Publications with Abstracts by Alfred Hübler (1990)
* Publications in journals with a strict referee process
- * E.A. Jackson, A. Hübler, Periodic Entrainment of Chaotic Logistic Map Dynamics, Physica D 44, 407-420 (1990)
Abstract: Consider the map dynamics xn+1 = F(xn;c), with a control parameter c. Let the governing set {gn | n=0,1,2,…} be a desired periodic dynamic set (gN+n≡gn). It is noted that the non-autonomous system xn+1 = F(xn;c) + Gn has such a solution, xn = gn, if Gn = gn+1 - F(gn;c). In particular, the set of values {gn} might be obtained from the periodic solutions of gn+1 = F(gn,c*), using suitable values of c*. This study explores the values of c* which yield entrained solutions, their basin of entrainment, {x0 | limn→∞|xn - gn| = 0}, and more generally the basins of bounded solutions and their character, when F(x,c) = cx(1 - x), the logistic map. Of particular interest is the entrainment of chaotic dynamics, c = 4. Generally, the basin of entrainment for period-one governing, g0, is {x0 | 1 - g0 - 1/c < x0 < g0 + 1/c} for all allowed g0, (c - 1)/2c < g0 < (1 + c)/2c. There are bounded but non-entrained solutions if (2 + c)/2c > g0 > (1 + c)/2c, with the above basin, or if (c - 1)/2c > g0 > (c - 4)/2c, with the basin {x0 | g0 < x0 < 1 - g0}. The complimentary x0 region yields unbounded dynamics. Period-two governing values have two basins, depending on the initial value used for g0. These basins are investigated for the sets gn+1 = F(gn, c*). Other periodic sets, {gn} in Gn do not become disjoint in space. Period-four entrainment, for suitable c*, can have even larger basins. The unwanted occurrence of other disjoint basins of attraction, which interlace the basins of entrainment, as a function of both x0 and g0, is discussed. Other periodic and non-periodic responses to these periodic Gn are also studied.
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- * T. Eisenhammer, A. Hübler, T. Geisel, E. Lüscher, Scaling Behavior of the Maximum Energy Exchange between Coupled Anharmonic Oscillators, Phys.Rev. A 41, 3332-3342 (1990)
Abstract: The maximum energy exchange of two harmonically coupled nonlinear oscillators is investigated. We calculate the maximum energy exchange close to resonance and show that the corresponding resonance curves have a universal shape and become broader and smaller when the amplitude-frequency coupling becomes large. Since there is a large variety of nonlinear oscillators where the trajectories are nearly homothetic curves in a phase-space representation, we furthermore investigate the special situation where the oscillators are homothetic. We argue that in this case there is a scaling of the maximum energy exchange at resonance. Numerical investigations show that these relations remain valid if the oscillators are slightly damped or perturbed by random noise.
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- * J.L.Breeden, F. Dinkelacker, A. Hübler, Noise in the Modeling and Control of Dynamical Systems, Phys.Rev. A 42, 5827-5835 (1990)
Abstract: We demonstrate how noise can be an effective tool in modeling systems whose experimental data sets would normally be limited to a small region of the reconstructed state space. In fact, for systems with stable fixed points, using noise to extend the accessible stat-space volume may be the only possibility for constructing a model. We find that noise can also be useful in modeling limit cycles when multiple systems generate the same closed trajectory and the model that represents the true dynamics is desired. We discuss the implications of our method for nonlinear control theory about which important questions on the effects of noise in real-time modeling have arisen.
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- V. Thurner, W. Eberl, A. Hübler, N.Packard. E. Lüscher, Determination of the Absolute Maximum of Polynominals by Algebraic Bisection, to appear in Helv.Phys.Acta
Abstract: Many theoretical approaches to physical phenomena lead to optimization problems, typically to the problem of finding the absolute maximum of a polynomial [e.g. modeling if complex systems, spin glasses]. Well-known methods like gradient dynamics or simulated annealing have the disadvantage of not being definitely reliable in finding the absolute extremum. We present a new method where we divide the polynomial's interval into several parts, prove that some of these parts cannot contain the absolute maximum and eliminate those parts from further calculations.
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- W. Eberl, A. Hübler. N. Packard, E. Lüscher, Unique Models for Stochastic Dynamics, to appear in Helv.Phys.Acta
Abstract: Starting from a noisy set of experimental data, we reconstruct maps in order to describe and control the dynamics of the corresponding system. When including long-time correlations and colored noise the problem of modeling leads to the problem of finding the absolute maximum of a likelihood-function.
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- * J. Breeden, A. Hübler, Reconstructing Equations of Motion from Experimental Data with Unobserved Variables, Phys.Rev. A 42, 5817-5826 (1990)
Abstract: We have developed a method for reconstructing equations of motion where all the necessary variables have not been observed. This technique can be applied to systems with one or several such hidden variables, and can be used to reconstruct maps or differential equations. The effects of experimental noise are discussed through specific examples. The control of nonlinear systems containing variables is also discussed.
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- * B. Plapp, A. Hübler, Nonlinear Resonances and Suppression of Chaos in the rf-Biased Josephson Junction, Phys.Rev.Lett 65, 2303-2306 (1990)
Abstract: The response of rf-biased Josephson junctions to special aperiodic driving forces is studied through theory and numerical simulation. It is shown that aperiodic driving forces of very small amplitude can transform the junction from a stationary state into a rotation state. In addition, it can be shown that the resulting dynamics is not chaotic, in contrast to the generic dynamics resulting from a sinusoidal driving force. We discuss possible experimental applications.
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- * F. Ohle, P. Lehmann, E. Roesch, H. Eckelmann, A. Hübler, G. Mayer-Kress, P.W. Milonni, Description of Transient States of Karman Vortex Streets by Low Dimensional Differential Equations, Phys. Fluids A 2(4), 479-481 (1990)
Abstract: Aperiodic time series of hot-wired signals can be described as trajectories in a state space representation. The flow vector field is calculated by numerical differentiation of these trajectories and then each component of the flow vector field is approximated by a polynomial of order p. This approximation provides a model for the dynamics of the von Karman vortex street by a low-dimensional system of ordinary differential equations. At a Reynold number of 14 a compact description of the complex dynamics of the vortex street by a set of only ten parameters can be obtained. It will be shown that these parameters are independent of the probe position for distances greater than two-and-one-half cylinder diameters.
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