Publications with Abstracts by Alfred Hübler (1992)

1. * R. Wackerbauer, A. Hübler, G. Mayer-Kress, Algebraic Calculation of Stroboscopic Maps of Ordinary, Nonlinear Differential Equations, Physica D 60, 335-337 (1992)
   Abstract: The relation between the parameters of a differential equation and corresponding discrete maps is becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbations methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. An iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from OEDs with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is a polynomial in the state variables and if the ODE has a fixed point at the origin. For several nonlinear systems approximations of different orders are investigated.

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2. A. Hübler, Modeling and Control of Complex Systems: Paradigms and Applications, in Modeling Complex Phenomena, L. Lam, A. Naroditsky(Eds.), Springer, New York, pp.5-65 (1992)
   Abstract: In many cases, the dynamics of high dimensional nonlinear systems can be estimated from a low dimensional model. Nearly all variables are slaved by a few order parameters. If the complex system is perturbed by an external force on order to control it or to investigate it with a spectroscopic method, slaved variables can be stimulated and the prediction of the response from the low dimensional model may be impossible. We show that it is generally possible to predict the response of the complex system and to control it, if the external forces are resonant perturbations of the low dimensional model. We present this issue in terms of a few paradigms including the principle of the dynamical key, the principle of optimal interaction and the principle of matching in the framework of other paradigms in complex systems research. We discuss open problems as well as possible industrial applications.

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3. * S. Krempl, T. Eisenhammer, A. Hübler, G. Mayer-Kress, P.W. Milonni, Optimal Stimulation of a Conservative Nonlinear Oscillator: Classical and Quantum-mechanical Calculations, Phys.Rev.Lett. 69, 430-434 (1992)
   Abstract: A new method for nonlinear polychromatic resonant stimulation of conservative nonlinear oscillators is introduced. As an example we consider a Morse potential that serves as a model for the HF molecule. Numerical results show that a large energy transfer to such a conservative oscillator is possible under optimal stimulation with small driving fields. This makes selective excitation of specific modes possible. The classically determined optimal force was also applied to the corresponding quantum system, with similar results in energy tranfer and dissociation.

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4. J. Wang, T. Meyer, A. Hübler, The Production of Solitons by Optimal Driving Forces, 1991 Lect. in Complex Systems, SFI Studies in the Sciences of Complexity, Lect.Vol.IV, Eds. L. Nadel & D. Stein, Addison-Wesley 1992
   Abstact: In general, nonlinear waves are not stable in a chain of finite length. Since they have a finite lifetime, it is important to investigate the production of nonlinear waves, e.g. the production of solitons. A general feature of nonlinear waves is the amplitude frequency coupling, which causes the excitation by sinusoidal driving forces to be very inefficient. The response is usually very complex in addition. We present a method to calculate special driving forces, which generates nonlinear waves very efficiently. The response to these driving forces is very simple.

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