Kazuhisa Tomita

Chaotic response of nonlinear oscillators

Abstract A review is given of the chaotic response of nonlinear oscillators which is a typical example of chaotic, or turbulent phase, now attracting attention in various fields of research. Resorting to the quasi-one-dimensional character of the stroboscopic phase portrait of the chosen model, one-dimensional description and analysis have been presented at some length. This is the simplest case exhibiting chaos, and concepts and tools treating the problem are most abundant. Correlation spectra, invariant measure and Liapunov number are all of use. In addition an example of statistical mechanics describing chaos is presented. The associated variation principle has some analogy with that in equilibrium statistical thermodynamics; however, the quantity to be maximized is the rate of information loss rather than the information loss tself. A typical example of the period doubling route to chaos is found in our model. The phenomenological renormalization theory is described which puts this route on a universal basis. Examples of more general two-dimensional stroboscopic portrait are given and discussed using the concept of homo- (and hetero-) clinicity. Two perturbation theoretic approaches are described to locate the onset of homo-clinicity in the parameter space.

Chaotic response of a limit cycle

External periodic modulation of a nonlinear oscillator may lead to chaotic behavior. This phenomenon is attributed to the existence of a strange attractor, which embodies essentially a folding motion as is met within the Bernoulli shift or the bakers transformation. The results obtained for the Brussels model are discussed from this viewpoint.

Chaos in the Belousov-Zhabotinsky reaction in a flow system

Abstract Chaotic output has been obtained from a simplified model of the Belousov-Zhabotinsky reaction under stirred flow conditions. Both phase portraits and correlation spectra indicate that there exists a region of chaos between two different types of limit cycle.

Stroboscopic phase portrait and strange attractors

Abstract External periodic modulation of a nonlinear oscillator may lead to a chaotic output behaviour. This phenomenon is attributed to the existence of a strange attractor, which embodies essentially a folding motion as is met in Bernoulli shift or the Bakers transformation.

Entrainment of a Limit Cycle by a Periodic External Excitation

Entrainment of a limit cycle by a periodic external excitation 1s investigated with the Prigogine-Lefever·Nicolis model for chemical reaction. In the neighbourhood of the marginal point a quasi-harmonic theory is developed and the theoretical prediction is compared with the numerical computation. The stability of the entrained oscillation is examined in particular. The instabilities are classified into two types, i.e., hard- and soft-mode instabilities, by the use of the Floquet exponents which are, calculated by a non-perturbational method. The hard·mode instability corresponds to the limit of entrainment and the amplitude is subject to a modulation beyond the threshold. The frequency of modulation is estimated from the Floquet exponent, which is compared with the results of numerical computation. The soft. mode instability corresponds to a jump phenomenon in the amplitude. The smaller amplitude branch is liable to a modulation due to a superimposed hard-mode instability. As a whole, a reasonable agreement is obtained between numerical and theoretical results.

Green's Function Theory of Magnetic Relaxation. I General Formulation

It is shown that the method of two-time Greens function may be used as a systematic way of investigating the behaviour of relaxation in magnetic materials. A stochastic assumption has he en introduced to terminate the chain of higher order Greens functions. As a result the macroscopic parameter of relaxation in a localized spin assembly is expressed by the interaction constant and the spatial correlation of spin deviations at a single time in an equilibrium assembly. The temperature dependence of the latter quantity may also be calculated by the same method. Only the general formulation is presented in part I, and examples will be given in subsequent publications on the width of resonance in ferroand antiferromagnetic substances.

Irreversible circulation and the undampled spiking in lasers

Abstract Our new theory of fluctuation was applied to the undamped spiking in lasers with single mode saturable absorption. It was found that the irreversible circulation α diverges at the thresholds of hard mode instabilities, which leads to limit cycle.

On the State of Solid Hydrogen

Co-operative appearance of the rotation of hydrogen molecules in the solid phase is described using a semi-classical theory. The order of magnitude of the restriction to the rotation which is experienced by a molecule at the lowest temperature is estimated to be about 4°K using several different types of experiments, and the results are in reasonable agreement with each other. For the case of pure ortho-hydrogen a calculation of the potential energy and the anisotropy based on intermolecular forces is carried out in the Appendix.

A General Theory of Magnetic Double Resonance

A general theory is presented for describing a system which consists of two interacting different species

Bifurcations of the coupled logistic map

The coupled logistic map in this paper is a system of two identical logistic maps coupled symmetricaIiy. It is an example in which a simple system reveals a very complicated dynamics. We investigate its bifurcations by perturbative methods and numerical calculations and seek for the mechanism for the complicated behavior.