Showing papers in "Nonlinear Dynamics in 2009"

Development of fractional order capacitors based on electrolyte processes

Abstract: In recent years, significant research in the field of electrochemistry was developed. The performance of electrical devices, depending on the processes of the electrolytes, was described and the physical origin of each parameter was established. However, the influence of the irregularity of the electrodes was not a subject of study and only recently this problem became relevant in the viewpoint of fractional calculus. This paper describes an electrolytic process in the perspective of fractional order capacitors. In this line of thought, are developed several experiments for measuring the electrical impedance of the devices. The results are analyzed through the frequency response, revealing capacitances of fractional order that can constitute an alternative to the classical integer order elements. Fractional order electric circuits are used to model and study the performance of the electrolyte processes.

Chaos and Hopf bifurcation of a finance system

Abstract: The complex dynamical behavior of a finance system is investigated in this paper The Ruelle–Takens route to chaos and strange nonchaotic attractors (SNA) are found through numerical simulations Then the system with time-delayed feedback is considered and the stability and Hopf bifurcation of the controlled system are investigated This research has important theoretical and practical meanings

Investigation on effect of joint clearance on dynamics of four-bar mechanism

Abstract: As a result of design, manufacturing and assembly processes or a wear effect, clearances are inevitable at the joints of mechanisms. In this study, dynamic response of mechanism having revolute joints with clearance is investigated. A four-bar mechanism having two joints with clearance is considered as a model mechanism. A neural network was used to model several characteristics of joint clearance. Kinematic and dynamic analyses were achieved using continuous contact mode between journal and bearing. A genetic algorithm was also used to determine the appropriate values of design variables for reducing the additional vibration effect due primarily to the joint clearance. The results show that the optimal adjusting of suitable design variables gives a certain decrease in shaking forces and their moments on the mechanism frame.

Robust adaptive sliding mode attitude maneuvering and vibration damping of three-axis-stabilized flexible spacecraft with actuator saturation limits

Abstract: This paper presents a dual-stage control system design method for flexible spacecraft attitude maneuvering control by use of on-off thrusters and active vibration suppression by embedded smart material as actuator. As a stepping stone, an adaptive sliding mode controller with the assumption of knowing the upper bounds of the lumped perturbation is designed that ensures exponential convergence or uniform ultimate boundedness (UUB) of the attitude control system in the presence of bounded parameter variation/disturbances and control input saturation as well. Then this adaptive controller is redesigned such that the need for knowing the upper bound in advance is eliminated. Lyapunov analysis shows that this modified adaptive controller can also guarantee the exponential convergence or UUB of the system. For actively suppressing the induced vibration, linear quadratic regulator (LQR) based positive position feedback control method is presented. Numerical simulations are performed to show that rotational maneuver and vibration suppression are accomplished in spite of the presence of disturbance torque/parameter uncertainty and saturation input.

Lubricated revolute joints in rigid multibody systems

Abstract: The main purpose of this work is to present a general methodology for modeling lubricated revolute joints in constrained rigid multibody systems. In the dynamic analysis of journal-bearings, the hydrodynamic forces, which include both squeeze and wedge effects, generated by the lubricant fluid, oppose the journal motion. The hydrodynamic forces are obtained by integrating the pressure distribution evaluated with the aid of Reynolds’ equation, written for the dynamic regime. The hydrodynamic forces built up by the lubricant fluid are evaluated from the system state variables and included into the equations of motion of the multibody system. Numerical examples are presented in order to demonstrate the use of the methodologies and procedures described in this work.

Adaptive robust fuzzy control for a class of uncertain chaotic systems

Abstract: In this paper, the output feedback control of uncertain chaotic systems is addressed via an adaptive robust fuzzy approach Fuzzy logic systems are employed to approximate uncertain nonlinear functions in the chaotic systems Because only partial information of the system’s states is needed to be known, an observer is given to estimate the unmeasured states Compared with the existing results in the observer design, the prior knowledge on dynamic uncertainties is relaxed and a class of more general chaotic systems is considered as well as robustness to the approximation error is improved It can be proven that the closed-loop system is stable in the sense that all the variables are bounded Simulation example for the unified chaotic systems is given to verify the effectiveness of the proposed method

Predator cannibalism can give rise to regular spatial pattern in a predator–prey system

Abstract: One of the central issues in ecology is the study of spatial pattern in the distribution of organisms. Thus, in this paper, spatial pattern of a predator–prey system with predator cannibalism is considered. By mathematical analysis, we obtain the condition for emerging Turing pattern formation. Furthermore, numerical simulations reveal that large variety of different spatiotemporal dynamics emerge as the consequence of the interaction of Holling type II with predator cannibalism. The obtained results show predator cannibalism has great influence on the spatial pattern formation. In other words, the regular pattern is induced by predator cannibalism. Moreover, we find that although the environment is heterogeneous, the system still exhibits Turing pattern, which means the pattern is self-organized. It may help us better understand the dynamics of predator–prey interaction in a real environment.

On the hyperchaotic complex Lü system

Abstract: The aim of this paper is to introduce the new hyperchaotic complex Lu system. This system has complex nonlinear behavior which is studied and investigated in this work. Numerically the range of parameter values of the system at which hyperchaotic attractors exist is calculated. This new system has a whole circle of equilibria and three isolated fixed points, while the real counterpart has only three isolated ones. The stability analysis of the trivial fixed point is studied. Its dynamics is more rich in the sense that our system exhibits both chaotic and hyperchaotic attractors, as well as periodic and quasi-periodic solutions and solutions that approach fixed points. The nonlinear control method based on Lyapunov function is used to synchronize the hyperchaotic attractors. The control of these attractors is studied. Different forms of hyperchaotic complex Lu systems are constructed using the state feedback controller and complex periodic forcing.

Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation

Abstract: We investigate the dynamics and control of a nonlinear oscillator that is described mathematically by a Variable Order Differential Equation (VODE). The dynamic problem in question arises from the dynamical analysis of a variable viscoelasticity oscillator. The dynamics of the model and the behavior of the variable order differintegrals are shown in variable phase space for different parameters. Two different controllers are developed for the VODEs under study in order to track an arbitrary reference function. A generalization of the van der Pol equation using the VODE formulation is analyzed under the light of the methods introduced in this work.

Chaos in the fractional-order Volta’s system: modeling and simulation

Abstract: This paper deals with a new fractional-order chaotic system. It is based on the concept of Volta’s system, where the mathematical model of Volta’s system contains fractional-order derivatives. This system has simple structure and can display a double-scroll attractor. The behavior of the integer-order and the fractional-order Volta’s system with total order less than three which exhibits chaos is presented as well. Computer simulations are cross-verified by the numerical calculation and the Matlab/Simulink models.

Generalized projective synchronization of the fractional-order Chen hyperchaotic system

Abstract: In this paper, we numerically investigate the hyperchaotic behaviors in the fractional-order Chen hyperchaotic systems. By utilizing the fractional calculus techniques, we find that hyperchaos exists in the fractional-order Chen hyperchaotic system with the order less than 4. We found that the lowest order for hyperchaos to have in this system is 3.72. Our results are validated by the existence of two positive Lyapunov exponents. The generalized projective synchronization method is also presented for synchronizing the fractional-order Chen hyperchaotic systems. The present technique is based on the Laplace transform theory. This simple and theoretically rigorous synchronization approach enables synchronization of fractional-order hyperchaotic systems to be achieved and does not require the computation of the conditional Lyapunov exponents. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.

Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients

Abstract: This paper obtains an exact solitary wave solution of the Korteweg–de Vries equation with power law nonlinearity with time-dependent coefficients of the nonlinear as well as the dispersion terms. In addition, there are time-dependent damping and dispersion terms. The solitary wave ansatz is used to carry out the analysis. It is only necessary for the time-dependent coefficients to be Riemann integrable. As an example, the solution of the special case of cylindrical KdV equation falls out.

3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system

Abstract: This article introduces a new chaotic system of 3-D quadratic autonomous ordinary differential equations, which can display 2-scroll chaotic attractors. Some basic dynamical behaviors of the new 3-D system are investigated. Of particular interest is that the chaotic system can generate complex 3-scroll and 4-scroll chaotic attractors. Finally, bifurcation analysis shows that the system can display extremely rich dynamics. The obtained results clearly show that this is a new chaotic system which deserves further detailed investigation.

Simulation and stability analysis of impacting systems with complete chattering

Abstract: This paper considers dynamical systems that are derived from mechanical systems with impacts. In particular we will focus on chattering—accumulation of impacts—for which local discontinuity mappings will be derived. We will first show how to use these mappings in simulation schemes, and secondly how the mappings are used to calculate the stability of limit cycles with chattering by solving the first variational equations.

Dynamics of an inertial two-neuron system with time delay

Abstract: In this paper, we considered a delayed differential equation modeling two-neuron system with both inertial terms and time delay. By analyzing the distribution of the eigenvalues of the corresponding transcendental characteristic equation of its linearized equation, local stability criteria are derived for various model parameters and time delay. By choosing time delay as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcation. Furthermore, the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Also, resonant codimension-two bifurcation is found to occur in this model. Some numerical examples are finally given for justifying the theoretical results. Chaotic behavior of this inertial two-neuron system with time delay is found also through numerical simulation, in which some phase plots, waveform plots, power spectra and Lyapunov exponent are computed and presented.

Oil whirl, oil whip and whirl/whip synchronization occurring in rotor systems with full-floating ring bearings

Abstract: High-speed rotors are often supported in floating ring bearings because of their good damping behavior. In contrast to conventional hydrodynamic bearings with a single oil film, full-floating ring bearings consist of two oil films: An inner and an outer oil film. As single oil-film bearings, full-floating ring bearings also show the typical fluid-film-induced instabilities (self-excited vibrations). Both inner and outer oil films can become unstable and exhibit oil whirl/whip instabilities. The paper at hand considers a Laval (Jeffcott) rotor, which is symmetrically supported in full-floating ring bearings, and investigates the occurring oil whirl/whip effects by means of run-up simulations. It is shown that the inner oil film, which usually becomes unstable first, gives rise to a limit-cycle oscillation with an exactly circular rotor orbit, if gravity and imbalance are neglected. Interesting is the instability generated by the outer oil film. The calculations demonstrate that instability in the outer oil film does not lead to a simple circular limit-cycle orbit. Whirl/whip-induced limit-cycle oscillations generated by the outer oil film are more complex and entail a coupled circumferential and radial motion, although the mechanical problem is radially symmetric, if gravity and imbalance are neglected. Thus, whirl/whip instability in the outer fluid film may be interpreted as symmetry breaking. Finally, a further kind of bifurcation/instability occurring in rotors supported in full-floating ring bearings—called Total Instability in this paper—is analyzed. It is shown that Total Instability is caused by synchronization of two limit cycles, namely synchronization of the inner and outer oil whirl/whip. Total Instability is of practical interest and observed in real technical rotor systems, and frequently leads to complete rotor damage.

Comments on a nonlinear and nonideal electromechanical damping vibration absorber, Sommerfeld effect and energy transfer

Abstract: In this paper, we investigated the effectiveness of the linear electromechanical vibration absorber (LEVA) and a nonlinear electromechanical vibration absorber (NEVA) in the vibration attenuation for nonideal structures (NIS). This electromechanical damping device consists of an electrical system coupled magnetically to a mechanical structure under a nonideal excitation. An analysis of the effects of the parameters of coupling and of nonlinear coefficients with increasing of constant torque of the DC motor is presented.

Constrained controllability of semilinear systems with delays

Abstract: In the paper, finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with multiple point time-variable delays in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local relative controllability are formulated and proved. It is generally assumed that the values of admissible controls are in a convex and closed cone with the vertex at zero. The special case of constant multiple point delays is also discussed. Moreover, some remarks and comments on the existing results for controllability of nonlinear and semilinear dynamical systems are also presented.

Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations

Abstract: This paper presents the fractional order Euler–Lagrange equations and the transversality conditions for fractional variational problems with fractional integral and fractional derivatives defined in the sense of Caputo and Riemann–Liouville. A fractional Hamiltonian formulation was developed and some illustrative examples were treated in detail.

On stability, persistence, and Hopf bifurcation in fractional order dynamical systems

Abstract: This is a preliminary study about the bifurcation phenomenon in fractional order dynamical systems. Persistence of some continuous time fractional order differential equations is studied. A numerical example for Hopf-type bifurcation in a fractional order system is given, hence we propose a modification of the conditions of Hopf bifurcation. Local stability of some biologically motivated functional equations is investigated.

Exponential synchronization of chaotic neural networks: a matrix measure approach

Abstract: Based on matrix measure and Halanay inequality, exponential synchronization of a class of chaotic neural networks with time-varying delays is investigated. Without constructing Lyapunov function, some simple but generic criteria for exponential synchronization of chaotic neural networks are derived. It is shown that the obtained results are easy to verify and simple to implement in practice. Two examples are given to illustrate the effectiveness of the presented synchronization scheme.

Global sliding mode control and application in chaotic systems

Abstract: This paper is concerned with the stabilization problem for a class of nonlinear systems. Using the global sliding mode control approach, a novel robust control law is established to make the state of system stable and to improve the robustness and the stability of system. A new reaching law is introduced to reduce the chattering. Finally, the method is applied to chaotic systems and an example of the chaotic system is given to illustrate the advantage of the proposed method.

Interaction of torsion and lateral bending in aircraft nose landing gear shimmy

Abstract: In this paper we consider the onset of shimmy oscillations of an aircraft nose landing gear. To this end we develop and study a mathematical model with torsional and lateral bending modes that are coupled through a wheel-mounted elastic tyre. The geometric effects of a positive rake angle are fully incorporated into the resulting five-dimensional ordinary differential equation model. A bifurcation analysis in terms of the forward velocity and the vertical force on the gear reveals routes to different types of shimmy oscillations. In particular, we find regions of stable torsional and stable lateral shimmy oscillations, as well as transient quasiperiodic shimmy where both modes are excited.

Chaos synchronization of two different chaotic complex Chen and Lü systems

Abstract: This paper investigates the phenomenon of chaos synchronization of two different chaotic complex systems of the Chen and Lu type via the methods of active control and global synchronization. In this regard, it generalizes earlier work on the synchronization of two identical oscillators in cases where the drive and response systems are different, the parameter space is larger, and the dimensionality increases due to the complexification of the dependent variables. The idea of chaos synchronization is to use the output of the drive system to control the response system so that the output of the response system converges to the output of the drive system as time increases. Lyapunov functions are derived to prove that the differences in the dynamics of the two systems converge to zero exponentially fast, explicit expressions are given for the control functions and numerical simulations are presented to illustrate the success of our chaos synchronization techniques. We also point out that the global synchronization method is better suited for synchronizing identical chaotic oscillators, as it has serious limitations when applied to the case where the drive and response systems are different.

Frequency domain analysis of a dimensionless cubic nonlinear damping system subject to harmonic input

Abstract: The effects of cubic nonlinear damping on the system output spectrum are theoretically studied through a dimensionless mass–spring damping system model subject to a harmonic input, based on the Volterra series approximation. It is theoretically shown that the cubic nonlinear damping has little effect on the system output spectrum at high or low frequencies but drives the system output spectrum to be an alternative series at the natural frequency 1 such that the system output spectrum can be suppressed by the cubic damping.

Nonlinear dynamic response and modeling of a bi-stable composite plate for applications to adaptive structures

Abstract: This paper discusses the formulation and validation of a low order model to capture the dynamics of a bi-stable composite plate, focusing on the dynamics around its stable states. More specifically, the model aims to capture the complex nonlinear subharmonic behavior observed in the dynamic response of the plate. A system identification approach is used to derive simplified equations of motion for the system. Experimental frequency response diagrams are obtained to characterize the observed dynamics in the identification process. Simulations using the identified model are presented showing excellent agreement with the experimentally observed behavior. A theoretical validation of the model is carried out studying the stability of the modes where subharmonic response was observed. Stability boundaries were computed using averaging techniques showing good agreement with experimental results.

Analytical bifurcation analysis of a rotor supported by floating ring bearings

Abstract: Like with other types of fluid bearings, rotors supported by floating ring bearings may become unstable with increasing speed of rotation due to self-excited vibrations. In order to study the effects of the nonlinear bearing forces, within this contribution a perfectly balanced symmetric rotor is considered which is supported by two identical floating ring bearings. Here, the bearing forces are modeled by applying the short bearing theory for both fluid films. A linear stability analysis about the static equilibrium position of the rotor shows that for a critical revolution speed the real part of an eigenvalue pair changes its sign. By means of a center manifold reduction it is shown that this destabilization of the steady state is due to a Hopf-bifurcation. Furthermore, the type of this bifurcation is determined as well as the existence and stability of limit-cycles. Notably it is found that depending on the parameters of the floating ring bearing subcritical as well as supercritical bifurcations may occur. Additionally, the analytical results obtained from the center manifold reduction are compared to numerical results by a continuation method. In conclusion, the influences of bearing design parameters on the stability and on the limit-cycles are discussed.

Comparison of near- and far-fault ground motion effect on the nonlinear response of dam–reservoir–foundation systems

Abstract: In this paper, it is aimed to compare the near- and far-fault ground motion effects on the nonlinear dynamic response of dams including dam–reservoir–foundation interaction. Two different types of dams, which are concrete arch and concrete faced rockfill dams, are selected to investigate the near- and far-fault ground motion effects on the dam responses. The behavior of reservoir water is taken into account using Lagrangian approach. The Drucker–Prager material model is employed in nonlinear analyses. Near and far-fault strong ground motion records, which have approximately identical peak ground accelerations, of Loma Prieta (1989) earthquake are selected for the analyses. Displacements, maximum and minimum principal stresses are determined using the finite element method. The displacements and principal stresses obtained from the analyses of dams subjected to each fault effect are compared with each other. It is clearly seen that there is more seismic demand on displacements and stresses when the dam is subjected to near-fault ground motion.

Nonlinear dynamics and chaos methods in neurodynamics and complex data analysis

Abstract: In this paper, we review modern nonlinear dynamical methods used in neuroscience and complex data analysis. We start with the general description of nonlinear dynamics, its geometrical (and topological) picture, as well as its extreme case, deterministic chaos, including its most popular models and methods: Lorenz attractor, Lyapunov exponents, and Kolmogorov–Sinai entropy.

Synchronization of four coupled van der Pol oscillators

Abstract: It is possible that self-excited vibrations in turbomachine blades synchronize due to elastic coupling through the shaft. The synchronization of four coupled van der Pol oscillators is presented here as a simplified model. For quasilinear oscillations, a stability condition is derived from an analysis based on linearizing the original equation around an unperturbed limit cycle and transforming it into Hill’s equation. For the nonlinear case, numerical simulations show the existence of two well-defined regions of phase relationships in parameter space in which a multiplicity of periodic attractors is embedded. The size of these regions strongly depends on the values of the oscillator and coupling constants. For the coupling constant below a critical value, there exists a region in which a diversity of phase-shift attractors is present, whereas for values above the critical value an in-phase attractor is predominant. It is observed that the presence of an anti-phase attractor in the subcritical region is associated with sudden changes in the period of the coupled oscillators. The convergence of the coupled system to a particular periodic attractor is explored using several initial conditions. The study is extended to non-identical oscillators, and it is found that there is synchronization even over a wide range of difference among the oscillator constants.