Kazuyuki Yagasaki

Chaos in a pendulum with feedback control

We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikovs method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.

The method of Melnikov for perturbations of multi-degree-of-freedom Hamiltonian systems

We develop a Melnikov-type global perturbation technique for detecting the existence of transverse homoclinic orbits and occurrence of homoclinic bifurcations in periodic perturbations of multi-degree-of-freedom Hamiltonian systems. The unperturbed system is assumed to have a saddle-centre whose stable and unstable manifolds do not coincide but intersect in a lower-dimensional manifold, and does not have to be completely integrable. Other Melnikov-type methods do not apply in this situation. We also apply our technique to a four-degree-of-freedom model of nonplanar vibrations of a forced, buckled beam.

Dynamics of a weakly nonlinear system subjected to combined parametric and external excitation

We study the dynamics of a weakly nonlinear single-degree-of-freedom system subjected to combined parametric and external excitation. The averaging method is used to establish the existence of invariant tori and analyze their stability. Furthermore, by applying the Melnikov technique to the average system it is shown that there exist transverse homoclinic orbits resulting in chaotic dynamics

Second-order averaging and chaos in quasiperiodically forced weakly nonlinear oscillators

Abstract Using the averaging method and Melnikovs technique, we study the chaotic dynamics of a class of quasiperiodically forced weakly nonlinear oscillators. Averaging is carried out to second order and the existence of transverse homoclinic tori resulting in chaotic dynamics for quasiperiodically forced oscillators is detected. We also give an example for a “Duffing-type” weakly nonlinear oscillator subjected to combined parametric and external forcing in the case of double resonance in which primary external resonance and subharmonic or superharmonic parametric resonance exist simultaneously.

The Melnikov theory for subharmonics and their bifurcations in forced oscillations

The subharmonic Melnikov theory for periodic perturbations of planar Hamiltonian systems is improved. An approximation to the associated Poincare map in action-angle coordinates is explicitly constructed, and existence, stability, and bifurcation theorems for subharmonics are obtained. In particular, simple formulas for determining the stability of subharmonics and invariant circles bifurcating from them at Hopf bifurcations are obtained, and a degenerate resonance case, which was not appropriately treated in previous references, is discussed. Furthermore, the weak nonlinearity case, in which the unperturbed system is linear, is studied. The results are also useful to describe dynamics near the unperturbed centers in strongly nonlinear systems. Several examples are given to illustrate our theory.

CONTROLLING CHAOS IN A PENDULUM SUBJECTED TO FEEDFORWARD AND FEEDBACK CONTROL

We consider a pendulum subjected to feedforward and feedback control, in which chaotic motions occur when the feedback gain is small. We apply two control techniques recently proposed by Ott and his coworkers so that the pendulum can exhibit the desired motion. The techniques use some dynamical properties relating to chaos. Numerical examples are given and the effectiveness of these control techniques is demonstrated.

A new approach for controlling chaotic dynamical systems

Abstract The chaos control method proposed by Ott and his coworkers and now called the OGY method has attracted much attention. However, in some applications this technique requires a very long time until the control applies and it is not so effective. In this Letter, we present a new chaos control method in which this problem is improved. The main difference from the OGY method is the use of nonlinear approximations for chaotic dynamical systems and stable manifolds of the target points. We give an example for the Henon map to demonstrate the effectiveness of the present method. Our method is also shown to be robust to modeling errors like the OGY method.

Influence of metal plating treatment on the electric response of Nafion

A potentially promising material as a high performance electroactive polymer gel actuator, Nafion, is known for its fast and large bending upon an applied voltage. Not long ago, it was reported that copper plating on Nafion enhances the degree of its bending. We performed the combinational metal plating treatment on Nafion surfaces with silver, copper and nickel, and the performances of metal plated Nafions—bending curvature and generated force—upon an applied voltage were quantitatively evaluated. From the obtained results, it was speculated that the hydrated mobile ions play substantial roles for the large bending of Nafion as has been widely believed and for the enhancement of its generated force. In addition to them, nickel plating on Nafion surfaces was found to enable Nafion to exhibit a large force without a significant force decay owing to the low-elastic property of nickel layers.

Periodic and Homoclinic Motions in Forced, Coupled Oscillators

We study periodic and homoclinic motions in periodically forced, weakly coupled oscillators with a form of perturbations of two independent planar Hamiltonian systems. First, we extend the subharmonic Melnikov method, and give existence, stability and bifurcation theorems for periodic orbits. Second, we directly apply or modify a version of the homoclinic Melnikov method for orbits homoclinic to two types of periodic orbits. The first type of periodic orbit results from persistence of the unperturbed hyperbolic periodic orbit, and the second type is born out of resonances in the unperturbed invariant manifolds. So we see that some different types of homoclinic motions occur. The relationship between the subharmonic and homoclinic Melnikov theories is also discussed. We apply these theories to the weakly coupled Duffing oscillators.

A simple feedback control system: Bifurcations of periodic orbits and chaos

We consider a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small time constant, so that the feedback control system can be approximated by a periodically forced oscillator. It was previously shown by Melnikovs method that transverse homoclinic and heteroclinic orbits exist and chaos may occur in certain parameter regions. Here we study local bifurcations of harmonics and subharmonics using the second-order averaging method and Melnikovs method. The Melnikov analysis was performed by numerically computing the Melnikov functions. Numerical simulations and experimental measurements are also given and are compared with the previous and present theoretical predictions. Sustained chaotic motions which result from homoclinic and heteroclinic tangles for not only single but also multiple hyperbolic periodic orbits are observed. Fairly good agreement is found between numerical simulation and experimental results.