A New Reliable Numerical Method for Computing Chaotic Solutions of Dynamical Systems: The Chen Attractor Case

This paper describes a new reliable numerical method for computing chaotic solutions of dynamical systems and, in special cases, is applied to Chen strange attractor. The numerical precision of the computation is finely mastered. We introduce a modification of the method of power series for the construction of approximate solutions of the Chen system together with forward/backward control of the precision. As a test for the method, we obtained the region of convergence of series and researched the behavior of the trajectories on this attractor. The results of a numerical experiment are presented.

[1]  A. Gibbons,et al.  A Program for the Automatic Integration of Differential Equations using the Method of Taylor Series , 1960, Comput. J..

[2]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[3]  I. Babuska,et al.  Numerical processes in differential equations , 1968 .

[4]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

[5]  M. Hénon.  A two-dimensional mapping with a strange attractor , 1976 .

[6]  James A. Yorke,et al.  Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model , 1979 .

[7]  C. Sparrow.  The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors , 1982 .

[8]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[9]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[10]  W. Tucker.  The Lorenz attractor exists , 1999 .

[11]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[12]  Guanrong Chen,et al.  Bifurcation Analysis of Chen's equation , 2000, Int. J. Bifurc. Chaos.

[13]  D. A. Kaloshin,et al.  Search for and Stabilization of Unstable Saddle Cycles in the Lorenz System , 2001 .

[14]  Warwick Tucker,et al.  Foundations of Computational Mathematics a Rigorous Ode Solver and Smale's 14th Problem , 2022 .

[15]  J. Butcher.  Numerical Methods for Ordinary Differential Equations: Butcher/Numerical Methods , 2005 .

[16]  Ravi P. Agarwal,et al.  Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems , 2008 .

[17]  René Lozi,et al.  Can we trust in numerical computations of chaotic solutions of dynamical systems , 2013 .

[18]  Zdenek Beran,et al.  Characteristics of the Chen Attractor , 2013, NOSTRADAMUS.

[19]  Alexander N. Pchelintsev.  Construction of Periodic Solutions of One Class Nonautonomous Systems of Differential Equations , 2013 .

[20]  Alexander N. Pchelintsev.  Numerical and physical modeling of the dynamics of the Lorenz system , 2014 .

[21]  Zbigniew Galias,et al.  Is the Hénon attractor chaotic? , 2015, Chaos.

[22]  V. Stepanov,et al.  Qualitative theory of differential equations , 1960 .