Linear Systems of Ordinary Differential Equations With Periodic and Quasi-Periodic Coefficients

Content: 
Edited by
Pages ii-iii

Copyright page
Page vi

Author's Comments
Page vii

Introduction
Pages xiii-xxi

1. Functions of a Single Matrix
Pages 1-22

2. Auxiliary Theorems
Pages 22-33

3. Functions of Several Matrices and of a Countable Set of Matrices
Pages 33-36

4. Classes of Systems of Linear Differential Equations That Can Be Integrated in Closed Form
Pages 36-41

5. Other Systems of Linear Differential Equations That Are Integrable in Closed Form
Pages 41-44

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)
Pages 44-49

7. Solution of the Poincaré-Lappo-Danilevskiy Problem
Pages 49-56

8. Formulation of Certain Problems of Linear Systems of Differential Equations with Real Periodic Coefficients
Pages 56-60

9. Solution of the Problems Posed in Section 8 on the Basis of Real Functions
Pages 60-68

10. Expansion of an Exponential Matrix in a Series of Powers of a Parameter
Pages 68-75

11. Determination of the Coefficients in the Series Expansion of an Exponential Matrix
Pages 75-82

12. Approximate Integration of Equation (10.1)
Pages 82-85

13. The Case in Which P0(t), P1(t),…, Pm (t) in Equation (10.1) Are Constants
Pages 85-89

14. The Case in Which Po is Constant and exp P0t is a Periodic Matrix in Equation (10.1)
Pages 89-90

15. An Example Illustrating Section 14
Pages 90-101

16. Canonical Systems [8, 9, 12, 13, 31, 33, 34, 67, 68]
Pages 101-105

17. The System (16.3) With P0 = P1 =… = Pm–1 = 0
Pages 105-106

18. Artem'yev's Problem
Pages 106-109

19. The Theory of Reducible Systems
Pages 109-112

20. Shtokalo's Method
Pages 112-116

21. Determination of the Coefficients of the Series (20.22) and (20.23) by Shtokalo's Method
Pages 116-120

22. Approximate Solutions Obtained by Shtokalo's Method
Pages 120-122

23. Inequalities Following from Shtokalo's Method
Pages 122-124

24. Shtokalo's Theorem. Inequalities Involving Approximate Solutions Found by Shtokalo's Method (for Linear and Nonlinear Systems). Particular Problems
Pages 124-129

25. Other Approximate Forms of Solutions That Arise From Shtokalo's and Bogolyubov's Methods
Pages 129-132

26. Demidovich's Problem
Pages 132-134

27. Another Formulation of Certain Problems and Consequences of Them
Pages 134-140

28. Solution of the Problems in Section 8 by Use of the Method of Solving the Poincaré—Lappo-Danilevskiy Problem and Lyapunov's Contributions
Pages 140-147

29. Remarks on Bounded and Periodic Solutions of a System of Two Differential Equations With Periodic Coefficients
Pages 147-154

30. Periodic and Bounded Solutions of the Systems of Equations Considered in Sections 3 and 4
Pages 155-157

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
Pages 157-168

32. Periodic Solutions of a System of Two Equations When Condition (3.6) is Satisfied
Pages 168-169

33. Lyapunov's Equation
Pages 169-175

34. (33.1) The Case in Which Equation (33.9) Has Roots | P1 | = | P2 | = 1. The Finding of Periodic Solutions
Pages 175-184

35. Regions of Values of the Parameters Appearing in Equation (33.1) in Which There Are Bounded and Periodic Solutions
Pages 184-197

36. Periodic Solutions of a Linear Homogeneous System of Differential Equations
Pages 197-201

37. An Equation of the Form (33.1) With Variable-Sign Function p (t)
Pages 202-210

38. Starzhinskiy's Transformation
Pages 210-213

39. Transformation of an Arbitrary System of Two Equations into a Canonical System
Pages 213-217

40. The Case in Which (39.7) is of the Form z22 = 0
Pages 217-221

41. The Transformation of n Linear Equations into a Canonical System
Pages 221-222

42. Necessary and Sufficient Conditions for a Polynomial to Have Roots Located on the Unit Circle
Pages 222-224

43. Investigation of the Roots of the Polynomial (42.1) as Functions of a Parameter Appearing in the Coefficient ak
Pages 224-227

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
Pages 228-230

45. A Sufficient Condition for the Integral Matrix of the Non-canonical System (44.1) to Possess the Property that X (t, z) → || 0 || as t → ∞
Pages 230-231

46. Another Method of Solving Artem'yev's Problem
Pages 231-232

47. Supplement to the Theory of Implicit Functions as Studied in (32, 73, 97)
Pages 232-243

48. Two Implicit Functions
Pages 243-248

49. The Construction of Functions (*) Defined by Equations (48.4) and (48.5)
Pages 248-253

Appendix
Pages 254-261

Bibliography
Pages 262-269

Index
Pages 270-271