Numerical Bifurcation Theory for High-Dimensional Neural Models
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The Lorenz System : Hidden Boundary of Practical Stability and the Lyapunov Dimension
This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Kuznetsov, N. V.; Mokaev, T. N.; Kuznetsova, O. A.; Kudryashova, E. V. Title: The Lorenz system : hidden boundary of practical stability and the Lyapunov dimension Year: 2020 Version: Published version Copyright: © 2020 the Authors Rights: CC BY 4.0 Rights url: https://creativecommons.org/licenses/by/4.0/ Please cite the original version: Kuznetsov, N. V., Mokaev, T. N., Kuznetsova, O. A., & Kudryashova, E. V. (2020). The Lorenz system : hidden boundary of practical stability and the Lyapunov dimension. Nonlinear Dynamics, 102(2), 713-732. https://doi.org/10.1007/s11071-020-05856-4 Nonlinear Dyn https://doi.org/10.1007/s11071-020-05856-4 ORIGINAL PAPER The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension N. V. Kuznetsov · T. N. Mokaev · O. A. Kuznetsova · E. V. Kudryashova Received: 16 March 2020 / Accepted: 29 July 2020 © The Author(s) 2020 Abstract On the example of the famous Lorenz Keywords Global stability · Chaos · Hidden attractor · system, the difficulties and opportunities of reliable Transient set · Lyapunov exponents · Lyapunov numerical analysis of chaotic dynamical systems are dimension · Unstable periodic orbit · Time-delayed discussed in this article. For the Lorenz system, the feedback control boundaries of global stability are estimated and the difficulties of numerically studying the birth of self- excited and hidden attractors, caused by the loss of 1 Introduction global stability, are discussed. The problem of reliable numerical computation of the finite-time Lyapunov In 1963, meteorologist Edward Lorenz suggested an dimension along the trajectories over large time inter- approximate mathematical model (the Lorenz system) vals is discussed. -
Numerical Bifurcation Analysis N 6329
Numerical Bifurcation Analysis N 6329 71. Popov G (2004) KAM theorem for Gevrey Hamiltonians. Ergod Books and Reviews Theor Dynam Syst 24:1753–1786 Braaksma BLJ, Stolovitch L (2007) Small divisors and large multi- 72. Ramis JP (1994) Séries divergentes et théories asymptotiques. pliers (Petits diviseurs et grands multiplicateurs). Ann l’institut Panor Synth pp 0–74 Fourier 57(2):603–628 73. Ramis JP, Schäfke R (1996) Gevrey separation of slow and fast Broer HW, Levi M (1995) Geometrical aspects of stability theory for variables. Nonlinearity 9:353–384 Hill’s equations. Arch Rat Mech An 131:225–240 74. Roussarie R (1987) Weak and continuous equivalences for fam- Gaeta G (1999) Poincaré renormalized forms. Ann Inst Henri ilies of line diffeomorphisms. In: Dynamical systems and bifur- Poincaré 70(6):461–514 cation theory, Pitman research notes in math. Series Longman Martinet J, Ramis JP (1982) Problèmes des modules pour les èqua- 160:377–385 tions différentielles non linéaires du premier ordre. Publ IHES 75. Sanders JA, Verhulst F, Murdock J (1985) Averaging methods 5563–164 in nonlinear dynamical systems. Revised 2nd edn, Appl Math Martinet J Ramis JP (1983) Classification analytique des équations Sciences 59, 2007. Springer différentielles non linéaires résonnantes du premier ordre. Ann 76. Siegel CL (1942) Iteration of analytic functions. Ann Math Sci École Norm Suprieure Sér 416(4):571–621 43(2):607–612 Vanderbauwhede A (2000) Subharmonic bifurcation at multiple 77. Siegel CL, Moser JK (1971) Lectures on celestial mechanics. resonances. In: Elaydi S, Allen F, Elkhader A, Mughrabi T, Saleh Springer, Berlin M (eds) Proceedings of the mathematics conference, Birzeit, 78. -
Krauskopf, B., Osinga, HM., Doedel, EJ., Henderson, ME., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., & Junge, O
Krauskopf, B., Osinga, HM., Doedel, EJ., Henderson, ME., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., & Junge, O. (2004). A survey of methods for computing (un)stable manifolds of vector fields. http://hdl.handle.net/1983/82 Early version, also known as pre-print Link to publication record in Explore Bristol Research PDF-document University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ A survey of methods for computing (un)stable manifolds of vector fields B. Krauskopf & H.M. Osinga Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR, UK E.J. Doedel Department of Computer Science, Concordia University, 1455 Boulevard de Maisonneuve O., Montr´eal Qu´ebec, H3G 1M8 Canada M.E. Henderson IBM Research, P.O. Box 218, Yorktown Heights, NY 10598, USA J. Guckenheimer & A. Vladimirsky Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853{4201, USA M. Dellnitz & O. Junge Institute for Mathematics, University of Paderborn, D-33095 Paderborn, Germany Preprint of May 2004 Keywords: stable and unstable manifolds, numerical methods, Lorenz equation. Abstract The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable mani- fold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example | the two-dimensional stable man- ifold of the origin in the Lorenz system. -
Arxiv:2106.10725V2 [Nlin.CD] 23 Jun 2021
Analytical and numerical study of the hidden boundary of practical stability: complex versus real Lorenz systems 1, 2, 3, 1 1, 4 5 6 N.V. Kuznetsov, ∗ T.N. Mokaev, A.A.-H. Shoreh, A. Prasad, and M.D. Shrimali 1Faculty of Mathematics and Mechanics, St. Petersburg State University, Peterhof, St. Petersburg, Russia 2Faculty of Information Technology, University of Jyv¨askyl¨a,Jyv¨askyl¨a,Finland 3Institute for Problems in Mechanical Engineering RAS, Russia 4Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt 5Department of Physics & Astrophysics, Delhi University, India 6Central University of Rajasthan, Ajmer, India (Dated: June 25, 2021) This work presents the continuation of the recent article "The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension" [1], published in the Nonlinear Dynamics journal. In this work, in comparison with the results for classical real-valued Lorenz system (henceforward { Lorenz system), the problem of analytical and numerical identification of the boundary of global stability for the complex-valued Lorenz system (henceforward { complex Lorenz system) is studied. As in the case of the Lorenz system, to estimate the inner boundary of global stability the possibility of using the mathematical apparatus of Lyapunov functions (namely, the Barbashin-Krasovskii and LaSalle theorems) is demonstrated. For additional analysis of homoclinic bifurcations in complex Lorenz system a special analytical approach by Vladimirov is utilized. To outline the outer boundary of global stability and identify the so-called hidden boundary of global stability, possible birth of hidden attractors and transient chaotic sets is analyzed. Keywords: chaos, Lorenz system, complex Lorenz system, boundary of global stability, hidden attractors, transient chaos I.