Linear Systems of Ordinary Differential Equations, with Periodic and Quasi-Periodic Coefficients

Name: Linear Systems of Ordinary Differential Equations, with Periodic and Quasi-Periodic Coefficients
Brand: Academic Press
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Academic Press

Erschienen am 14. Mai 2014

295 Seiten

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978-0-08-095535-3 (ISBN)

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Front Cover
Linear Systems of Ordinary Differential Equations
Copyright Page
Contents
Author's Comments
Introduction
Chapter 1. Functions of a Single Matrix
Chapter 2. Auxiliary Theorems
Chapter 3. Functions of Several Matrices and of a Countable Set of Matrices
Chapter 4. Classes of Systems of Linear Differential Equations That Can Be Integrated in Closed Form
Chapter 5. Other Systems of Linear Differential Equations That Are Integrable in Closed Form
Chapter 6. The Construction of Solutions of Certain Linear Systems of Differential Equations in the Form of a Series of Several Matrices (of a Series of Compositions)
Chapter 7. Solution of the Poincaré-Lappo-Danilevskiy Problem
Chapter 8. Formulation of Certain Problems of Linear Systems of Differential Equations with Real Periodic Coefficients
Chapter 9. Solution of the Problems Posed in Section 8 on the Basis of Real Functions
Chapter 10. Expansion of an Exponential Matrix in a Series of Powers of a Parameter
Chapter 11. Determination of the Coefficients in the Series Expansion of an Exponential Matrix
Chapter 12. Approximate Integration of Equation (10.1)
Chapter 13. The Case in Which P0 (t), P1 (t), ..., Pm (t) in Equation (10.1) Are Constants
Chapter 14. The Case in Which P0 is Constant and expP0t is a Periodic Matrix in Equation (10.1)
Chapter 15. An Example Illustrating Section 14
Chapter 16. Canonical Systems
Chapter 17. The System (16.3) With P0 = P1 = ... = Pm-1=0
Chapter 18. Artem'yev's Problem
Chapter 19. The Theory of Reducible Systems
Chapter 20. Shtokalo's Method
Chapter 21. Determination of the Coefficients of the Series (20.22) and (20.23) by Shtokalo's Method [10, 38]
Chapter 22. Approximate Solutions Obtained by Shtokalo's Method
Chapter 23. Inequalities Following from Shtokalo's Method
Chapter 24. Shtokalo's Theorem. Inequalities Involving Approximate Solutions Found by Shtokalo's Method (for Linear and Nonlinear Systems). Particular Problems
Chapter 25. Other Approximate Forms of Solutions That .Arise From Shtokalo's and Bogolyubov's Methods
Chapter 26. Demidovich's Problem
Chapter 27. Another Formulation of Certain Problems and Consequences of Them
Chapter 28. Solution of the Problems in Section 8 by Use of the Method of Solving the Poincaré-Lappo-DaniIevskiy Problem and Lyapunov's Contributions
Chapter 29. Remarks on Bounded and Periodic Solutions of a System of Two Differential Equations With Periodic Coefficients
Chapter 30. Periodic and Bounded Solutions of the Systems of Equations Considered in Sections 3 and 4
Chapter 31. Questions Involving the Boundedness and Periodicity of Solutions of a System of Two Linear Differential Equations With the Aid of a Special Exponential Substitution Obtained by Lappo-Danilevskiy
Chapter 32. Periodic Solutions of a System of Two Equations When Condition (3.6) is Satisfied
Chapter 33. Lyapunov's Equation
Chapter 34. (33.1) The Case in Which Equation (33.9) Has Roots | p1| = | p2| = 1. The Finding of Periodic Solutions.
Chapter 35. Regions of Values of the Parameters Appearing in Equation (33.1) in Which There Are Bounded and Periodic Solutions
Chapter 36. Periodic Solutions of a Linear Homogeneous System of Differential Equations
Chapter 37. An Equation of the Form (33.1) With Variable-Sign Function p (t)
Chapter 38. Starzhinskiy's Transformation
Chapter 39. Transformation of an Arbitrary System of Two Equations into a Canonical System
Chapter 40. The Case in Which (39.7) is of the Form z22 = 0
Chapter 41. The Transformation of n Linear Equations into a Canonical System
Chapter 42. Necessary and Sufficient Conditions for a Polynomial to Have Roots Located on the Unit Circle
Chapter 43. Investigation of the Roots of the Polynomial (42.1) as Functions of a Parameter Appearing in the Coefficient ak
Chapter 44. Questions Regrading the Stability and Boundedness of Solutions of Linear Systems of Differential Equations With Periodic Coefficients on the Basis of the Methods of Section 43
Chapter 45. A Sufficient Condition for the Integral Matrix of the Noncanonical System (44.1) to Possess the Property that X ( t, e ) ||0|| as t 8
Chapter 46. Another Method of Solving Artem'yev's Problem
Chapter 47. Supplement to the Theory of Implicit Functions as Studied in (32, 73, 97)
Chapter 48. Two Implicit Functions
Chapter 49. The Construction of Functions (*) Defined by Equations (48.4) and (48.5)
Appendix
Bibliogaphy
Index

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