Planar Dynamical Systems

Yirong Liu, Jibin Li and Wentao Huang, Zhejiang Normal University, Jinhua, Zhejiang, P. R. China.

• 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 . . . . . . . . . . . . . . . . . . . . . . . . . 1022.7 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) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946 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 . . . . . . . . . . . . . . . . . . . 2698.3.1 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 . . . . . . . . . . 371Bibliography 377