Approximation procedures in nonlinear oscillation theory:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin u.a.
<<de>> Gruyter
1994
|
| Series: | De Gruyter series in nonlinear analysis and applications
2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 272 S. |
| ISBN: | 3110141329 |
Staff View
MARC
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| 100 | 1 | |a Bobylev, Nikolaj A. |e Verfasser |0 (DE-588)1157367208 |4 aut | |
| 245 | 1 | 0 | |a Approximation procedures in nonlinear oscillation theory |c Nikolai A. Bobylev ; Yuri M. Burman ; Sergey K. Korovin |
| 264 | 1 | |a Berlin u.a. |b <<de>> Gruyter |c 1994 | |
| 300 | |a XI, 272 S. | ||
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Record in the Search Index
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| adam_text | Contents
Preface v
Chapter I: Basic Concepts 1
§ 1 Equations for oscillatory systems 1
1.1 First order systems 1
1.2 Linear periodic systems 2
1.3 Higher order systems 4
§ 2 The shift operator and first return function 8
2.1 Shift operator 8
2.2 Periodic solutions of autonomous systems 9
§ 3 Integral and integrofunctional operators for periodic problem .... 11
3.1 Completely continuous operators 11
3.2 Spaces of functions 12
3.3 Linear integral operators and their properties 13
3.4 A superposition operator 13
3.5 The Hammerstein operator 14
3.6 Integral and integrofunctional equations 14
3.7 The Frechet derivative 16
3.8 Periodic problem for systems of automatic control 17
§ 4 The harmonic balance method 20
4.1 Forced oscillations 20
4.2 Free oscillations 20
4.3 Oscillations in the systems of automatic control 21
4.4 Harmonic balance method and general theory of projection methods 22
§ 5 The method of mechanical quadratures 25
5.1 The Galerkin method with perturbations 25
5.2 Quadrature processes 26
5.3 The method of mechanical quadratures in looking for periodic
solutions 27
§ 6 The collocation method 29
6.1 The operator scheme of the collocation method 29
6.2 The collocation method in approximation of oscillatory regimes . 31
6.3 On choice of interpolation nodes 32
viii Contents
§ 7 The method of finite differences 33
7.1 Formulae of numeric differentiation 33
7.2 Discretization of the differential equations 33
7.3 Reduction of the method of finite differences to the Galerkin method 34
§ 8 Factor methods 37
8.1 Discrete convergence 37
8.2 Discrete compactness 38
8.3 Discrete convergent sequences of operators 38
8.4 The method of mechanical quadratures 39
8.5 The method of finite differences 40
Chapter II: Existence theorems for oscillatory regimes 43
§ 1 Smooth manifolds and differential forms 43
1.1 Smooth manifolds 43
1.2 Tangent spaces 44
1.3 Orientation 46
1.4 Manifolds with boundaries 46
1.5 Exterior forms 47
1.6 Outer product 47
1.7 Differential forms 48
1.8 Integration of differential forms 49
1.9 Outer differentiation 51
§ 2 Degree of a mapping 51
2.1 The Sard theorem 52
2.2 Lemmas about one dimensional manifolds 52
2.3 The degree of a mapping 53
2.4 The degree of a mapping (the second approach) 57
2.5 Relation between two definitions of the degree of a mapping . . 60
2.6 Properties of the degree of a mapping 61
2.7 The degree of continuous mappings 64
§ 3 Rotation of vector fields 65
3.1 Vector fields 65
3.2 Homotope fields. Criteria of homotopy 65
3.3 Rotation of a vector field 66
3.4 Properties of the rotation 66
§ 4 Completely continuous vector fields 73
4.1 Finite dimensional approximations 73
4.2 Soundness of the definition of the rotation 75
4.3 Properties of the rotation 77
Contents jx
§ 5 Fixed point principles and solution of operator equations 80
5.1 The Brouwer theorem 80
5.2 The Browder theorem 81
5.3 The Schauder theorem 86
5.4 The Leray Schauder principle 87
5.5 The contraction mapping principle 88
5.6 Operator equations in products of Banach spaces 90
§ 6 Forced oscillations in systems with weak nonlinearities 94
6.1 Systems with bounded nonlinearities 94
6.2 Systems of automatic control 95
§ 7 Oscillations in systems with strong nonlinearities. Directing functions
method 98
7.1 Points of r irreversibility 98
7.2 Directing functions 100
7.3 The full system of directing functions 103
7.4 Regular directing functions 105
7.5 Construction of directing functions 105
Chapter III: Convergence of numerical procedures 107
§1 Projection methods 107
1.1 The Galerkin method procedure 107
1.2 The nondegenerate case 107
1.3 Topological principle of convergence of the Galerkin method . . Ill
1.4 Convergence of the Galerkin method with perturbations .... 113
1.5 Projection procedures in the Hilbert space 115
1.6 Estimates of the convergence rate 117
§2 Factor methods 119
2.1 Regular and compact convergence 119
2.2 The a posteriori estimate lemma 119
2.3 The convergence of the factor method (the nondegenerate case) . 121
2.4 The topological principle of convergence of the factor method . . 124
2.5 Convergence of the factor method for equations with a linear main
part 129
2.6 Additional remarks 130
§ 3 Convergence of the harmonic balance method and the collocation
method in the problem of periodic oscillations 148
3.1 Local convergence of the harmonic balance method 148
3.2 The topological principle of convergence 153
3.3 Convergence of the harmonic balance method in the case of
nondifferentiable nonlinearities 154
x Contents
3.4 Convergence of the harmonic balance method in the problem of
forced oscillations of the systems of automatic control 156
3.5 Local convergence of the collocation method 158
3.6 A global convergence of the collocation method 162
3.7 The harmonic balance and the collocation methods for periodic
oscillations of multi circuit systems of automatic control .... 164
3.8 The feasibility of the harmonic balance and the collocation methods 167
§ 4 Convergence of the method of mechanical quadratures 171
4.1 Convergent quadrature processes 171
4.2 Local convergence of the method of mechanical quadratures . . 174
4.3 The global convergence of the method of mechanical quadratures 179
4.4 Convergence of the method of mechanical quadratures for general
nonlinear systems 181
§5 Convergence of the method of finite differences 188
5.1 Convergent formulae of numeric differentiation 188
5.2 Discretization of differential equations 190
5.3 Network spaces and connecting mappings 191
5.4 A convergence theorem for differentiable nonlinearities .... 192
5.5 Auxiliary statements 192
5.6 Proof of Theorem 5.1 194
5.7 The topological principle of convergence 197
5.8 Additional remarks 198
§ 6 Numerical procedures of approximate construction of oscillatory regimes
in autonomous systems 198
6.1 Specific features of the problem 198
6.2 The method of the functional parameter 199
6.3 The method of additional constraints 201
6.4 The degeneracy dimension 205
6.5 The convergence theorem 205
6.6 The harmonic balance method in search for oscillations of
autonomous systems of automatic control 208
6.7 The functional characteristic 213
6.8 The topological principle of convergence 214
6.9 Additional remarks 215
§7 Affinity theory 215
7.1 Formulation of the problem 215
7.2 Domains with the same core 217
7.3 An affinity theorem for nonautonomous systems 217
7.4 The affinity theorem for nonautonomous systems of automatic
control 220
7.5 Autonomous systems. The topological index of a cycle 223
Contents xi
7.6 Affinity theorem for autonomous systems of automatic control . . 224
7.7 Additional remarks 232
§ 8 Effective convergence criteria for numerical procedures 235
8.1 Use of affinity theory for proving the convergence of numerical
procedures 235
8.2 The directing functions method and convergence of numerical
procedures 235
8.3 Stability of oscillatory regimes and convergence of numerical
procedures 237
§ 9 Effective estimates of the convergence rate for the harmonic balance
method 240
9.1 A posteriori error estimates 240
9.2 Quasi linear systems 241
9.3 Systems of automatic control 245
9.4 Stability analysis for the periodic solution 248
Notes on the References 251
References 257
Index 271
|
| any_adam_object | 1 |
| author | Bobylev, Nikolaj A. Bûrman, Yûrî M. Korovin, Sergej K. 1945- |
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| language | English |
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| physical | XI, 272 S. |
| publishDate | 1994 |
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| publisher | <<de>> Gruyter |
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| series | De Gruyter series in nonlinear analysis and applications |
| series2 | De Gruyter series in nonlinear analysis and applications |
| spelling | Bobylev, Nikolaj A. Verfasser (DE-588)1157367208 aut Approximation procedures in nonlinear oscillation theory Nikolai A. Bobylev ; Yuri M. Burman ; Sergey K. Korovin Berlin u.a. <<de>> Gruyter 1994 XI, 272 S. txt rdacontent n rdamedia nc rdacarrier De Gruyter series in nonlinear analysis and applications 2 Approximation, théorie de l' ram Oscillations non linéaires ram approximation non linéaire inriac oscillation non linéaire inriac théorie oscillation inriac Approximation theory Nonlinear oscillations Nichtlineare Schwingung (DE-588)4042100-4 gnd rswk-swf Integraloperator (DE-588)4131247-8 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf Nichtlineare Schwingung (DE-588)4042100-4 s Integraloperator (DE-588)4131247-8 s Approximation (DE-588)4002498-2 s DE-604 Bûrman, Yûrî M. Verfasser (DE-588)1066291519 aut Korovin, Sergej K. 1945- Verfasser (DE-588)121849007 aut De Gruyter series in nonlinear analysis and applications 2 (DE-604)BV005530011 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006449127&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bobylev, Nikolaj A. Bûrman, Yûrî M. Korovin, Sergej K. 1945- Approximation procedures in nonlinear oscillation theory De Gruyter series in nonlinear analysis and applications Approximation, théorie de l' ram Oscillations non linéaires ram approximation non linéaire inriac oscillation non linéaire inriac théorie oscillation inriac Approximation theory Nonlinear oscillations Nichtlineare Schwingung (DE-588)4042100-4 gnd Integraloperator (DE-588)4131247-8 gnd Approximation (DE-588)4002498-2 gnd |
| subject_GND | (DE-588)4042100-4 (DE-588)4131247-8 (DE-588)4002498-2 |
| title | Approximation procedures in nonlinear oscillation theory |
| title_auth | Approximation procedures in nonlinear oscillation theory |
| title_exact_search | Approximation procedures in nonlinear oscillation theory |
| title_full | Approximation procedures in nonlinear oscillation theory Nikolai A. Bobylev ; Yuri M. Burman ; Sergey K. Korovin |
| title_fullStr | Approximation procedures in nonlinear oscillation theory Nikolai A. Bobylev ; Yuri M. Burman ; Sergey K. Korovin |
| title_full_unstemmed | Approximation procedures in nonlinear oscillation theory Nikolai A. Bobylev ; Yuri M. Burman ; Sergey K. Korovin |
| title_short | Approximation procedures in nonlinear oscillation theory |
| title_sort | approximation procedures in nonlinear oscillation theory |
| topic | Approximation, théorie de l' ram Oscillations non linéaires ram approximation non linéaire inriac oscillation non linéaire inriac théorie oscillation inriac Approximation theory Nonlinear oscillations Nichtlineare Schwingung (DE-588)4042100-4 gnd Integraloperator (DE-588)4131247-8 gnd Approximation (DE-588)4002498-2 gnd |
| topic_facet | Approximation, théorie de l' Oscillations non linéaires approximation non linéaire oscillation non linéaire théorie oscillation Approximation theory Nonlinear oscillations Nichtlineare Schwingung Integraloperator Approximation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006449127&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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