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Differential Equations with Symbolic Computation

E-BookPDF1 - PDF WatermarkE-Book
374 Seiten
Englisch
Birkhäuser Baselerschienen am16.03.20062005
This book presents the state-of-the-art in tackling differential equations using advanced methods and software tools of symbolic computation. It focuses on the symbolic-computational aspects of three kinds of fundamental problems in differential equations: transforming the equations, solving the equations, and studying the structure and properties of their solutions.mehr
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Produkt

KlappentextThis book presents the state-of-the-art in tackling differential equations using advanced methods and software tools of symbolic computation. It focuses on the symbolic-computational aspects of three kinds of fundamental problems in differential equations: transforming the equations, solving the equations, and studying the structure and properties of their solutions.
Details
Weitere ISBN/GTIN9783764374297
ProduktartE-Book
EinbandartE-Book
FormatPDF
Format Hinweis1 - PDF Watermark
FormatE107
Erscheinungsjahr2006
Erscheinungsdatum16.03.2006
Auflage2005
ReiheTrends in Mathematics
Seiten374 Seiten
SpracheEnglisch
IllustrationenVIII, 374 p.
Artikel-Nr.1424949
Rubriken
Genre9200

Inhalt/Kritik

Inhaltsverzeichnis
1;Preface;6
2;Contents;8
3;Symbolic Computation of Lyapunov Quantities and the Second Part of Hilbert s Sixteenth Problem;10
3.1;1. Introduction;10
3.2;2. Small-Amplitude Limit Cycle Bifurcations;13
3.3;3. Symbolic Computation;17
3.4;4. Results;22
3.5;5. A New Algorithm;24
3.6;6. Conclusions and Further Work;28
3.7;References;29
4;Estimating Limit Cycle Bifurcations from Centers;32
4.1;1. Introduction;32
4.2;2. The Basic Technique;34
4.3;3. Higher Order Perturbations;39
4.4;4. Using Symmetries;42
4.5;5. Conclusion;43
4.6;References;43
5;Conditions of Infinity to be an Isochronous Center for a Class of Differential Systems;46
5.1;1. Introduction;46
5.2;2. The Isochronous Center at Infinity and Some Preliminary Results;47
5.3;3. Singular Point Values and Center Conditions;53
5.4;4. Period Constants and Isochronous Center Conditions;55
5.5;References;61
6;Darboux Integrability and Limit Cycles for a Class of Polynomial Differential Systems;64
6.1;1. Introduction and Statement of the Results;64
6.2;2. Proof of Theorem 1.1;68
6.3;3. Existence of Limit Cycles in the Class;70
6.4;4. The Appendix;72
6.5;References;73
7;Time-Reversibility in Two-Dimensional Polynomial Systems;76
7.1;1. Introduction;76
7.2;2. Invariants of the Rotation Group;82
7.3;3. Invariants and Time-Reversibility;86
7.4;4. A Computational Algorithm;89
7.5;References;91
8;On Symbolic Computation of the LCE of N-Dimensional Dynamical Systems;94
8.1;1. Introduction;94
8.2;2. On the LCE of N-Dimensional Dynamical Systems;96
8.3;3. Representing and Manipulating a Class of Non-Algebraic Objects;102
8.4;4. Symbolic Computation of the LCE;106
8.5;5. Final Remarks;111
8.6;References;112
8.7;Appendix;113
9;Symbolic Computation for Equilibria of Two Dynamic Models;118
9.1;1. Introduction;118
9.2;2. Equilibria of the Multi-molecular Reaction;120
9.3;3. Equilibria of the Rotor Oscillation;125
9.4;References;128
10;Attractive Regions in Power Systems by Singular Perturbation Analysis;130
10.1;1. Introduction;130
10.2;2. Singular Perturbed Systems;131
10.3;3. Decomposition of Multimachine Dynamics;133
10.4;4. A Detailed Analysis for Single Machine System;137
10.5;5. Conclusion;146
10.6;References;147
11;Algebraic Multiplicity and the Poincar´e Problem;152
11.1;1. Introduction;152
11.2;2. Algebraic Multiplicity and the Poincar´ e Problem;154
11.3;3. Application to 2D Lotka-Volterra System;164
11.4;References;165
11.5;Acknowledgment;166
12;Formalizing a Reasoning Strategy in Symbolic Approach to Differential Equations;168
12.1;1. Introduction;168
12.2;2. Preliminary;170
12.3;3. The Construction of Ideals for Polynomial Sets with Arbitrarily Many Variables;171
12.4;4. AModel for Incremental Computations;172
12.5;5. Well Limit Behavior of Term Rewriting Systems;173
12.6;6. Automated Reasoning and the Construction of Ideals for Polynomial Sets in K[x1, . . . , xn, . . .];174
12.7;7. Procedure Scheme DISCOVER;175
12.8;8. An Example;176
12.9;References;179
13;Looking for Periodic Solutions of ODE Systems by the Normal Form Method;182
13.1;1. Introduction;182
13.2;2. Problem Formulation;183
13.3;3. The Normal Form Method;185
13.4;4. General Scheme of Investigation by the Normal Form Method;189
13.5;5. Second Order Systems as a Transparent Example;189
13.6;6. Examples of the Fourth Order ODEs;194
13.7;7. Conclusions;206
13.8;References;206
13.9;Acknowledgments;209
14;Algorithmic Reduction and Rational General Solutions of First Order Algebraic Differential Equations;210
14.1;1. Introduction;210
14.2;2. Rational General Solutions of First Order Algebraic Differential Equations;211
14.3;3. Reduction of First Order Algebraic Differential Equations;213
14.4;4. Algorithm and Example;217
14.5;5. Conclusion;218
14.6;References;219
15;Factoring Partial Differential Systems in Positive Characteristic;222
15.1;Introduction;222
15.2;1. Preliminaries;224
15.3;2. Partial Di.erential Systems in Positive Characteristic;227
15.4;3. Simultaneous Reduction of Commuting Matrices;233
15.5;4. Factoring Partial Di.erential Systems in Positive Characteristic;237
15.6;5. Two Other Generalizations;241
15.7;References;242
15.8;Appendix: Classi.cation of Partial Di.erential Modules in Positive Characteristic;244
16;On the Factorization of Differential Modules;248
16.1;1. Introduction;248
16.2;2. Preliminaries;249
16.3;3. Reduction of the Factorization Problem;250
16.4;4. Factorization Algorithm;259
16.5;5. Conclusion and Future Work;262
16.6;References;262
17;Continuous and Discrete Homotopy Operators and the Computation of Conservation Laws;264
17.1;1. Introduction;265
17.2;2. Examples of Nonlinear PDEs;267
17.3;3. Key Definitions - Continuous Case;268
17.4;4. Conserved Densities and Fluxes of Nonlinear PDEs;270
17.5;5. Tools from the Calculus of Variations;272
17.6;6. Removing Divergences and Divergence-Equivalent Terms;279
17.7;7. Application: Conservation Laws of Nonlinear PDEs;281
17.8;8. Examples of Nonlinear DDEs;288
17.9;9. Dilation Invariance and Uniformity in Rank for DDEs;289
17.10;10. Conserved Densities and Fluxes of Nonlinear DDEs;290
17.11;11. Discrete Euler and Homotopy Operators;291
17.12;12. Application: Conservation Laws of Nonlinear DDEs;294
17.13;13. Conclusion;296
17.14;References;297
17.15;Acknowledgements and Dedication;299
18;Partial and Complete Linearization of PDEs Based on Conservation Laws;300
18.1;1. Introduction;300
18.2;2. Notation;301
18.3;3. Conservation Laws with Arbitrary Functions;302
18.4;4. The Procedure;303
18.5;5. Scope of the Procedure;305
18.6;6. Computational Aspects;307
18.7;7. An Example Requiring the Introduction of a Potential;308
18.8;8. Inhomogeneous Linear DEs;310
18.9;9. An Example of a Triangular Linear System;311
18.10;10. Summary;312
18.11;Appendix;313
18.12;References;315
19;CONSLAW: A Maple Package to Construct the Conservation Laws for Nonlinear Evolution Equations;316
19.1;1. Introduction;316
19.2;2. Computer Algebraic Algorithm and Routines;318
19.3;3. Applications to Several Nonlinear PDEs;325
19.4;4. Summary;330
19.5;Appendix. The Usage of CONSLAW;330
19.6;References;333
20;Generalized Diferential Resultant Systems of Algebraic ODEs and Differential Elimination Theory;336
20.1;1. Introduction;336
20.2;2. Differential Algebra Preliminaries;337
20.3;3. Algebraic Elimination Theory;337
20.4;4. Differential Elimination Theory;341
20.5;References;349
21;On Good Bases of Algebraico-Differential Ideals;352
21.1;1. Introduction;352
21.2;2. Problem 1 in the Polynomial Case;353
21.3;3. Extension to Algebrico-Differential Systems;354
21.4;4. Problem 2 in the Polynomial Case;355
21.5;5. Problem 2 in the Algebrico-Differential Case;356
21.6;6. Example: Pommaret s Devil Problem;357
21.7;References;359
22;On the Construction of Groebner Basis of a Polynomial Ideal Based on Riquier - Janet Theory;360
22.1;0. Introduction;360
22.2;1. Tuples of Integers;362
22.3;2. Well-Arranged Basis of a Polynomial Ideal;365
22.4;3. Well-Behaved Basis of a Polynomial Ideal;367
22.5;4. Identi.cation of Well-Behaved Basis with Groebner Basis;369
22.6;5. An Example;373
22.7;References;376
23;Index;378mehr
Leseprobe
Estimating Limit Cycle Bifurcations from Centers (p. 23)

Colin Christopher

Abstract.

We consider a simple computational approach to estimating the cyclicity of centers in various classes of planar polynomial systems. Among the results we establish are confirmation of , Zoladek's result that at least 11 limit cycles can bifurcate from a cubic center, a quartic system with 17 limit cycles bifurcating from a non-degenerate center, and another quartic system with at least 22 limit cycles globally.

Mathematics Subject Classification (2000). 34C07.

Keywords. Limit cycle, center, multiple bifurcation.

1. Introduction
The use of multiple Hopf bifurcations of limit cycles from critical points is now a well-established technique in the analysis of planar dynamical systems. For many small classes of systems, the maximum number, or cyclicity, of bifurcating limit cycles is known and has been used to obtain important estimates on the general behavior of these systems.

In particular, quadratic systems can have at most three such limit cycles [1], symmetric cubic systems (those without quadratic terms) and projective quadratic systems at most five [11, 15, 8]. Results are also known explicitly for several large classes of Lienard systems [3].

The idea behind the method is to calculate the successive coeficients ?i in the return map for the vector field about a non-degenerate monodromic critical point.
The cyclicity can then be found from examining these coeficients and their common zeros. The terms ?2k are merely analytic functions of the previous ?i, so the only interesting functions are the ones of the form ?2i+1. If ?2k+1 is the first non-zero one of these, then at most k limit cycles can bifurcate from the origin, and, provided we have suficient choice in the coeficients ?i, we can also obtain that many limit cycles in a simultaneous bifurcation from the critical point.

We call the functions ?2i+1 the Liapunov quantities of the system. If all the ?2i+1 vanish then the critical point is a center. It is possible to analyze this case also, but to do fully requires a more intimate knowledge, not only of the common zeros of the polynomials ?i, but also of the ideal they generate in the ring of coeficients. The papers [1, 15] cover the case of a center also. We call the set of coe.cients for which all the ?i vanish the center variety.

In the cases we consider here, when ?0 = 0, the remaining coeficients are polynomials in the parameters of the system. By the Hilbert Basis Theorem, the center variety is then an algebraic set. Unfortunately, although the calculation of the Liapunov quantities is straight forward, the computational complexity of finding their common zeros grows very quickly. The result is that some very simple systems have remained intractable (to direct calculation at least) at present, for example, the system of Kukles' [9]:mehr