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E-raamat: Stability of Dynamical Systems

(The University of Western Ontario, London, Ontario, Canada), (The University of Hong Kong, Hong Kong), (Huazhong University of Science and Technology, Wuhan, China)
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Stability of Dynamical SystemsSuurem pilt
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The main purpose of developing stability theory is to examine dynamic responses of a system to disturbances as the time approaches infinity. It has been and still is the object of intense investigations due to its intrinsic interest and its relevance to all practical systems in engineering, finance, natural science and social science. This monograph provides some state-of-the-art expositions of major advances in fundamental stability theories and methods for dynamic systems of ODE and DDE types and in limit cycle, normal form and Hopf bifurcation control of nonlinear dynamic systems.

· Presents comprehensive theory and methodology of stability analysis
· Can be used as textbook for graduate students in applied mathematics, mechanics, control theory, theoretical physics, mathematical biology, information theory, scientific computation
· Serves as a comprehensive handbook of stability theory for practicing aerospace, control, mechanical, structural, naval and civil engineers

Arvustused

"The book is a collection of a host of results with a variety of real world applications and practical examples. It is a joy to read. The wide variety of results presented in this exposition will make it a useful text from graduate students of humanities, sciences and engineering This book can serve a very useful purpose and will be a valuable addition to the literature." --Mathematical Reviews, April 2011

Muu info

"The book is a collection of a host of results with a variety of real world applications and practical examples. It is a joy to read. The wide variety of results presented in this exposition will make it a useful text from graduate students of humanities, sciences and engineering... This book can serve a very useful purpose and will be a valuable addition to the literature." - Mathematical Reviews, April 2011
Preface v
Chapter
1. Fundamental Concepts and Mathematical Tools		 1
1.1. Fundamental theorems of ordinary differential equations
1
1.2. Lyapunov function
4
1.3. K-class function
7
1.4. Dini derivative
10
1.5. Differential and integral inequalities
13
1.6. A unified simple condition for stable matrix, p.d. matrix and M matrix
16
1.7. Definition of Lyapunov stability
21
1.8. Some examples of stability relation
24
Chapter
2. Linear Systems with Constant Coefficients		 35
2.1. NASCs for stability and asymptotic stability
35
2.2. Sufficient conditions of Hurwitz matrix
43
2.3. A new method for solving Lyapunov matrix equation: BA + AT B = C
53
2.4. A simple geometrical NASC for Hurwitz matrix
61
2.5. The geometry method for the stability of linear control systems
69
Chapter
3. Time-Varying Linear Systems		 77
3.1. Stabilities between homogeneous and nonhomogeneous systems
77
3.2. Equivalent condition for the stability of linear systems
80
3.3. Robust stability of linear systems
84
3.4. The expression of Cauchy matrix solution
90
3.5. Linear systems with periodic coefficients
95
3.6. Spectral estimation for linear systems
100
3.7. Partial variable stability of linear systems
104
Chapter
4. Lyapunov Direct Method		 111
4.1. Geometrical illustration of Lyapunov direct method
112
4.2. NASCs for stability and uniform stability
113
4.3. NASCs for uniformly asymptotic and equi-asymptotic stabilities
119
4.4. NASCs of exponential stability and instability
127
4.5. Sufficient conditions for stability
130
4.6. Sufficient conditions for asymptotic stability
139
4.7. Sufficient conditions for instability
152
4.8. Summary of constructing Lyapunov functions
162
Chapter
5. Development of Lyapunov Direct Method		 167
5.1. LaSalle's invariant principle
167
5.2. Comparability theory
171
5.3. Lagrange stability
177
5.4. Lagrange asymptotic stability
185
5.5. Lagrange exponential stability of the Lorenz system
188
5.6. Robust stability under disturbance of system structure
196
5.7. Practical stability
200
5.8. Lipschitz stability
203
5.9. Asymptotic equivalence of two dynamical systems
208
5.10. Conditional stability
218
5.11. Partial variable stability
224
5.12. Stability and boundedness of sets
235
Chapter
6. Nonlinear Systems with Separate Variables		 241
6.1. Linear Lyapunov function method
241
6.2. General nonlinear Lyapunov function with separable variable
253
6.3. Systems which can be transformed to separable variable systems
263
6.4. Partial variable stability for systems with separable variables
268
6.5. Autonomous systems with generalized separable variables
278
6.6. Nonautonomous systems with separable variables
280
Chapter
7. Iteration Method for Stability		 285
7.1. Picard iteration type method
285
7.2. Gauss–Seidel type iteration method
290
7.3. Application of iteration method to extreme stability
302
7.4. Application of iteration method to stationary oscillation
307
7.5. Application of iteration method to improve frozen coefficient method
309
7.6. Application of iteration method to interval matrix
315
Chapter
8. Dynamical Systems with Time Delay		 321
8.1. Basic concepts
321
8.2. Lyapunov function method for stability
324
8.3. Lyapunov function method with Razumikhin technique
330
8.4. Lyapunov functional method for stability analysis
338
8.5. Nonlinear autonomous systems with various time delays
341
8.6. Application of inequality with time delay and comparison principle
350
8.7. Algebraic method for LDS with constant coefficients and time delay
356
8.8. A class of time delay neutral differential difference systems
362
8.9. The method of iteration by parts for large-scale neural systems
366
8.10. Stability of large-scale neutral systems on C1 space
373
8.11. Algebraic methods for GLNS with constant coefficients
378
Chapter
9. Absolute Stability of Nonlinear Control Systems		 389
9.1. The principal of centrifugal governor and general Lurie systems
389
9.2. Lyapunov—Lurie type V function method
394
9.3. NASCs of negative definite for derivative of Lyapunov—Lurie type function
399
9.4. Popov's criterion and improved criterion
402
9.5. Simple algebraic criterion
407
9.6. NASCs of absolute stability for indirect control systems
420
9.7. NASCs of absolute stability for direct and critical control system
434
9.8. NASCs of absolute stability for control systems with multiple non-linear controls
442
9.9. NASCs of absolute stability for systems with feedback loops
454
9.10. Chaos synchronization as a stabilization problem of Lurie system
459
9.11. NASCs for absolute stability of time-delayed Lurie control systems
469
Chapter
10. Stability of Neural Networks		 487
10.1. Hopfield energy function method
487
10.2. Lagrange globally exponential stability of general neural network
491
10.3. Extension of Hopfield energy function method
493
10.4. Globally exponential stability of Hopfield neural network
502
10.5. Globally asymptotic stability of a class of Hopfield neural networks
515
10.6. Stability of bidirectional associative memory neural network
530
10.7. Stability of BAM neural networks with variable delays
534
10.8. Exp. stability and exp. periodicity of DNN with Lipschitz type activation function
541
10.9. Stability of general ecological systems and neural networks
550
10.10. Cellular neural network
563
Chapter
11. Limit Cycle, Normal Form and Hopf Bifurcation Control		 591
11.1. Introduction
591
11.2. Computation of normal forms and focus values
594
11.2.1 The Takens method
594
11.2.2 A perturbation method
597
11.2.3 The singular point value method
599
11.2.4 Applications
602
11.3. Computation of the SNF with parameters
613
11.3.1 General formulation
614
11.3.2 The SNF for single zero
622
11.3.3 The SNF for Hopf bifurcation
624
11.4. Hopf bifurcation control
626
11.4.1 Continuous-time systems
627
11.4.2 Discrete maps
640
11.4.3 2-D lifting surface
658
References 671
Subject Index 697


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