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E-grāmata: Frequency-domain Approach To Hopf Bifurcation Analysis: Continuous Time-delayed Systems

(Univ Nacional Del Sur, Argentina), (City Univ Of Hong Kong, China), (Univ Nacional De Sur, Agrentina)
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Frequency-domain Approach To Hopf Bifurcation Analysis: Continuous Time-delayed SystemsPalielināt
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Drawing on system control theory, Gentile, Moiola, and Chen explain an effective frequency-domain approach to computing and analyzing several types of standard bifurcation conditions for general continuous-time nonlinear dynamical systems. They present a pictorial gallery of local bifurcation diagrams for such nonlinear systems under simultaneous variations of several system parameters, and derive some higher-order harmonic balance approximation formulas for analyzing the oscillatory dynamics in small neighborhoods of certain types of Hopf degenerate Hopf bifurcations. Annotation ©2020 Ringgold, Inc., Portland, OR (protoview.com)

"This book is devoted to the study of an effective frequency-domain approach, based on systems control theory, to compute and analyze several types of standard bifurcation conditions for general continuous-time nonlinear dynamical systems. A very rich pictorial gallery of local bifurcation diagrams for such nonlinear systems under simultaneous variations of several system parameters is presented. Some higher-order harmonic balance approximation formulas are derived for analyzing the oscillatory dynamics in small neighborhoods of certain types of Hopf and degenerate Hopf bifurcations. The frequency-domain approach is then extended to the large class of delay-differential equations, where the time delays can be either discrete or distributed. For the case of discrete delays, two alternatives are presented, depending on the structure of the underlying dynamical system, where the more general setting is then extended to the case of distributed time-delayed systems. Some representative examples in engineering and biology are discussed."

Preface vii
1 Stability and bifurcation analysis
1(26)
1.1 Preliminaries
1(12)
1.1.1 Equilibrium points and periodic solutions
2(2)
1.1.2 Stability
4(3)
1.1.3 Bifurcations
7(6)
1.2 Overview of bifurcations
13(11)
1.2.1 A single eigenvalue at 0
13(3)
1.2.2 A pair of complex conjugate eigenvalues
16(2)
1.2.3 The Hopf bifurcation theorem
18(3)
1.2.4 Degeneracies
21(3)
1.3 Classical methods for bifurcation analysis
24(3)
1.3.1 The center manifold theorem
24(1)
1.3.2 The normal form theory
25(2)
2 The Hopf bifurcation theorem in the frequency domain
27(30)
2.1 Introduction
27(2)
2.2 Formulation of the frequency-domain approach
29(12)
2.2.1 Feedback control representation of the system
29(2)
2.2.2 Stability analysis: The Nyquist criterion
31(2)
2.2.3 Choosing an adequate representation
33(1)
2.2.4 Emergence of a periodic solution
34(7)
2.3 Advantages of the frequency-domain approach
41(2)
2.4 Application examples of the graphical Hopf theorem
43(14)
2.4.1 The normal form of the Hopf bifurcation
43(3)
2.4.2 The tunnel-diode oscillator
46(3)
2.4.3 The continuous-flow stirred tank reactor
49(8)
3 Analysis of static and multiple bifurcations
57(32)
3.1 Introduction
57(1)
3.2 Formulation of elementary bifurcation conditions on the parameter plane
58(4)
3.3 Applications of the frequency-domain formulas
62(10)
3.3.1 The saddle-node bifurcation
62(2)
3.3.2 The transcritical bifurcation
64(1)
3.3.3 The hysteresis bifurcation
65(2)
3.3.4 The pitchfork bifurcation
67(2)
3.3.5 Static bifurcations in chemical reactor models
69(3)
3.4 Formulation of multiple crossings and determination of degeneracies
72(4)
3.5 Applications of the frequency-domain formulas to multiple bifurcations
76(10)
3.6 Multiplicity of equilibrium solutions in the parameter space
86(3)
4 Degenerate Hopf bifurcations
89(66)
4.1 Introduction
89(2)
4.2 Degenerate Hopf bifurcations of co-dimension one on the parameter plane
91(17)
4.2.1 Local bifurcation diagrams
92(6)
4.2.2 Application examples
98(10)
4.3 Multiple Hopf bifurcation points in the parameter space
108(13)
4.4 Degenerate Hopf bifurcations and the singularity theory
121(11)
4.5 Degenerate Hopf bifurcations and feedback systems
132(7)
4.6 Degenerate Hopf bifurcations: The graphical Hopf theorem
139(10)
4.6.1 Degenerate Hopf bifurcations of the H0m type
139(5)
4.6.2 Degenerate Hopf bifurcations of the Hno type
144(5)
4.7 Some applications
149(6)
5 Higher-order Hopf bifurcation formulas
155(40)
5.1 Introduction
155(1)
5.2 Approximations of periodic solutions by higher-order formulas
156(11)
5.2.1 The algorithm
159(1)
5.2.2 Some applications
160(7)
5.3 Continuation of periodic solutions: degenerate cases
167(9)
5.4 Local bifurcation diagrams: The graphical Hopf theorem
176(3)
5.5 Algorithms for recovering periodic solutions
179(16)
5.5.1 The original formulation
179(1)
5.5.2 The modified scheme
180(1)
5.5.3 An iterative graphical Hopf method
181(3)
5.5.4 Study of the van der Pol system
184(7)
5.5.5 Harmonic distortion
191(4)
6 Hopf bifurcation in continuous-time systems with discrete-time delays
195(74)
6.1 Introduction
195(2)
6.2 Preliminaries: Retarded functional differential equations
197(9)
6.2.1 Existence and uniqueness of solutions
199(1)
6.2.2 Stability notions for retarded functional differential equations
199(1)
6.2.3 Linear delay-differential equations
200(2)
6.2.4 Linearised stability criterion
202(4)
6.3 Hopf bifurcation in retarded functional differential equations
206(3)
6.4 The graphical Hopf bifurcation theorem and delay-differential equations
209(28)
6.4.1 Feedback systems with a single delay in the loop
209(5)
6.4.2 Systems with delay in the linear block and in the nonlinear feedback
214(12)
6.4.3 An alternative approach
226(11)
6.5 Some applications
237(23)
6.5.1 Dynamics of baroreflex control of heart rate
237(8)
6.5.2 Internet congestion control
245(15)
6.6 Analysis of delay equations of the neutral type
260(9)
7 Hopf bifurcation in continuous-time systems with distributed time delays
269(40)
7.1 Introduction
269(2)
7.2 Discrete versus distributed delays
271(3)
7.3 A common approach: equivalent models
274(2)
7.4 The graphical Hopf method for distributed delay systems: The general case
276(6)
7.5 Application examples
282(27)
7.5.1 A simple scalar system
282(13)
7.5.2 A system of coupled neurons
295(14)
8 Degenerate bifurcations in time-delayed systems
309(28)
8.1 Conditions for degenerate bifurcations in time-delayed systems
309(4)
8.2 Applications
313(24)
8.2.1 A nonlinear feedback control system with two time-delays
313(2)
8.2.2 Variable structure control and the Smith predictor
315(1)
8.2.3 Cascading time-delayed feedback integrators
316(8)
8.2.4 Analysis of the 1:2 resonant double Hopf bifurcation
324(1)
8.2.5 Campbell-LeBlanc system
324(13)
Appendix A Higher-order Hopf bifurcation formulas: Part I 337(10)
Appendix B Higher-order Hopf bifurcation formulas: Part II 347(2)
Appendix C Higher-order Hopf bifurcation formulas: Part III 349(2)
Bibliography 351(18)
Subject Index 369(4)
Author Index 373
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