Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations (Springer Monographs in Mathematics)

The Book Is Dedicated To The Construction Of Particular Solutions Of Systems Of Ordinary Differential Equations In The Form Of Series That Are Analogous To Those Used In Lyapunov's First Method. A Prominent Place Is Given To Asymptotic Solutions That Tend To An Equilibrium Position, Especially In The Strongly Nonlinear Case, Where The Existence Of Such Solutions Can't Be Inferred On The Basis Of The First Approximation Alone. The Book Is Illustrated With A Large Number Of Concrete Examples Of Systems In Which The Presence Of A Particular Solution Of A Certain Class Is Related To Special Properties Of The System's Dynamic Behavior. It Is A Book For Students And Specialists Who Work With Dynamical Systems In The Fields Of Mechanics, Mathematics, And Theoretical Physics.--publisher's Website. Semi-quasihomogeneous Systems Of Differential Equations. Formal Asymptotic Particular Solutions Of Semiquasihomogeneous Systems Of Differential Equations ; Problems Of Convergence ; Exponentialmethods For Finding Nonexponential Solutions ; Examples ; Group Theoretical Interpretation. -- The Critical Case Of Pure Imaginary Roots. Asymptotic Solutions Of Autonomous Systems Of Differential Equations In The Critical Case Of M Pairs Of Pure Imaginary And N 2m Zero Roots Of The Characteristic Equation ; Periodic And Quasiperiodic Systems ; Hamiltonian Systems. -- Singular Problems. Asymptotic Solutions Of Autonomous Systems Of Differential Equations In The Critical Case Of Zero Roots Of The Characteristic Equation ; Concerning Iterated Logarithms ; Systems Implicit With Respect To Higher Derivatives And Kuznetsov's Theory. -- Inversion Problem For The Lagrange Theorem On The Stability Of Equilibrium And Related Problems. On Energy Criteria For Stability ; Regular Problems ; Singular Problems. -- Nonexponential Asymptotic Solutions Of Systems Of Functional-differential Equations -- Arithmetic Properties Of The Eigenvalues Of The Kovalevsky Matrix And Conditions For The Nonintegrability Of Semi-quasihomogeneous Systems Of Ordinary Differential Equations. Valery V. Kozlov, Stanislav D. Furta. Translation Of The 2nd Russian Original Edition Entitled Asimptotiki Reshenij Sil'no Nelinejnykh Sistem Differentsial'nykh Uravnenij, Published ... In 2009--t.p. Verso. Includes Bibliographical References (pages 249-257) And Index. Cover......Page 1 Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations......Page 4 Translator’s Note......Page 6 Preface......Page 8 Contents......Page 10 Introduction......Page 12 1.1 Formal Asymptotic Particular Solutions of Semi-quasihomogeneous Systems of Differential Equations......Page 21 1.2 Problems of Convergence......Page 33 1.3 Exponential Methods for Finding Nonexponential Solutions......Page 44 1.4 Examples......Page 61 1.5 Group Theoretical Interpretation......Page 75 2.1 Asymptotic Solutions of Autonomous Systems of Differential Equations in the Critical Case of m Pairs of Pure Imaginary and n-2m Zero Roots of the Characteristic Equation......Page 96 2.2 Periodic and Quasiperiodic Systems......Page 111 2.3 Hamiltonian Systems......Page 127 3.1 Asymptotic Solutions of Autonomous Systems of Differential Equations in the Critical Case of Zero Roots of the Characteristic Equation......Page 150 3.2 Concerning Iterated Logarithms......Page 162 3.3 Systems Implicit with Respect to Higher Derivatives and Kuznetsov's Theory......Page 171 4.1 On Energy Criteria for Stability......Page 187 4.2 Regular Problems......Page 207 4.3 Singular Problems......Page 218 Appendix A Nonexponential Asymptotic Solutions of Systems of Functional-Differential Equations......Page 233 Appendix B Arithmetic Properties of the Eigenvalues of the Kovalevsky Matrix and Conditions for the Nonintegrability of Semi-quasihomogeneous Systems of Ordinary Differential Equations......Page 246 Literature......Page 266 Index......Page 275 The book is dedicated to the construction of particular solutions of systems of ordinary differential equations in the form of series that are analogous to those used in Lyapunov' s first method. A prominent place is given to asymptotic solutions that tend to an equilibrium position, especially in the strongly nonlinear case, where the existence of such solutions can' t be inferred on the basis of the first approximation alone. The book is illustrated with a large number of concrete examples of systems in which the presence of a particular solution of a certain class is related to special properties of the system' s dynamic behavior. It is a book for students and specialists who work with dynamical systems in the fields of mechanics, mathematics, and theoretical physics

Preface Semi-quasihomogeneous systems of ordinary differential equations 2. The critical case of purely imaginary kernels 3. Singular problems 4. The inverse problem for the Lagrange theorem on the stability of equilibrium and other related problems Appendix A. Nonexponential asymptotic solutions of systems of functional-differential equations Appendix B. Arithmetic properties of the eigenvalues of the Kovalevsky matrix and conditions for the nonintegrability of semi-quasihomogeneous systems of ordinary di¤erential equations Bibliography.

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