The response of the superconducting sheath state of Pb-2wt% In to low-frequency (20-500 cps), smallamplitude (10~2 Oe <^0<20 Oe) ac fields has been studied. Direct observation of the wave form shows that screening currents are developed in the sheath up to a critical value J c , when the field begins to penetrate. The screening current continues to increase until a saturation value J 8 is reached, at which point the current in the sheath stays constant for the remainder of the half-cycle. Fourier analysis of a model wave form for which J m =Jc= z Js indicates the harmonic content, and nonlinear nature, of the response. Measurements of the real and imaginary parts of the ac permeability, /*' and /*", at the fundamental frequency as a function of the ac field amplitude at constant dc field shows that the transition is characterized by only one parameter, dependent on the dc field, and that the model gives an excellent description of the nonlinearity. Welldefined quantitative critical-current data can be obtained for comparison with the theories of Abrikosov, Park, and Fink and Barnes. The relationship between these observations and other measurements of // and M" as a function of dc field at constant ac field amplitude is discussed. It is shown that the transition is sensitive to misalignment of the specimen relative to the dc field. The change in the transition as the frequency is changed is explored, and a preliminary conclusion is that J 8 is much more sensitive to frequency than J c .
A model of a simple nonlinear physical system, the driven diode resonator comprised of an oscillator, resistor, inductor, and diode in series, is shown to reduce exactly to a one-dimensional, noninvertible map. With use of a model of the diode which includes the forward bias voltage, reverse recovery time, and junction capacitance, the response of the system is calculated exactly. The solution exhibits the period-doubling route to chaos with universal scaling. PACS numbers: 05.40.+j, 02.90.+p, 47.25.-c Period doubling and chaotic behavior were recently reported by Linsay' and by Testa, Pyrex, and Jeffries' for the response of a driven anharmonic resonator consisting of a series circuit composed of a resistance, inductance, and a Varactor diode as shown in Fig. 1(a). This diode resonator was shown to follow a patterned route to chaos in good agreement with the universal behavior found in iterated, unimodal, one-dimensional maps. ' ' Several recent experiments on a variety of nonlinear physical systems have revealed similar patterned chaotic behavior"' and there is an active effort to develop an understanding of the dynamics of complex nonlinear physical systems by using much simpler models which are universally applicable to large classes of these systems. "" It is important to have a clear understanding of the physical conditions present in the nonlinear system which lead to its approximate description by a simple model. such as a one-dimensional map. It is the purpose of this paper to present a realistic physical model of the nonlinear diode resonator and to show that this model. provides, for the first time, an exact description of the response in terms of a one-dimensional, noninvertible mapping function which is explicitly defined.The previous work" on the diode resonator attributes the period doubling and chaotic behavior partially to the nonlinearity introduced by the voltage-dependent capacitance of the Varactor.However, Hunt' commented that another property of such diodes was responsible for the behavior; namely, the rather large reverse recovery time. We show here that both. a finite forward bias voltage and a finite reverse recovery time are required if the diode resonator is to exhibit chaotic behavior. We totally neglect the changing capacitance of the Varactor and believe that it is unimportant with regard to the salient features of the response. We assume that the diode will be{b) {c) FlG. 1. (a) Driven Varactor-diode resonator circuit. (b) When the diode is conducting, it is replaced by an emf=~f. (c) %(hen the diode is off, it is replaced by capacitance C. 1982 The American Physical Society 1295
A new ac technique is described which allows the experimental determination of the magnetic field profile near the sample surface in type-II superconductors with pinning. The technique, like a previous method of Campbell. depends only on the validity of the critical-state model and can separate the position and magnetic field dependence of the flux-pinning forces. The entire field profile is obtained from a measurement of the waveform of the response of the sample to a small ac magnetic field superimposed on the dc field. Profiles obtained in cold-worked Nb and Nb-Ti alloys are reported" and the results compared with previous measurements. We have found that the critical-state model does not adequately describe the results obtained on the Nb-Ti alloys when the magnetic field is less than 0.5Hez: 5392
We report experimental measurements and calculations using a model on a driven, dissipative, dynamical system which shows chaotic behavior. The system is the diode resonator composed of the series combination of a generator, inductor, and a p-n-junction diode. It is studied where there are sudden transient changes in the strange attractor, phenomena called crises by Grebogi, Ott, and Yorke, for which a universal scaling law exists. We verify the scaling law both experimentally and with model calculations. Furthermore, the Lyapunov exponent, a measure of sensitivity to initial conditions, is shown by both methods to increase rapidly but continuously through the crisis region. GENERAL INTRODUCTION QUANTITATIVE DESCRIPTION OF CRISISThe motion of driven nonlinear dissipative physical systems is often observed to settle into a state of sustained nonperiodic turbulent or chaotic behavior. It has been found that many of these systems show patterned routes to chaos which are well described by the universal behavior of iterated, unimodal, one-dimensional maps. Also, many features of the chaotic behavior of these systems are surprisingly well described by simple onedimensional maps. We report here the experimental observation of a "universal" scaling law in the chaotic behavior of the p-n-junction diode resonator in the vicinity of interior crisis. ' We discuss the behavior in terms of the exact one-dimensional mapping function which we have recently obtained from a model of the p-n-junction diode and in terms of an extension of that model to two dimensions.We also report values of the I.yapunov exponent, which gives a measure of the predictability for the system, in the neighborhood of interior crisis.Recently, Grebogi, Ott, and Yorke ' have discussed the occurrence of sudden qualitative changes in the chaotic dynamics of nonlinear systems in terms of one-and twodimensional quadratic maps. These changes occur at particular values of the "drive" parameter, where an unstable periodic orbit enters the region of phase space occupied by the orbit of the sustained chaotic state into which the system settles. The region of phase space into which the systern settles is called an attractor and Grebogi, Ott, and Yorke have assigned the term crisis to the phenomenon of the joining of the unstable orbit with the attractor. Recent reports ' of experimental observations of the driven nonlinear p-n-junction diode resonator show the qualitative changes in the attractor at crisis as described by Grebogi, Ott, and Yorke. In this paper we report both measurements and model calculations for the p-n-junction diode-resonator system which show the onset of intermittent, transient, chaotic behavior of the response as the amplitude of the drive voltage is increased beyond the critical value where crisis occurs.The response of a driven anharmonic p-n-junction diode resonator, composed of a resistance, inductance, and a diode in series with an oscillator, has been found to exhibit a pattern of period doubling ' and tangentbifurcation-intermittency...
Periodic and chaotic current oscillations are observed during electrodissolution of copper in a pH 3.5 sodium acetate/ acetic acid buffer under potentiostatic conditions using a rotating copper disk electrode. Periodic or chaotic oscillations are observed depending on the applied potential and electrode rotation rate. The oscillations arise after the formation and dissolution of an acetate salt film precursor to oxide passivation. The nature and composition of the surface films were examined using scanning electron microscopy and x‐ray powder diffraction data. Nonlinear dynamic analysis methods have been used to show the existence of deterministic chaos. Time series data of the chaotic oscillations were used to generate a return map that appears to be one‐dimensional. The chaotic attractor was reconstructed in a three‐dimensional state space using the method of time delays and the largest Lyapunov exponent was calculated and was positive for the chaotic oscillations indicating a sensitive dependence on initial conditions characteristic of deterministic chaos.
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