Dynamic instability of a non-uniform beam modeled in a two-degree-of-freedom system

The behaviour of a non-uniform beam loaded by a parallel or tangential compressive force respectively is analyzed by exploiting a two-degree-of-freedom dynamic model. The same system has been analyzed previously by Lee and Reissner from a static point of view and by Neer and Baruch by a dynamic approach using a one-degree-of-freedom model. The previous analyses revealed only part of the phenomena by the present approach.Here, for the tangential force a classical flutter instability is obtained and for the parallel force only static instability is possible.     

The generalized synchronization of a Quantum-CNN chaotic oscillator with different order systems

This paper presents a special kind of the generalized synchronization of different order systems, proved by Lyapunov asymptotical stability theorem. A sufficient condition is given for the asymptotical stability of the null solution of an error dynamics. The generalized synchronization developed may be applied to the design of secure communication. Finally, numerical results are studied for a Quantum-CNN oscillator synchronized with three different order systems respectively to show the effectiveness of the proposed synchronization strategy.     

Phenomena of subharmonic motions of oscillator with soft impacts

The excited, one degree of freedom mechanical system with soft impacts, characterised by triangle hysteresis loop, is investigated using numerical simulation. Small viscous damping is assumed. Phenomena of subharmonic motions are explained by regions of their existence and stability in the plane of dimensionless excitation frequency and static clearances. Bifurcation diagrams are evaluated during quasistationary changes of frequency by constant clearance.     

Approximate solutions and chaotic motions of a piecewise nonlinear–linear oscillator

A global symmetric period-1 approximate solution is analytically constructed for the non-resonant periodic response of a periodically excited piecewise nonlinear–linear oscillator. The approximate solutions are found to be in good agreement with the exact solutions that are obtained from the numerical integration of the original equations. In addition, the dynamic behaviour of the oscillator is numerically investigated with the help of bifurcation diagrams, Lyapunov exponents, Poincare maps, phase portraits and basins of attraction. The existence of subharmonic and chaotic motions and the coexistence of four attractors are observed for some combinations of the system parameters.     

Periodic and chaotic dynamics in an asymmetric elastoplastic oscillator

We consider the dynamics of a harmonically forced oscillator with an asymmetric elastic–perfectly plastic stiffness function. The computed bifurcation diagrams for the oscillator show regions of periodic motion, hysteresis and large regions of chaotic motion. These different regions of dynamical behaviour are plotted in a two-dimensional parameter space consisting of forcing amplitude and forcing frequency. Examples of the chaotic motion encountered are shown using a discontinuity crossing map. Comparisons are made with the symmetric oscillator by computing a typical bifurcation diagram and considering previously published results for the symmetric system. From this we conclude that the asymmetric system is dominated by a large region of chaotic motion whereas in the symmetric oscillator period one motion and coexisting period three motion predominates.     

On the dynamics of a periodic Colpitts oscillator forced by periodic and chaotic signals

In this paper, we have examined effects of forcing a periodic Colpitts oscillator with periodic and chaotic signals for different values of coupling factors. The forcing signal is generated in a master bias-tuned Colpitts oscillator having identical structure as that of the slave periodic oscillator. Numerically solving the system equations, it is observed that the slave oscillator goes to chaotic state through a period-doubling route for increasing strengths of the forcing periodic signal. For forcing with chaotic signal, the transition to chaos is observed but the route to chaos is not clearly detectable due to random variations of the forcing signal strength. The chaos produced in the slave Colpitts oscillator for a chaotic forcing is found to be in a phase-synchronized state with the forced chaos for some values of the coupling factor. We also perform a hardware experiment in the radio frequency range with prototype Colpitts oscillator circuits and the experimental observations are in agreement with the numerical simulation results.     

Regular and chaotic motions of a Chaplygin sleigh under periodic pulsed torque impacts

For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced three-dimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.     

Synchronization of van der Pol oscillator and Chen chaotic dynamical system

This paper addresses the synchronization problem of two different electronic circuits by using nonlinear control function. This technique is applied to achieve synchronization for the stable van der Pol oscillator and Chen chaotic dynamical system. Numerical simulations results are given to demonstrate the effectiveness of the proposed control method.     

Dynamical analysis of a compound oscillator with initial phase difference

The influence of initial phase difference as well as the amplitude of parametric excitation on the dynamics of a compound oscillator has been investigated in this paper. Based on the multiple time scale method, bifurcation sets of the averaged equations have been derived to divide the parameter space into regions associated with two types of phase trajectories. Imperfect cascading of period-doubling bifurcations have been observed, which may lead to different chaotic attractors. It is presented that, the three frequencies evolved in the vector field, though in resonance, may cause modulated effect on the structures of chaotic attractors.     

Bifurcation analysis in a limit cycle oscillator with delayed feedback

A kind of limit cycle oscillator with delayed feedback is considered. Firstly, the linear stability is investigated. According to the analysis results, the bifurcation diagram is drawn in the parameter plane. It is found that there are stability switches for time delay, and Hopf bifurcations when time delay crosses through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results found.     

Adaptive synchronization of a modified and uncertain chaotic Van der Pol-Duffing oscillator based on parameter identification

This paper considers the problem of adaptive synchronization and parameter identification of an uncertain chaotic oscillator. Using recent results on adaptive control, we design a controller which enables both the synchronization of two unidirectionally coupled modified Van der Pol-Duffing oscillators and the estimation of unknown parameters of the drive oscillator.     

The equilibria and stability of a dynamically symmetrical satellite with a two-degree-of-freedom powered gyroscope

For a dynamically symmetrical satellite carrying a two-degree-of-freedom powered gyroscope, all the relative equilibria in a circular orbit are found as a function of the angular momentum of the rotor and the angle between the precession axis of the gyroscope and the plane of dynamical symmetry. The case with no spring on the axis of the gyroscope frame and the case with a spring whose stiffness satisfies definite conditions are considered. The secular stability of the equilibria is investigated. For a system with dissipation in the axis of the gyroscope frame, the Barbashin–Krasovskii theorem is used to perform a detailed analysis, which enables the character of the Lyapunov stability of all the equilibria to be determined, with the exception of a few points. The results of a numerical solution of the problem of the optimal values of the system parameters, for which asymptotically stable equilibria are obtained with maximum speed, are presented.     

Defects analysis and improvement of a chaotic logic gate

We introduced one chaotic logic gate design and described its performance defects. These defects, namely nondeterminacy and variability of the output signal, were depicted and analysis for the reasons behind them was done. Further more, a new logic gate design by changing the variable under control is proposed. By successfully overcoming defects of the prototype, the new chaotic logic gate can be better applied to a complicated computing architecture.     

Nonlinear analysis of a novel three-scroll chaotic system

This paper reports the nonlinear dynamics of a novel three-scroll chaotic system. The local stability of hyperbolic equilibrium and non-hyperbolic equilibrium are investigated by using center manifold theorem. Pitchfork bifurcation, degenerate pitchfork bifurcation and Hopf bifurcation are analyzed when the parameters are varied in the space of parameter. For a suitable choice of the parameters, the existence of singularly degenerate heteroclinic cycles and Hopf bifurcation without parameters are also investigated. Some numerical simulations are given to support the analytic results.     

Synchronization of chaotic systems with parameter driven by a chaotic signal

Chaos control with driving parameter scheme in uncoupled identical chaotic oscillators is presented. By driving the parameter of chaotic systems using external chaotic signal, synchronization and anti-synchronization can be implemented. Numerical simulations show that either synchronization or anti-synchronization can appear depending significantly on initial condition and on driving strength. The proposed method is particularly suited for a variety of chaotic systems, which cannot couple with each other in engineering.     

Synchronization of chaotic systems with parameter driven by a chaotic signal

Chaos control with driving parameter scheme in uncoupled identical chaotic oscillators is presented. By driving the parameter of chaotic systems using external chaotic signal, synchronization and anti-synchronization can be implemented. Numerical simulations show that either synchronization or anti-synchronization can appear depending significantly on initial condition and on driving strength. The proposed method is particularly suited for a variety of chaotic systems, which cannot couple with each other in engineering.     

Bifurcation analysis and feedback control of a 3D chaotic system

In this paper, we analyze a three-dimensional differential system derived from the Chen system based on the first Lyapunov coefficient, and apply it to investigate the local bifurcation. And we present some insights on bifurcation and stability, also obtain some conditions for subcritical and supercritical. Finally, we give some numerical simulation studies of system in order to verify analytic results.     

Dynamical analysis and numerical simulation of a new Lorenz-type chaotic system

In this paper, a new 3D autonomous Lorenz-type chaotic system is modelled based on the condition that the system may generate chaos whereas it has only stable or non-hyperbolic equilibrium points. This system also includes some well-known Lorenz-like systems as its special cases, such as the diffusionless Lorenz system, the Burke-Shaw system and some other systems found. Although the new chaotic system is similar to other Lorenz-type systems in algebraic structure, they are topologically non-equivalent. This interesting fact motivates one to further investigate its dynamical behaviours, such as the number and the stability of equilibrium points, Hopf bifurcation and its direction, Poincaré maps, Lyapunov exponents and dissipativity, etc. Given numerical simulations not only verify the corresponding theoretically analytical results, but also demonstrate that this system possesses abundant and complex dynamical properties, which need further attention.     

Synchronization and anti-synchronization coexist in two-degree-of-freedom dissipative gyroscope with nonlinear inputs

This study demonstrates that synchronization and anti-synchronization can coexist in two-degree-of-freedom dissipative gyroscope system with input nonlinearity. Because of the nonlinear terms of the gyroscope system, the system exhibits complex motions containing regular and chaotic motions. Using the variable structure control technique, a novel control law is established which guarantees the hybrid projective synchronization including synchronization, anti-synchronization and projective synchronization even when the control input nonlinearity is present. By Lyapunov stability theory with control terms, two suitable sliding surfaces are proposed to ensure the stability of the controlled closed-loop system in sliding mode, and two variable structure controllers (VSC) are designed to guarantee the hitting of the sliding surfaces. Numerical simulations are presented to verify the proposed synchronization approach.