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 |  موضوع: كتاب Analysis and Control of Complex Dynamical Systems السبت 19 أكتوبر 2024, 11:36 am | |
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Analysis and Control of Complex Dynamical Systems
Robust Bifurcation, Dynamic Attractors, and Network Complexity
Kazuyuki Aihara , Jun-ichi Imura , Tetsushi Ueta
Editors
و المحتوى كما يلي :
Contents
Part I Robust Bifurcation and Control
1 Dynamic Robust Bifurcation Analysis . 3
Masaki Inoue, Jun-ichi Imura, Kenji Kashima
and Kazuyuki Aihara
1.1 Introduction 3
1.2 Problem Formulation: Dynamic Robust
Bifurcation Analysis 5
1.3 Equilibrium, Stability/Instability, and Robustness Analysis 7
1.3.1 Equilibrium Analysis 8
1.3.2 Robust Hyperbolicity Analysis . 11
1.3.3 Robust Bifurcation Analysis . 14
1.4 Examples of Robust Bifurcation Analysis . 15
1.4.1 Robustness Analysis of Saddle-Node Bifurcation 15
1.4.2 Robustness Analysis of Hopf Bifurcation 16
1.5 Conclusion . 17
References 17
2 Robust Bifurcation Analysis Based on Degree of Stability . 21
Hiroyuki Kitajima, Tetsuya Yoshinaga, Jun-ichi Imura
and Kazuyuki Aihara
2.1 Introduction 21
2.2 System Description and Robust Bifurcation Analysis 22
2.2.1 Continuous-Time Systems 22
2.2.2 Discrete-time Systems . 23
2.2.3 Robust Bifurcation Analysis . 24
2.3 Method of Robust Bifurcation Analysis . 25
2.4 Numerical Examples 27
2.4.1 Equilibrium Point 27
ix2.4.2 Periodic Solution 29
2.5 Conclusion . 30
References 31
3 Use of a Matrix Inequality Technique for Avoiding
Undesirable Bifurcation . 33
Yasuaki Oishi, Mio Kobayashi and Tetsuya Yoshinaga
3.1 Introduction 33
3.2 Considered Problem 34
3.3 Proposed Method 35
3.4 Extension 36
3.5 Example 37
3.6 Avoidance of Chaos 38
3.6.1 Method for Chaos Avoidance 38
3.6.2 Experimental Result 39
3.7 Conclusion . 40
References 40
4 A Method for Constructing a Robust System Against
Unexpected Parameter Variation . 41
Hiroyuki Kitajima and Tetsuya Yoshinaga
4.1 Introduction 41
4.2 Method . 42
4.2.1 Dynamical System . 42
4.2.2 Search for Optimal Parameter Values 43
4.3 Results . 45
4.3.1 Discrete-Time System . 45
4.3.2 Continuous-Time System . 46
4.4 Conclusion . 47
References 48
5 Parametric Control to Avoid Bifurcation Based
on Maximum Local Lyapunov Exponent . 49
Ken’ichi Fujimoto, Tetsuya Yoshinaga, Tetsushi Ueta
and Kazuyuki Aihara
5.1 Introduction 49
5.2 Problem Statement . 50
5.3 Proposed Method 51
5.4 Experimental Results 52
5.5 Conclusion . 54
References 55
x Contents6 Threshold Control for Stabilization of Unstable Periodic
Orbits in Chaotic Hybrid Systems 57
Daisuke Ito, Tetsushi Ueta, Takuji Kousaka, Jun-ichi Imura
and Kazuyuki Aihara
6.1 Introduction 57
6.2 Design of Controller with Perturbation
of the Threshold Value 59
6.3 A Simple Chaotic System 63
6.3.1 Numerical Simulation . 65
6.3.2 Circuit Implementation 67
6.4 Izhikevich Model 68
6.4.1 Controller . 69
6.4.2 Numerical Simulation . 70
6.5 Conclusion . 71
References 72
Part II Dynamic Attractor and Control
7 Chaotic Behavior of Orthogonally Projective
Triangle Folding Map . 77
Jun Nishimura and Tomohisa Hayakawa
7.1 Introduction 77
7.2 Orthogonally Projective Triangle Folding Map 78
7.3 Tetrahedron Map 81
7.3.1 Fixed Point and Periodic Point Analysis
on the Boundary of D . 82
7.4 Extended Fixed Point and Periodic Point Analysis
for Tetrahedron Map 83
7.4.1 Geometric Interpretation of the Triangle
Folding Map . 83
7.4.2 Periodic Points of the Tetrahedron Map . 87
7.4.3 Chaos by the Tetrahedron Map . 88
7.5 Conclusion . 90
References 90
8 Stabilization Control of Quasi-periodic Orbits . 91
Natushiro Ichinose and Motomassa Komuro
8.1 Introduction 91
8.2 Properties of Quasi-periodic Orbit on Invariant
Closed Curve . 92
8.3 Unstable Quasi-periodic Orbit 94
8.4 External Force Control 96
8.5 Delayed Feedback Control 98
Contents xi8.6 Pole Assignment Method . 103
8.7 Conclusions 105
References 106
9 Feedback Control Method Based on Predicted Future
States for Controlling Chaos 109
Miki U. Kobayashi, Tetsushi Ueta and Kazuyuki Aihara
9.1 Introduction 109
9.2 Method . 111
9.3 Application 112
9.3.1 Logistic Map . 112
9.3.2 Hénon Map 116
9.4 Conclusions 118
References 119
10 Ultra-discretization of Nonlinear Control Systems
with Spatial Symmetry 121
Masato Ishikawa and Takuto Kita
10.1 Introduction 121
10.2 Basic Properties on the Hexagonal Cellular Space 123
10.2.1 Coordinate Settings . 123
10.2.2 Basics of Difference Calculus in Concern . 124
10.3 Locomotion Under Nonholonomic Constraints 125
10.3.1 Derivation of the Continuous Single-Cart Model . 125
10.3.2 Derivation of the Discrete Version 126
10.3.3 Holonomy and the Lie Bracket Motion . 128
10.4 Connected Rigid Bodies: Locomotion Under
both Nonholonomic and Holonomic Constraints . 129
10.4.1 Cart-Trailer Systems 129
10.4.2 Derivation of the Discrete Version 131
10.5 Reachability Issues . 134
10.5.1 Definitions . 135
10.5.2 Application 135
10.6 Other Possibilities of Cellular Tesselation . 137
10.7 Conclusion . 139
References 140
11 Feedback Control of Spatial Patterns
in Reaction-Diffusion Systems . 141
Kenji Kashima and Toshiyuki Ogawa
11.1 Introduction 141
11.2 Pattern Formation by Global Feedback . 143
11.2.1 Turing Instability 143
xii Contents11.2.2 Interpretation of Turing Instability
by Global Feedback 144
11.2.3 0:1:2-Mode Interaction 148
11.2.4 Wave Instability . 150
11.2.5 Summary 153
11.3 Selective Stabilization of Turing Patterns 153
11.3.1 Reaction-Diffusion Systems . 153
11.3.2 Problem Formulation 155
11.3.3 Feedback Control of Center Manifold Dynamics . 156
11.3.4 Numerical Example . 158
11.3.5 Summary 159
References 159
12 Control of Unstabilizable Switched Systems . 161
Shun-ichi Azuma, Tomomi Takegami and Yoshito Hirata
12.1 Introduction 161
12.2 Problem Formulation 162
12.2.1 Unstabilizable Switched Systems . 162
12.2.2 Divergence Delay Problem 163
12.3 Discrete Abstraction of Switched Systems . 163
12.4 Divergence Delay Control Based on Discrete Abstraction 164
12.5 Application to Optimal Scheduling Intermittent Androgen
Suppression for Treatment of Prostate Cancer . 166
12.5.1 Mathematical Model of ISA . 166
12.5.2 Sub-optimal Scheduling Based
on Discrete Abstraction 167
12.6 Conclusion . 168
References 169
Part III Complex Networks and Modeling for Control
13 Clustered Model Reduction of Large-Scale
Bidirectional Networks 173
Takayuki Ishizaki, Kenji Kashima, Jun-ichi Imura
and Kazuyuki Aihara
13.1 Introduction 173
13.2 Preliminaries . 175
13.3 Clustered Model Reduction . 177
13.3.1 Problem Formulation 177
13.3.2 Exact Clustered Model Reduction . 178
13.3.3 Approximation Error Evaluation for Clustered
Model Reduction 181
Contents xiii13.4 Numerical Example: Application to Complex Networks 185
13.5 Conclusion . 188
References 188
14 Network Structure Identification from a Small Number
of Inputs/Outputs 191
Masayasu Suzuki, Jun-ichi Imura and Kazuyuki Aihara
14.1 Introduction 191
14.2 Characteristic-Polynomial-Based NW Structure Identification
using Knock-Out 193
14.2.1 Problem Formulation 193
14.2.2 Representation Using the Generalized
Frequency Variable . 194
14.2.3 Identification Method . 195
14.3 Identification of a Transfer Characteristic Among
Measurable Nodes 200
14.3.1 Network System and Its Dynamical
Structure Function 200
14.3.2 Reconstruction of Dynamical Structure Function
from the Transfer Function of the NW System 204
14.4 Conclusions and Discussions 207
References 207
Index 209
xiv Contents
Index
Symbols
1:2 resonance, 147
A
Activator-inhibitor, 145
Aggregation matrix, 177
Almost periodic, 91
Alternans, 29
Angle addition formulae, 123
Aperiodic, 91
Avoiding bifurcations, 49
B
Barabási-Albert model, 185
Basins of attraction, 28, 66, 70
Bidirectional network, 175
Bifurcation, 3
Bisection method, 96
BvP (Bonhöffer-van der Pol) equations, 27
C
Cart-trailer systems, 129
Causality, 202
Cellular automata, 122
Cellular tesselation, 137
Chaos control, 109
Chaos periodic orbit, 109
Chaotic attractor, 64, 68
Chaotic dynamical system, 111
Characteristic equation, 23, 24, 50, 60
Characteristic multiplier, 24, 43, 50
Characteristic polynomial, 103, 196
Closed-loop system, 201
Cluster reducibility, 174
Cluster set, 177
Clustered model reduction, 174
Clusters, 174
Conditional L2 gain, 14
Conditional Lyapunov exponents, 97
Continued fraction, 93
Continuous gradient method, 26
Control energy, 61
Control input, 61
Control structure function, 201
Controllability Lie algebra, 126
Controllable, 194
Controllable subspace, 177
Controller Hessenberg transformation, 176
Convergent, 94
Coprime integers, 93
Coupled map lattice, 94
D
Degree of stability, 25
Delayed feedback control, 98
Difference calculus, 124
Discrete abstraction, 163
Discrete nonholonomic constraint, 127
Discrete-time systems, 121
Divergence delay problem, 163
Duffing’s equations, 46
Dynamic uncertainties, 4
Dynamical structure function, 201
E
Energy-saving control, 62
Equivalence relation, 79
Erdö-Rényi model, 186
Erdö-Rényi network, 174
External force control, 96
F
Fixed point, 60, 113
G
Gene-knock-out procedure, 192
Generalized frequency variable, 194
Global feedback, 144
Gradient system, 52
Gray-box models, 191
Grey code property, 87
HH∞
norm, 4
Hénon map, 52
Hénon map, 37, 116
Hexagonal cellular space, 123
Holme-Kim model, 185, 186
Holonomic constraints, 130
Holonomy, 128
Hopf bifurcation, 16
Householder transformation, 176
Hybrid systems, 58
Hyperbolic, 23
I
Identifiability, 191
Identification, 191
Imperfect bifurcation, 7
Input/output data, 194
Intermittent androgen suppression (ISA),
166
Intermittent hormone therapy, 72
Internal structure function, 201
Invariant closed curve, 91
Invariant torus, 91
Irrational rotation, 92
Izhikevich model, 68
J
Jacobi’s identity for determinants, 197
Jacobians, 129
Jumping dynamics, 68
K
Kawakami map, 45
Kronecker product, 193
Krylov projection method, 173
LL
∞ norm, 4
L∞-induced norm, 174
L2-induced norm, 174
Large-scale network, 191
Left shift operation, 86
Lie algebra rank condition, 122
Lie bracket motion, 128
Limiter, 70
Linear interpolation, 101
Local expansion rate, 38
Local section, 59
Logistic map, 94
Longest path problem, 165
Lower LFT representation, 12
Luo-Rudy (LR) model, 29
Lyapunov exponent, 97
M
Margin to bifurcations, 41
Matrix inequality, 35
Maximum local Lyapunov exponent
(MLLE), 50
Maximum Lyapunov exponent, 38
Maximum singular value, 4
Method of analogues, 111, 112
Metzler, 174
Minimization problem, 51
Moore neighborhood, 137
N
Neimark-Sacker bifurcation, 95
Network structure-preserving model reduction, 173
Neumann neighborhood, 137
Nonholonomic constraint, 125
Nonlinear time series analysis, 110
Normal vector, 44
Nyquist stability criterion, 12
O
Observable, 194
Odd-number limitation, 110
OGY method, 109
Optimal parameter values, 43
Optimization problem, 25, 35
Orbital instability, 119
Orthogonally projective triangle folding
map, 78, 80Index 211
P
Parametric controller, 50
Penalty function method, 35
Period-doubling bifurcation, 95
Permutation matrix, 177
Planar locomotion, 122
Poincaré map, 23, 60
Poincaré section, 58
Pole assignment, 58, 103
Poles, 4
Positive systems, 176
Positive tridiagonal realization, 176
Positive tridiagonal structure, 175
Positive tridiagonalization, 175
Potential bifurcation region, 6
Prediction-based feedback control, 111
Prostate cancer, 166
Prostate-specific antigen, 72
Q
Quasi-periodic orbit, 91
R
Reaction-diffusion systems, 141, 175
Recurrence time, 91
Reducible, 178
Reducible clusters, 174
Relative degree, 198
Robust bifurcation analysis, 6, 24
Robust hyperbolicity, 11
Robust hyperbolicity condition, 12
Rotating wave, 152
Rotational number, 92
S
Saddle-node bifurcation, 15
Scale-free, 185
Scale-free network, 174
Self recovery, 21
Semi-passivity, 143
Simple interrupt chaotic system, 63
Single cart, 125
Small-gain theorem, 13
Small-world, 185
SO(2) symmetry, 143, 147, 152
Spectral radius, 51
Spectrum consensus, 159
Spectrum radius, 4
Stability analysis, 113
Stability index, 34
Standing wave, 152
Stepwise reachability, 134
Strictly proper, 198
Structured balanced truncation, 174
Sudden cardiac death, 29
Switched systems, 162
Switching control, 113
Synchronization, 97
T
Tetrahedron map, 81
-reducible, 181
Threshold value, 58
Time-delayed control, 109
Topologically conjugate, 92
Transfer function, 5, 174, 194
Tridiagonal structure, 176
Turing instability, 142
U
Ultra-discretization, 121
Uncontrollability, 174
Unpredictability, 110
Unstabilizable, 162
Unstable manifolds, 11
Unstable periodic orbit (UPO), 57
Upper LFT representation, 8
V
Ventricular muscle cell, 29
W
Wave instability, 151
Weighted adjacency matrix, 200
Weighted graph Laplacian, 175, 193
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