Global Analysis of Nonlinear Dynamical Systems

This chapter discusses recent applications and algorithm developments of the cell mapping methods, which were created by C. S. Hsu in 1980s for global analysis of nonlinear dynamical systems. Such systems can have multiple steady-state responses including equilibrium states, periodic motions, chaotic attractors as well as domains of attraction of these steady-state responses. Without the cell mapping methods, these dynamical responses would have been far more difficult to obtain. Since the creation of them, the cell mapping methods have enjoyed attention from the research communities. New extensions of the methods and new algorithms including parallel computing have been developed in the past few decades. The cell mapping methods have also been applied to global analysis and control design of deterministic, stochastic and fuzzy dynamical systems. Representative examples of new applications are presented in this chapter.

[1]  C. Hsu.  GLOBAL ANALYSIS OF DYNAMICAL SYSTEMS USING POSETS AND DIGRAPHS , 1995 .

[2]  Zhi-Chang Qin,et al.  Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems , 2015 .

[3]  V. L. Zakovorotny,et al.  Response analysis of Van der Pol oscillator subjected to harmonic and random excitations , 2014 .

[4]  Qingjie Cao,et al.  Resonances of the SD oscillator due to the discontinuous phase , 2011 .

[5]  C. Hsu,et al.  The Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-Time Gaussian Approximation , 1990 .

[6]  Jian-Qiao Sun,et al.  Stochastic response and bifurcation of periodically driven nonlinear oscillators by the generalized cell mapping method , 2016 .

[7]  Jiang Jun,et al.  An Effective Numerical Procedure to Determine Saddle-Type Unstable Invariant Limit Sets in Nonlinear Systems , 2012 .

[8]  C. Hsu,et al.  Cumulant-Neglect Closure Method for Nonlinear Systems Under Random Excitations , 1987 .

[9]  C. Hsu,et al.  A Global Analysis of an Harmonically Excited Spring-Pendulum System With Internal Resonance , 1994 .

[10]  G. Cai,et al.  A new path integration procedure based on Gauss-Legendre scheme , 1997 .

[11]  Ling Hong,et al.  Crises and chaotic transients studied by the generalized cell mapping digraph method , 1999 .

[12]  C. Hsu,et al.  A Probabilistic Theory of Nonlinear Dynamical Systems Based on the Cell State Space Concept , 1982 .

[13]  Xu Wei.  DEVELOPMENT AND STUDY ON CELL MAPPING METHODS , 2013 .

[14]  J. Yorke,et al.  Crises, sudden changes in chaotic attractors, and transient chaos , 1983 .

[15]  U. Sandler,et al.  Fuzzy dynamics as an alternative to statistical mechanics , 1999, Fuzzy Sets Syst..

[16]  Ling Hong,et al.  Codimension two bifurcations of nonlinear systems driven by fuzzy noise , 2006 .

[17]  Yaakov Friedman,et al.  Evolution of systems under fuzzy dynamic laws , 1996, Fuzzy Sets Syst..

[18]  Benson H. Tongue,et al.  A higher order method of interpolated cell mapping , 1988 .

[19]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application , 1980 .

[20]  Robin S. Langley,et al.  Numerical methods for calculating the response of a deterministic and stochastically excited Duffing oscillator , 2016 .

[21]  Chunbiao Gan,et al.  Noise-Induced Chaos in Duffing Oscillator with Double Wells , 2006 .

[22]  Marco Laumanns,et al.  Computing Gap Free Pareto Front Approximations with Stochastic Search Algorithms , 2010, Evolutionary Computation.

[23]  Zhang Yue,et al.  Solving processes for a system of first-order fuzzy differential equations , 1998, Fuzzy Sets Syst..

[24]  Ling Hong,et al.  Fuzzy Responses and Bifurcations of a Forced Duffing Oscillator with a Triple-Well Potential , 2015, Int. J. Bifurc. Chaos.

[25]  Stefano Lenci,et al.  A Global Dynamics Perspective for System Safety From Macro- to Nanomechanics: Analysis, Control, and Design Engineering , 2015 .

[26]  U. Ruckert,et al.  Multiobjective optimization for transistor sizing of CMOS logic standard cells using set-oriented numerical techniques , 2009, 2009 NORCHIP.

[27]  Meng Guang,et al.  Response statistic of strongly non-linear oscillator to combined deterministic and random excitation , 2004 .

[28]  James J. Buckley,et al.  Applications of fuzzy chaos to fuzzy simulation , 1998, Fuzzy Sets Syst..

[29]  A. Naess,et al.  Response statistics of nonlinear, compliant offshore structures by the path integral solution method , 1993 .

[30]  Benson H. Tongue,et al.  Interpolated Cell Mapping of Dynamical Systems , 1988 .

[31]  M. Wehner.  Numerical Evaluation of Path Integral Solutions to Fokker-Planck Equations with Application to Void Formation. , 1983 .

[32]  Michael Dellnitz,et al.  Chapter 5 - Set Oriented Numerical Methods for Dynamical Systems , 2002 .

[33]  Celso Grebogi,et al.  Archetypal oscillator for smooth and discontinuous dynamics. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Oliver Schütze,et al.  Set Oriented Methods for the Numerical Treatment of Multiobjective Optimization Problems , 2013, EVOLVE.

[35]  Wei Xu,et al.  Global Bifurcation Analysis of a Duffing-Van der Pol Oscillator with Parametric Excitation , 2014, Int. J. Bifurc. Chaos.

[36]  Yoshiyuki Suzuki,et al.  Stochastic averaging of strongly non-linear oscillators under combined harmonic and white-noise excitations , 2000 .

[37]  Andrew J. Dick,et al.  A parallelized multi-degrees-of-freedom cell mapping method , 2014 .

[38]  Michael Dellnitz,et al.  An adaptive subdivision technique for the approximation of attractors and invariant measures , 1998 .

[39]  James J. Buckley,et al.  Fuzzy differential equations , 2000, Fuzzy Sets Syst..

[40]  B. H. Tongue,et al.  On obtaining global nonlinear system characteristics through interpolated cell mapping , 1987 .

[41]  Shaopu Yang,et al.  The codimension-two bifurcation for the recent proposed SD oscillator , 2009 .

[42]  Thomas Sattel,et al.  On Set-oriented Numerical Methods for Global Analysis of Non-smooth Mechanical Systems , 2007 .

[43]  M. Dellnitz,et al.  A subdivision algorithm for the computation of unstable manifolds and global attractors , 1997 .

[44]  W. F. Wu,et al.  CUMULANT-NEGLECT CLOSURE FOR NON-LINEAR OSCILLATORS UNDER RANDOM PARAMETRIC AND EXTERNAL EXCITATIONS , 1984 .

[45]  Benson H. Tongue,et al.  A theoretical basis for interpolated cell mapping , 1988 .

[46]  Celso Grebogi,et al.  The limit case response of the archetypal oscillator for smooth and discontinuous dynamics , 2008 .

[47]  O. Junge,et al.  A set oriented approach to global optimal control , 2004 .

[48]  Jong Yeoul Park,et al.  Fuzzy differential equations , 2000, Fuzzy Sets Syst..

[49]  B. Tongue,et al.  APPLICATION OF INTERPOLATED CELL MAPPING TO AN ANALYSIS OF THE LORENZ EQUATIONS , 1995 .

[50]  Hai-Tao Zhu,et al.  Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations , 2015 .

[51]  Z. E. Musielak,et al.  High-Dimensional Chaos in dissipative and Driven Dynamical Systems , 2009, Int. J. Bifurc. Chaos.

[52]  Ueda,et al.  Safe, explosive, and dangerous bifurcations in dissipative dynamical systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[53]  Jun Jiang,et al.  A method of point mapping under cell reference for global analysis of nonlinear dynamical systems , 1994 .

[54]  F.H. Bursal,et al.  A New Method of Nonlinear System Identification using Interpolated Cell Mapping , 1992, 1992 American Control Conference.

[55]  Jian-Qiao Sun,et al.  Stochastic dynamics and control , 2006 .

[56]  Celso Grebogi,et al.  Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[57]  Jian-Qiao Sun,et al.  Random vibration analysis of a non-linear system with dry friction damping by the short-time gaussian cell mapping method , 1995 .

[58]  Ling Hong,et al.  Chapter 3 Global Bifurcations of Complex Nonlinear Dynamical Systems with Cell Mapping Methods , 2006 .

[59]  Abraham Kandel,et al.  Numerical solutions of fuzzy differential equations , 1999, Fuzzy Sets Syst..

[60]  Xiaole Yue,et al.  Stochastic bifurcations in the SD (smooth and discontinuous) oscillator under bounded noise excitation , 2013 .

[61]  Lars Grüne,et al.  A set oriented approach to optimal feedback stabilization , 2005, Syst. Control. Lett..

[62]  W. K. Lee,et al.  Domains of Attraction of a Forced Beam by Interpolated Mapping , 1994 .

[63]  C. S. Hsu,et al.  A Statistical Study of Generalized Cell Mapping , 1988 .

[64]  A. H. Stroud,et al.  Numerical Quadrature and Solution of Ordinary Differential Equations: A Textbook for a Beginning Course in Numerical Analysis , 1974 .

[65]  Kai Lai Chung,et al.  Markov Chains with Stationary Transition Probabilities , 1961 .

[66]  Z. Ge,et al.  A MODIFIED INTERPOLATED CELL MAPPING METHOD , 1997 .

[67]  H. Risken,et al.  The Fokker-Planck Equation: Methods of Solution and Application, 2nd ed. , 1991 .

[68]  Yuji Yoshida.  A continuous-time dynamic fuzzy system. (I). A limit theorem , 2000, Fuzzy Sets Syst..

[69]  Fu-Rui Xiong,et al.  Global Analysis of Nonlinear Dynamics , 2018, Cell Mapping Methods.