Computation of focus values with applications

Computation of focus (or focal) values for nonlinear dynamical systems is not only important in theoretical study, but also useful in applications. In this paper, we compare three typical methods for computing focus values, and give a comparison among these methods. Then, we apply these methods to study two practical problems and Hilbert's 16th problem. We show that these different methods have the same computational complexity. Finally, we discuss the “minimal singular point value” problem.

[1]  Pei Yu,et al.  Analysis on Double Hopf Bifurcation Using Computer Algebra with the Aid of Multiple Scales , 2002 .

[2]  Jimmy J. M. Tan,et al.  On some super fault-tolerant Hamiltonian graphs , 2004, Appl. Math. Comput..

[3]  Ali H. Nayfeh,et al.  The Method of Normal Forms , 2011 .

[4]  Jaume Llibre,et al.  Limit cycles of a class of polynomial systems , 1988, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[5]  Liu Yi-Reng,et al.  THEORY OF VALUES OF SINGULAR POINT IN COMPLEX AUTONOMOUS DIFFERENTIAL SYSTEMS , 1990 .

[6]  Yirong Liu,et al.  A cubic system with twelve small amplitude limit cycles , 2005 .

[7]  Jian-Qiao Sun,et al.  Bifurcation and chaos in complex systems , 2006 .

[8]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[9]  Yen-chʿien Yeh,et al.  Theory of Limit Cycles , 2009 .

[10]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[11]  Jibin Li,et al.  Hilbert's 16th Problem and bifurcations of Planar Polynomial Vector Fields , 2003, Int. J. Bifurc. Chaos.

[12]  Yirong Liu,et al.  CENTER AND ISOCHRONOUS CENTER AT INFINITY IN A CLASS OF PLANAR DIFFERENTIAL SYSTEMS , 2008 .

[13]  Pei Yu,et al.  SIMPLEST NORMAL FORMS OF HOPF AND GENERALIZED HOPF BIFURCATIONS , 1999 .

[14]  Pei Yu,et al.  A perturbation method for computing the simplest normal forms of dynamical systems , 2003 .

[15]  N. G. Lloyd.  THEORY OF LIMIT CYCLES (Translations of Mathematical Monographs 66) , 1988 .

[16]  D. Hilbert.  Mathematical Problems , 2019, Mathematics: People · Problems · Results.

[17]  B. Drachman,et al.  Computation of normal forms , 1990 .

[18]  Pei Yu,et al.  COMPUTATION OF NORMAL FORMS VIA A PERTURBATION TECHNIQUE , 1998 .

[19]  Jan A. Sanders,et al.  Further reduction of the Takens-Bogdanov normal form , 1992 .

[20]  Pei Yu,et al.  Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields , 2005 .

[21]  Henri Poincaré,et al.  New methods of celestial mechanics , 1967 .

[22]  S. Smale.  Mathematical problems for the next century , 1998 .

[23]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[24]  William F. Langford,et al.  Degenerate Hopf bifurcation formulas and Hilbert's 16th problem , 1989 .

[25]  P. Yu,et al.  Closed-Form Conditions of bifurcation Points for General Differential Equations , 2005, Int. J. Bifurc. Chaos.

[26]  Pei Yu,et al.  Twelve Limit Cycles in a cubic Case of the 16TH Hilbert Problem , 2005, Int. J. Bifurc. Chaos.

[27]  Pei Yu,et al.  The simplest normal form of Hopf bifurcation , 2003 .

[28]  Chen Hai-bo,et al.  Linear recursion formulas of quantities of singular point and applications , 2004, Appl. Math. Comput..

[29]  P. Yu,et al.  SYMBOLIC COMPUTATION OF NORMAL FORMS FOR RESONANT DOUBLE HOPF BIFURCATIONS USING A PERTURBATION TECHNIQUE , 2001 .

[30]  James Murdock,et al.  Normal Forms and Unfoldings for Local Dynamical Systems , 2002 .

[31]  N. N. Bautin,et al.  On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type , 1954 .

[32]  Pei Yu.  Chapter 1 Bifurcation, Limit Cycle and Chaos of Nonlinear Dynamical Systems , 2006 .

[33]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[34]  N. G. Lloyd,et al.  New Directions in Dynamical Systems: Limit Cycles of Polynomial Systems – Some Recent Developments , 1988 .

[35]  Floris Takens,et al.  Unfoldings of certain singularities of vectorfields: Generalized Hopf bifurcations , 1973 .

[36]  Richard C. Churchill,et al.  Unique normal forms for planar vector fields , 1988 .

[37]  Alan S. Perelson,et al.  Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order Through Fluctuations.G. Nicolis , I. Prigogine , 1978 .

[38]  Scott D. Sudhoff,et al.  Analysis of Electric Machinery and Drive Systems , 1995 .

[39]  Floris Takens,et al.  Singularities of vector fields , 1974 .