Dynamics from Multivariable Longitudinal Data Wed, 19 Mar 2014 13:30:27 +0000 http://www.hindawi.com/journals/jndy/2014/901838/ We introduce a method of analysing longitudinal data in variables and a population of observations. Longitudinal data of each observation is exactly coded to an orbit in a two-dimensional state space . At each time, information of each observation is coded to a point , where is the physical condition of the observation and is an ordering of variables. Orbit of each observation in is described by a map that dynamically rearranges order of variables at each time step, eventually placing the most stable, least frequently changing variable to the left and the most frequently changing variable to the right. By this operation, we are able to extract dynamics from data and visualise the orbit of each observation. In addition, clustering of data in the stable variables is revealed. All possible paths that any observation can take in are given by a subshift of finite type (SFT). We discuss mathematical properties of the transition matrix associated to this SFT. Dynamics of the population is a nonautonomous multivalued map equivalent to a nonstationary SFT. We illustrate the method using a longitudinal data of a population of households from Agincourt, South Africa. Maria Vivien Visaya and David Sherwell Copyright © 2014 Maria Vivien Visaya and David Sherwell. All rights reserved. Markov Chain Model to Explain the Dynamics of Human Depression Tue, 18 Mar 2014 08:48:59 +0000 http://www.hindawi.com/journals/jndy/2014/107164/ Depression is one of the major concerns of the present generation. A Markov chain model has been used to portray and investigate this curse. Long-term behaviour of the model has been discussed. Different types of treatment strategies have been considered in this paper to identify the most powerful measure of keeping this disease from its spread in the society. This paper also focuses on the usefulness of the drugs available at present for the treatment of this disease. Souvik Bhattacharya Copyright © 2014 Souvik Bhattacharya. All rights reserved. Dynamic Analyses of Urban Expressway Network with Mesoscopic Traffic Flow Model Integrated Variable Speed Limits Tue, 18 Mar 2014 07:04:52 +0000 http://www.hindawi.com/journals/jndy/2014/867236/ Urban congestion is a major and costly problem in many cities both in China and other countries. The purpose of building urban expressway is to alleviate the growing traffic pressure. In this paper, the mesoscopic traffic flow models are improved by variable speed limits strategy for the dynamic of vehicles on urban expressway network. The models include static queuing model, the velocity model, and the movement model of the vehicle. Moreover the method of the simulation is also proposed. So that we can get the corresponding variable speed limits values and aid traffic managers in making decisions to develop a network traffic flow control strategy. In the end, the elevated expressway of Jinan city is used as a simulation example. We investigated the performance of the transport system with averaged density, speed, and flow on link. We also analysed the dynamic of the traffic system on expressway network at different demand levels. The simulation results show that the models are feasible and effective and the variable speed limits strategy can successfully alleviate the traffic congestion in some extent. The operational efficiency of the transportation system is greatly improved. Shu-Bin Li, Bai-Bai Fu, and Wen-Xiu Dang Copyright © 2014 Shu-Bin Li et al. All rights reserved. Bifurcation Analysis of a Delayed Predator-Prey Model with Holling Type III Functional Response and Predator Harvesting Mon, 03 Mar 2014 12:15:26 +0000 http://www.hindawi.com/journals/jndy/2014/543041/ This paper tries to highlight a delayed prey-predator model with Holling type III functional response and harvesting to predator species. In this context, we have discussed local stability of the equilibria, and the occurrence of Hopf bifurcation of the system is examined by considering the harvesting effort as bifurcation parameter along with the influences of harvesting effort of the system when time delay is zero. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by applying the normal form theory and the center manifold theorem. Lastly some numerical simulations are carried out to draw for the validity of the theoretical results. Uttam Das and T. K. Kar Copyright © 2014 Uttam Das and T. K. Kar. All rights reserved. Frequency Response of an Impacting Lap Joint Thu, 27 Feb 2014 10:05:21 +0000 http://www.hindawi.com/journals/jndy/2014/310834/ Damage or failure of a relatively small component can precipitate the failure of a larger part of a structure. The behavior of damaged or worn joints is of particular concern. To address contact/impact in structural systems, this work models a structural lap joint from first principles. A beam with four stops and gaps is used to simulate a loose or damaged lap joint, which also represents designed manufacturing clearances in mechanical systems. The goal is to generate frequency responses to identify the local shock effect due to impact. Spatial and temporal solutions are presented for an example case. Converged time histories were used to generate the impulse as a metric of frequency response. Facilitating mode contribution calculations, the metric of impulse proves to be an excellent indicator of complexities in the beam's motion due to excitation frequency. Noncontact regions, sticking motions, local extrema, grazing impacts, and aperiodicities are identifiable for specific operating parameters. These conditions indicate when harmful impact may occur that can ultimately cause local damage within a structure. Knowledge of dangerous operating conditions can better focus on inspection before propagation occurs. Amir M. Rahmani and Elizabeth K. Ervin Copyright © 2014 Amir M. Rahmani and Elizabeth K. Ervin. All rights reserved. Multiswitching Synchronization of a Driven Hyperchaotic Circuit Using Active Backstepping Tue, 18 Feb 2014 10:07:20 +0000 http://www.hindawi.com/journals/jndy/2014/918586/ An active backstepping technique is proposed for the realization of multiswitching synchronization of periodically forced hyperchaotic Van der Pol-Duffing oscillators. The active backstepping technique is a systematic design approach with recursive procedures that skillfully optimizes the choice of Lyapunov functions and active control technique. Using the active backstepping technique, the usual master-slave synchronization scheme is extended to study the synchronization of systems with different combinations of the slave states variables with master state variables. Our numerical results confirm the effectiveness of the proposed analytical technique. A. Ayotunde Ajayi, S. Kayode Ojo, E. Uchechukwu Vincent, and N. Abdullahi Njah Copyright © 2014 A. Ayotunde Ajayi et al. All rights reserved. A Review of Theoretical Perspectives in Cognitive Science on the Presence of Scaling in Coordinated Physiological and Cognitive Processes Mon, 10 Feb 2014 07:51:32 +0000 http://www.hindawi.com/journals/jndy/2014/962043/ Time series of human performances present fluctuations around a mean value. These fluctuations are typically considered as insignificant, and attributable to random noise. Over recent decades, it became clear that temporal fluctuations possess interesting properties, however, one of which the property of fractal 1/f scaling. 1/f scaling indicates that a measured process extends over a wide range of timescales, suggesting an assembly over multiple scales simultaneously. This paper reviews neurological, physiological, and cognitive studies that corroborate the claim that 1/f scaling is most clearly present in healthy, well-coordinated activities. Prominent hypotheses about the origins of 1/f scaling are confronted with these reviewed studies. It is concluded that 1/f scaling in living systems appears to reflect their genuine complex nature, rather than constituting a coincidental side-effect. The consequences of fractal dynamics extending from the small spatial and temporal scales (e.g., neurons) to the larger scales of human behavior and cognition, are vast, and impact the way in which relevant research questions may be approached. Rather than focusing on specialized isolable subsystems, using additive linear methodologies, nonlinear dynamics, more elegantly so, imply a complex systems methodology, thereby exploiting, rather than rejecting, mathematical concepts that enable describing large sets of natural phenomena. Maarten L. Wijnants Copyright © 2014 Maarten L. Wijnants. All rights reserved. A Note on Taylor-Eddy and Kovasznay Solutions of NS--Deconvolution and Leray--Deconvolution Models Mon, 20 Jan 2014 07:25:47 +0000 http://www.hindawi.com/journals/jndy/2014/959038/ We show that both the Taylor-eddy and Kovasznay exact solutions of the Navier-Stokes equations are also exact solutions of both the NS-α-deconvolution and Leray-α-deconvolution models, but with modified pressures that converge to the Navier-Stokes pressure solution as or the order of deconvolution tends to infinity. The existence of these exact model solutions will provide for better benchmark testing and validation of numerical codes and also shows that the models preserve these special structures. Leo G. Rebholz and Stacey A. Watro Copyright © 2014 Leo G. Rebholz and Stacey A. Watro. All rights reserved. Dynamic Sliding Mode Control Design Based on an Integral Manifold for Nonlinear Uncertain Systems Thu, 02 Jan 2014 15:27:38 +0000 http://www.hindawi.com/journals/jndy/2014/489364/ An output feedback sliding mode control law design relying on an integral manifold is proposed in this work. The considered class of nonlinear systems is assumed to be affected by both matched and unmatched uncertainties. The use of the integral sliding manifold allows one to subdivide the control design procedure into two steps. First a linear control component is designed by pole placement and then a discontinuous control component is added so as to cope with the uncertainty presence. In conventional sliding mode the control variable suffers from high frequency oscillations due to the discontinuous control component. However, in the present proposal, the designed control law is applied to the actual system after passing through a chain of integrators. As a consequence, the control input actually fed into the system is continuous, which is a positive feature in terms of chattering attenuation. By applying the proposed controller, the system output is regulated to zero even in the presence of the uncertainties. In the paper, the proposed control law is theoretically analyzed and its performances are demonstrated in simulation. Qudrat Khan, Aamer Iqbal Bhatti, and Antonella Ferrara Copyright © 2014 Qudrat Khan et al. All rights reserved. Analytical Homoclinic Solution of a Two-Dimensional Nonlinear System of Differential Equations Wed, 11 Dec 2013 09:38:55 +0000 http://www.hindawi.com/journals/jndy/2013/879040/ Analytical solution of the homoclinic orbit of a two-dimensional system of differential equations that describes the Hamiltonian part of the slow flow of a three-degree-of-freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment is described. J. O. Maaita and E. Meletlidou Copyright © 2013 J. O. Maaita and E. Meletlidou. All rights reserved. Suboptimal Control Strategies for Finite-Time Nonlinear Processes with Input Constraints Sun, 17 Nov 2013 16:09:46 +0000 http://www.hindawi.com/journals/jndy/2013/182034/ Novel techniques for the optimization and control of finite-time processes in real-time are pursued. These are developed in the framework of the Hamiltonian optimal control. Two methods are designed. The first one constructs the reference control trajectory as an approximation of the optimal control via the Riccati equations in an adaptive fashion based on the solutions of a set of partial differential equations called the and matrices. These allow calculating the Riccati gain for a range of the duration of the process and the final penalization . The second method introduces input constraints to the general optimization formulation. The notions of linear matrix inequalities allow us to recuperate the Riccati gain as in the first method, but using an infinite horizon optimization method. Finally, the performance of the proposed strategies is illustrated through numerical simulations applied to a batch reactor and a penicillin fed-batch reactor. Pablo S. Rivadeneira and Eduardo J. Adam Copyright © 2013 Pablo S. Rivadeneira and Eduardo J. Adam. All rights reserved. Stabilization of Driven Pendulum with Periodic Linear Forces Thu, 03 Oct 2013 15:46:09 +0000 http://www.hindawi.com/journals/jndy/2013/824701/ Using Kapitza method of averaging for arbitrary periodic forces, the pendulum driven by different forms of periodic piecewise linear forces is stabilized. These periodic piecewise linear forces are selected in the range to establish an exact comparison with harmonic forces. In this contest, the rectangular force was found to be the best, but this force is more effective when it has a time-dependent structure. This time-dependent structure is found by defining a parametric control on some other periodic piecewise linear forces. Babar Ahmad Copyright © 2013 Babar Ahmad. All rights reserved. Comparison of Two Impact Simulation Methods Used for Nonlinear Vibroimpact Systems with Rigid and Soft Impacts Tue, 24 Sep 2013 18:23:53 +0000 http://www.hindawi.com/journals/jndy/2013/485676/ This paper compares the use of two impact simulation methods for two-degree-of-freedom nonlinear vibroimpact systems with rigid and soft impacts. These methods are (I) impact simulation by boundary conditions with the use of Newton's restitution coefficient based on stereomechanic shock theory and (II) impact simulation by contact interaction force based on quasistatic Hertz's contact theory. It is shown that both methods are applied and give the coinciding results for system with elastic rigid impact under periodic external loading. Loading curves built by parameter continuation method are confirming this result. Impact simulation by the second method is also fulfilled for vibroimpact system with rigid impact under random external loading. For vibroimpact system with soft impact, the simulation of impact by the second method gives a better result. The application of linear elastic force as contact one is possible too but the use of Hertz's contact force is more preferable. The authors consider that the impact simulation by Hertz contact interaction force gives good results for nonlinear vibroimpact systems with impacts of any kind if all limitations with Hertz's law used are observed. V. A. Bazhenov, O. S. Pogorelova, and T. G. Postnikova Copyright © 2013 V. A. Bazhenov et al. All rights reserved. Dynamical Behaviour of a Tumor-Immune System with Chemotherapy and Optimal Control Sun, 14 Jul 2013 10:35:54 +0000 http://www.hindawi.com/journals/jndy/2013/608598/ We have considered a tumor growth model with the effect of tumor-immune interaction and chemotherapeutic drug. We have considered two immune components—helper (resting) T-cells which stimulate CTLs and convert them into active (hunting) CTL cells and active (hunting) CTL cells which attack, destroy, or ingest the tumor cells. In our model there are four compartments, namely, tumor cells, active CTL cells, helper T-cells, and chemotherapeutic drug. We have discussed the behaviour of the solutions of our system. The dynamical behaviour of our system by analyzing the existence and stability of the system at various equilibrium points is discussed elaborately. We have set up an optimal control problem relative to the model so as to minimize the number of tumor cells and the chemotherapeutic drug administration. Here we used a quadratic control to quantify this goal and have considered the administration of chemotherapy drug as control to reduce the spread of the disease. The important mathematical findings for the dynamical behaviour of the tumor-immune model with control are also numerically verified using MATLAB. Finally, epidemiological implications of our analytical findings are addressed critically. Swarnali Sharma and G. P. Samanta Copyright © 2013 Swarnali Sharma and G. P. Samanta. All rights reserved.