سیستم‌های خطی معادلات دیفرانسیل معمولی با ضرایب دوره‌ای و شبه‌دوره‌ای، جلد ۲۸ (ریاضیات در علم و مهندسی)

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 P 0 ( t ), P 1 ( t ),…, P m ( t ) in Equation (10.1) Are Constants Pages 85-89 14. The Case in Which Po is Constant and exp P 0 t 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 P 0 = P 1 =… = P m–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 | P 1 | = | P 2 | = 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 z 22 = 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 a k 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