Universality in the transition to chaos of dissipative systems |
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| Institution: | 1. FutureLab on Earth Resilience in the Anthropocene, Earth System Analysis, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Telegrafenberg A31, Potsdam 14473, Germany;2. Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany;3. Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, Stockholm 114 19, Sweden;4. GESIS – Leibniz Institute for the Social Sciences, Unter Sachsenhausen 6-8, Cologne 50667, Germany;5. International Political Economy and Environmental Politics, ETH Zurich, Switzerland;6. Department of Sociology and Human Geography, University of Oslo, Moltke Moes Vei 31, 0851 Oslo;7. FutureLab on Game Theory and Networks of Interacting Agents, Complexity Science, Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Telegrafenberg A31, Potsdam 14473, Germany;8. Institute of Sociology and Social Psychology, University of Cologne, Cologne, Germany;9. Department of Physics, Humboldt University of Berlin, Berlin, Germany;10. Global Systems Institute, University of Exeter, Exeter EX4 4QE, United Kingdom;1. Berlin Institute of Health at Charité, Universitätsmedizin Berlin, Charitéplatz 1, Berlin 10117, Germany;2. Department of Neurology with Experimental Neurology, Charité, Universitätsmedizin Berlin, Corporate member of Freie Universität Berlin and Humboldt Universität zu Berlin, Charitéplatz 1, Berlin 10117, Germany;3. Bernstein Focus State Dependencies of Learning and Bernstein Center for Computational Neuroscience, Berlin, Germany;4. Einstein Center for Neuroscience Berlin, Charitéplatz 1, Berlin 10117, Germany;5. Einstein Center Digital Future, Wilhelmstraße 67, Berlin 10117, Germany;6. Department of Electrical and Computer Engineering, National University of Singapore, Singapore;7. Yong Loo Lin School of Medicine, Centre for Sleep and Cognition and Centre for Translational Magnetic Resonance Research, Singapore;8. N.1 Institute for Health and Institute for Digital Medicine, National University of Singapore, Singapore;9. Integrative Sciences and Engineering Programme (ISEP), National University of Singapore, Singapore, Singapore;10. Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, United States;11. Department of Information and Communication Technologies, Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain;12. Institució Catalana de la Recerca i Estudis Avançats, Barcelona, Spain;13. Department of Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany;14. School of Psychological Sciences, Turner Institute for Brain and Mental Health, Monash University, Melbourne, Clayton, Australia |
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| Abstract: | The period-doubling bifurcation process for two-dimensional transforms exhibits a new class of universality when a small dissipation is taken into account. The effective Jacobian is then defined as a function of both the dissipation and the rank n of the cascade (cycle 2n). Numerical simulations of a simple mechanical system and numerical calculations on the Hénon mapping show that the decrement lies on a continuous curve as function of the effective Jacobian. A method using this result to understand experimental data is explained and a first order approximation of the renormalization process yields an analytic expression of the curve.Among the different transitions to chaos, the period-doubling bifurcation cascade 1, 2] has been extensively studied. This transition is characterized by an experimental convergence rate of the bifurcation threshold sequence to the accumulation point: the threshold of chaos. It is well known that the decrement of this bifurcation cascade can take different values. Each value corresponds to a specific class of systems which can be characterized by some general features of the system undergoing the transition 3, 4, 5]. We are concerned here with the two values; δ(I) = 4.699… the decrement of the well-known one-dimensional transform with a quadratic maximum 2] and δ(II) = 8.721 the decrement of a two-dimensional non-dissipative transforms 3]. These two classes of systems are generic in physics and the two values δ(I) and δ(II) are therefore relevant values of the decrement. However, these two exponents stand for the infinite dissipation case and the conservative one thus leaving out the general physical situation of a finite dissipation. Only hints of the effect of a small dissipation in a two-dimensional mapping have been given 6] before the work of Zisook 7].A thorough study of the effect of dissipation is set forth here. The first two sections deal with the physical model used to perform the numerical investigation and the “experimental” data thus obtained. A study of the renormalization process enables to generalise the relation δn(J)=δn−(J2), first given by Zisook in 7], to all transforms where the Jacobian does not depend on the linearization point in the phase space. Furthermore a first order approximation gives an excellent analytic expression of the universal function displaying the crossover of the decrement between δ(II) and δ(I). |
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