Planar Dynamical Systems
Selected Classical Problems
1st Edition
Published on 29. October 2014
XVIII, 371 pages
978-3-11-038914-2 (ISBN)
Description
In 2008, November 23-28, the workshop of "Classical Problems on Planar Polynomial Vector Fields " was held in the Banff International Research Station, Canada. Called "classical problems", it was concerned with the following: (1) Problems on integrability of planar polynomial vector fields. (2) The problem of the center stated by Poincaré for real polynomial differential systems, which asks us to recognize when a planar vector field defined by polynomials of degree at most n possesses a singularity which is a center. (3) Global geometry of specific classes of planar polynomial vector fields. (4) Hilbert's 16th problem. These problems had been posed more than 110 years ago. Therefore, they are called "classical problems" in the studies of the theory of dynamical systems. The qualitative theory and stability theory of differential equations, created by Poincaré and Lyapunov at the end of the 19th century, had major developments as two branches of the theory of dynamical systems during the 20th century. As a part of the basic theory of nonlinear science, it is one of the very active areas in the new millennium. This book presents in an elementary way the recent significant developments in the qualitative theory of planar dynamical systems. The subjects are covered as follows: the studies of center and isochronous center problems, multiple Hopf bifurcations and local and global bifurcations of the equivariant planar vector fields which concern with Hilbert's 16th problem. The book is intended for graduate students, post-doctors and researchers in dynamical systems. For all engineers who are interested in the theory of dynamical systems, it is also a reasonable reference. It requires a minimum background of a one-year course on nonlinear differential equations.
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Yirong Liu, Jibin Li and Wentao Huang, Zhejiang Normal University, Jinhua, Zhejiang, P. R. China.
Content
Preface i1 Basic Concept and Linearized Problem of Systems 11.1 Basic Concept and Variable Transformation . . . . . . . . . . 11.2 Resultant of the Weierstrass Polynomial and Multiplicity ofa Singular Point . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Quasi-Algebraic Integrals of Polynomial Systems . . . . . . . 121.4 Cauchy Majorant and Analytic Properties in a Neighborhoodof an Ordinary Point . . . . . . . . . . . . . . . . . . . . . . 171.5 Classification of Elementary Singular Points and LinearizedProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.6 Node Value and Linearized problem of the Integer-Ratio Node 331.7 Linearized Problem of the Degenerate Node . . . . . . . . . . 391.8 Integrability and Linearized Problem of Weak Critical SingularPoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.9 Integrability and Linearized Problem of the Resonant SingularPoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 Focal Values, Saddle Values and Singular Point Values 772.1 Successor Functions and Properties of Focal Values . . . . . . 772.2 Poincar´e Formal Series and Algebraic Equivalence . . . . . . 832.3 Linear Recursive Formulas for the Computation of SingularPoint Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.4 The Algebraic Construction of Singular Values . . . . . . . . 922.5 Elementary Generalized Rotation Invariants of the Cubic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.6 Singular Point Values and Integrability Condition of theQuadratic Systems . . . . . . . . . . . . . . . . . . . . . . . . 1002.6.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.7 Singular Point Values and Integrability Condition of the CubicSystems Having Homogeneous Nonlinearities . . . . . . . 1032.7.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 1043 Multiple Hopf Bifurcations 1073.1 The Zeros of Successor Functions in the Polar Coordinates . . 1073.2 Analytic Equivalence . . . . . . . . . . . . . . . . . . . . . . . 1113.3 Quasi Successor Function . . . . . . . . . . . . . . . . . . . . 1133.4 Bifurcations of Limit Circle of a Class of Quadratic Systems . 1194 Isochronous Center In Complex Domain 1234.1 Isochronous Centers and Period Constants . . . . . . . . . . 1234.2 Linear Recursive Formulas to Compute Period Constants . . 1294.3 Isochronous Center for a Class of Quintic System in the ComplexDomain . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.3.1 The Conditions of Isochronous Center under ConditionC1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.3.2 The Conditions of Isochronous Center under ConditionC2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.3.3 The Conditions of Isochronous Center under ConditionC3 . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.3.4 Non-Isochronous Center under Condition C4 and C.4 . 1424.4 The Method of Time-Angle Difference . . . . . . . . . . . . . 1424.5 The Conditions of Isochronous Center of the Origin for a CubicSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485 Theory of Center-Focus and Bifurcation of Limit Cycles atInfinity of a Class of System 1535.1 Definition of the Focal Values of Infinity . . . . . . . . . . . . 1535.2 Conversion of Questions . . . . . . . . . . . . . . . . . . . . . 1565.3 Method of Formal Series and Singular Point Value of Infinity 1595.4 The Algebraic Construction of Singular Point Values of Infinity1735.5 Singular Point Values at Infinity and Integrable Conditionsfor a Class of Cubic System . . . . . . . . . . . . . . . . . . . 1785.6 Bifurcation of Limit Cycles at Infinity . . . . . . . . . . . . . 1855.7 Isochronous Centers at Infinity of a Polynomial Systems . . . 1905.7.1 Conditions of Complex Center for System (5.7.6) . . . 1915.7.2 Conditions of Complex Isochronous Center for System(5.7.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6 Theory of Center-Focus and Bifurcations of Limit CyclesFor a Class of Multiple Singular Points 1996.1 Succession Function and Focal Values for a Class of MultipleSingular Points . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.2 Conversion of the Questions . . . . . . . . . . . . . . . . . . . 2016.3 Formal Series, Integral Factors and Singular Point Values fora Class of Multiple Singular Points . . . . . . . . . . . . . . . 2036.4 The Algebraic Structure of Singular Point Values of a Classof Multiple Singular Points . . . . . . . . . . . . . . . . . . . 2176.5 Bifurcation of Limit Cycles From a Class of Multiple SingularPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2196.6 Bifurcation of Limit Cycles Created from a Multiple SingularPoint for a Class of Quartic System . . . . . . . . . . . . . . 2216.7 Quasi Isochronous Center of Multiple Singular Point for aClass of Analytic System . . . . . . . . . . . . . . . . . . . . 2247 On Quasi Analytic Systems 2277.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277.2 Reduction of the Problems . . . . . . . . . . . . . . . . . . . . 2307.3 Focal Values, Periodic Constants and First Integrals of (7.2.3) 2327.4 Singular Point Values and Bifurcations of Limit Cycles ofQuasi-Quadratic Systems . . . . . . . . . . . . . . . . . . . . 2367.5 Integrability of Quasi-Quadratic Systems . . . . . . . . . . . . 2407.6 Isochronous Center of Quasi-Quadratic Systems . . . . . . . . 2437.6.1 The Problem of Complex Isochronous Centers Underthe Condition of C1 . . . . . . . . . . . . . . . . . . . 2437.6.2 The Problem of Complex Isochronous Centers Underthe Condition of C2 . . . . . . . . . . . . . . . . . . . 2467.6.3 The Problem of Complex Isochronous Centers Underthe Other Conditions . . . . . . . . . . . . . . . . . . 2507.7 Singular Point Values and Center Conditions for a Class ofQuasi-Cubic Systems . . . . . . . . . . . . . . . . . . . . . . . 2538 Local and Non-Local Bifurcations of Perturbed Zq- Equiv-ariant Hamiltonian Vector Fields 2578.1 Zq-Equivariant Planar Vector Fields and an Example . . . . . 2588.2 The Method of Detection Functions: Rough Perturbations ofZq-Equivariant Hamiltonian Vector Fields . . . . . . . . . . 2678.3 Bifurcations of Limit Cycles of a Z2- Equivariant PerturbedHamiltonian Vector Fields . . . . . . . . . . . . . . . . . . . 269 8.3.1 Hopf Bifurcation Parameter Values . . . . . . . . . . 2718.3.2 Bifurcations From Heteroclinic or Homoclinic Loops . 2738.3.3 The Values of Bifurcation Directions of Heteroclinicand Homoclinic Loops . . . . . . . . . . . . . . . . . . 2788.3.4 Analysis and Conclusions . . . . . . . . . . . . . . . . 2818.4 The Rate of Growth of Hilbert Number H(n) with n . . . . . 2858.4.1 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . 2858.4.2 A Correction to the Lower Bounds of H(2k -1) Givenin [Christopher and Lloyd, 1995] . . . . . . . . . . . . 2898.4.3 A New Lower Bound for H(2k - 1) . . . . . . . . . . . 2918.4.4 Lower Bound for H(3 × 2k-1 - 1) . . . . . . . . . . . 2939 Center-Focus Problem and Bifurcations of Limit Cycles fora Z2-Equivariant Cubic System 2999.1 Standard Form of a Class of System EZ23 . . . . . . . . . . . . 2999.2 Liapunov Constants, Invariant Integrals and the Necessaryand Sufficient Conditions of the Existence for the Bi-Center . 3019.3 The Conditions of Six-Order Weak Focus and Bifurcations ofLimit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 3149.4 A Class of EZ23 System With 13 Limit Cycles . . . . . . . . . 3189.5 Proofs of Lemma 9.4.1 and Theorem 9.4.1 . . . . . . . . . . . 3229.6 The Proofs of Lemma 9.4.2 and Lemma 9.4.3 . . . . . . . . . 3309.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33210 Center-Focus Problem and Bifurcations of Limit Cycles forThree-Multiple Nilpotent Singular Points 33910.1 Criteria of Center-Focus for a Nilpotent Singular Point . . . . 33910.2 Successor Functions and Focus Value of Three-MultipleNilpotent Singular Point . . . . . . . . . . . . . . . . . . . . . 34210.3 Bifurcation of Limit Cycles Created from Three-MultipleNilpotent Singular Point . . . . . . . . . . . . . . . . . . . . . 34610.4 The Classification of Three-Multiple Nilpotent Singularpoints and Inverse Integral Factor . . . . . . . . . . . . . . . 35410.5 Quasi-Lyapunov Constants For the Three-Multiple NilpotentSingular Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 36010.6 Proof of Theorem 10.5.2 . . . . . . . . . . . . . . . . . . . . . 36310.7 On the Computation of Quasi-Lyapunov Constants . . . . . . 36810.8 Bifurcations of Limit Cycles Created From a Three-MultipleNilpotent Singular Point of a Cubic System . . . . . . . . . . 371 Bibliography 377
