Dynamics of nonlinear time-delay systems:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2010
|
| Schriftenreihe: | Springer series in synergetics
Springer complexity |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Literaturangaben |
| Beschreibung: | XVII, 313 S. graph. Darst. 24 cm |
| ISBN: | 9783642149375 9783642149382 |
Internformat
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| 100 | 1 | |a Lakshmanan, Muthuswamy |d 1946- |e Verfasser |0 (DE-588)112070434 |4 aut | |
| 245 | 1 | 0 | |a Dynamics of nonlinear time-delay systems |c M. Lakshmanan ; D. V. Senthilkumar |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2010 | |
| 300 | |a XVII, 313 S. |b graph. Darst. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Springer series in synergetics | |
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| 700 | 1 | |a Senthilkumar, Dharmapuri Vijayan |e Verfasser |4 aut | |
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Datensatz im Suchindex
| _version_ | 1805095476808122368 |
|---|---|
| adam_text |
IMAGE 1
CONTENTS
1 DELAY DIFFERENTIAL EQUATIONS 1
1.1 INTRODUCTION 1
1.1.1 DDE WITH SINGLE CONSTANT DELAY 3
1.1.2 DDE WITH DISCRETE DELAYS 4
1.1.3 DDE WITH DISTRIBUTED DELAY 5
1.1.4 DDE WITH STATE-DEPENDENT DELAY 6
1.1.5 DDE WITH TIME-DEPENDENT DELAY 6
1.2 CONSTRUCTING THE SOLUTION FOR DDES WITH SINGLE CONSTANT DELAY . 7
1.2.1 LINEAR DELAY DIFFERENTIAL EQUATION 8
1.2.2 NUMERICAL SIMULATION OF DDES 10
1.2.3 NONLINEAR DELAY DIFFERENTIAL EQUATIONS 11
1.3 SALIENT FEATURES OF CHAOTIC TIME-DELAY SYSTEMS 13
REFERENCES 13
2 LINEAR STABILITY AND BIFURCATION ANALYSIS 17
2.1 INTRODUCTION 17
2.2 LINEAR STABILITY ANALYSIS 17
2.2.1 EXAMPLE: LINEAR DELAY DIFFERENTIAL EQUATION 19 2.3 A GEOMETRIC
APPROACH TO STUDY STABILITY 20
2.3.1 EXAMPLE: LINEAR DELAY DIFFERENTIAL EQUATION 21 2.4 A GENERAL
APPROACH TO DETERMINE LINEAR STABILITY OF EQUILIBRIUM POINTS 22
2.4.1 CHARACTERISTIC EQUATION 22
2.4.2 STABILITY CONDITIONS 22
2.4.3 STABILITY CURVES/SURFACES IN THE (R, A, B) PARAMETER SPACE 23
2.4.4 EXTENSION TO COUPLED DDES/COMPLEX SCALAR DDES 24 2.4.5 BIFURCATION
ANALYSIS 25
2.4.6 RESULTS OF STABILITY ANALYSIS 25
2.4.7 A THEOREM ON THE STABILITY OF EQUILIBRIUM POINTS 26 2.4.8 EXAMPLE:
LINEAR DELAY DIFFERENTIAL EQUATION 26 REFERENCES 29
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1004244274
DIGITALISIERT DURCH
IMAGE 2
DI CONTENTS
3 BIFURCATION AND CHAOS IN TIME-DELAYED PIECEWISE LINEAR DYNAMICAL
SYSTEM 31
3.1 INTRODUCTION 31
3.2 SIMPLE SCALAR FIRST ORDER PIECEWISE LINEAR DDE 32
3.2.1 FIXED POINTS AND LINEAR STABILITY 33
3.3 NUMERICAL STUDY OF THE SINGLE SCALAR PIECEWISE LINEAR TIME-DELAY
SYSTEM 36
3.3.1 DYNAMICS IN THE PSEUDOSPACE 36
3.3.2 TRANSIENTS 37
3.3.3 ONE AND TWO PARAMETER BIFURCATION DIAGRAMS 41 3.3.4 LYAPUNOV
EXPONENTS AND HYPERCHAOTIC REGIMES 43 3.4 EXPERIMENTAL REALIZATION USING
PSPICE SIMULATION 44 3.5 STABILITY ANALYSIS AND CHAOTIC DYNAMICS OF
COUPLED DDES 46
3.5.1 FIXED POINTS AND LINEAR STABILITY 46
3.6 NUMERICAL ANALYSIS OF THE COUPLED DDE 49
3.6.1 TRANSIENTS 50
3.6.2 ONE AND TWO PARAMETER BIFURCATION DIAGRAMS 51 REFERENCES 53
4 A FEW OTHER INTERESTING CHAOTIC DELAY DIFFERENTIAL EQUATIONS 55 4. 1
INTRODUCTION 55
4.2 THE MACKEY-GLASS SYSTEM: A TYPICAL NONLINEAR DDE 55 4.2.1
MACKEY-GLASS TIME-DELAY SYSTEM 55
4.2.2 FIXED POINTS AND LINEAR STABILITY ANALYSIS 56
4.2.3 TIME-DELAY R = 0 57
4.2.4 TIME-DELAY T 0 57
4.2.5 NUMERICAL SIMULATION: BIFURCATIONS AND CHAOS 62 4.2.6 EXPERIMENTAL
REALIZATION USING ELECTRONIC CIRCUIT 64 4.3 OTHER INTERESTING SCALAR
CHAOTIC TIME-DELAY SYSTEMS 67 4.3.1 A SIMPLE CHAOTIC DELAY DIFFERENTIAL
EQUATION 67
4.3.2 IKEDA TIME-DELAY SYSTEM 67
4.3.3 SCALAR TIME-DELAY SYSTEM WITH POLYNOMIAL NONLINEARITY 69 4.3.4
SCALAR TIME-DELAY SYSTEM WITH OTHER PIECEWISE LINEAR NONLINEARITIES 70
4.3.5 ANOTHER FORM OF SCALAR TIME-DELAY SYSTEM 73
4.3.6 EL NINO AND THE DELAYED ACTION OSCILLATOR 76
4.4 COUPLED CHAOTIC TIME-DELAY SYSTEMS 78
4.4.1 TIME-DELAYED CHUA'S CIRCUIT 78
4.4.2 SEMICONDUCTOR LASERS 79
4.4.3 NEURAL NETWORKS 81
REFERENCES 82
IMAGE 3
CONTENTS
IMPLICATIONS OF DELAY FEEDBACK: AMPLITUDE DEATH AND OTHER EFFECTS 85 5.1
INTRODUCTION 85
5.2 TIME-DELAY INDUCED AMPLITUDE DEATH 85
5.2.1 THEORETICAL STUDY: SINGLE OSCILLATOR 86
5.2.2 EXPERIMENTAL STUDY 89
5.3 AMPLITUDE DEATH WITH DISTRIBUTED DELAY IN COUPLED LIMIT CYCLE
OSCILLATORS 91
5.4 AMPLITUDE DEATH IN COUPLED CHAOTIC OSCILLATORS 93
5.5 AMPLITUDE DEATH WITH CONJUGATE (DISSIMILAR) COUPLING 96 5.6
AMPLITUDE DEATH WITH DYNAMIC COUPLING 98
5.7 TIME-DELAY INDUCED BIFURCATIONS 101
5.8 SOME OTHER EFFECTS OF DELAY FEEDBACK 102
REFERENCES 103
6 RECENT DEVELOPMENTS ON DELAY FEEDBACK/COUPLING: COMPLEX NETWORKS,
CHIMERAS, GLOBALLY CLUSTERED CHIMERAS AND SYNCHRONIZATION 105
6.1 INTRODUCTION 105
6.2 COMPLEX NETWORKS 105
6.3 CHIMERA STATES IN DELAY COUPLED IDENTICAL OSCILLATORS 108 6.3.1
DISCOVERY OF CHIMERA STATES 108
6.3.2 CHIMERA STATES IN DELAY COUPLED SYSTEMS ILL
6.4 CHIMERA STATES IN DELAY COUPLED SUBPOPULATIONS: GLOBALLY CLUSTERED
STATES 113
6.5 SYNCHRONIZATION IN COMPLEX NETWORKS WITH DELAY 117 6.6 CONTROLLING
USING TIME-DELAY FEEDBACK 118
6.6.1 PYRAGAS TIME-DELAY FEEDBACK CONTROL 119
6.6.2 TRANSIENT BEHAVIOR WITH TIME-DELAY FEEDBACK 122 6.7 FURTHER
DEVELOPMENTS 124
REFERENCES 125
7 COMPLETE SYNCHRONIZATION OF CHAOTIC OSCILLATIONS IN COUPLED TIME-DELAY
SYSTEMS 127
7.1 INTRODUCTION 127
7.2 COMPLETE SYNCHRONIZATION IN COUPLED TIME-DELAY SYSTEMS 129 7.3
STABILITY USING KRASOVSKII-LYAPUNOV THEORY 130
7.4 NUMERICAL CONFIRMATION 133
7.4.1 CASE 1 134
7.4.2 CASE 2 134
7.4.3 CASE 3 135
7.4.4 CASE 4 135
7.5 CONCLUSION 135
REFERENCES 136
IMAGE 4
XIV CONTENTS
8 TRANSITION FROM ANTICIPATORY TO LAG SYNCHRONIZATION VIA COMPLETE
SYNCHRONIZATION 139
8.1 INTRODUCTION 139
8.2 COUPLED SYSTEM AND THE GENERAL STABILITY CONDITION 139 8.3 COUPLED
PIECEWISE LINEAR TIME-DELAY SYSTEM AND STABILITY CONDITION: TRANSITION
FROM ANTICIPATORY TO LAG SYNCHRONIZATION . 141 8.3.1 ANTICIPATORY
SYNCHRONIZATION FOR X 2 X\ 142
8.3.2 COMPLETE SYNCHRONIZATION FOR X 2 = X\ 146
8.3.3 LAG SYNCHRONIZATION FOR X 2 X\ 147
8.3.4 INVERSE SYNCHRONIZATIONS 149
8.4 TRANSITION FROM ANTICIPATORY TO LAG VIA COMPLETE SYNCHRONIZATION:
MACKEY-GLASS AND IKEDA SYSTEMS 153 8.4.1 ANTICIPATORY SYNCHRONIZATION
FOR X 2 X\ 154
8.4.2 COMPLETE SYNCHRONIZATION FOR X 2 = X\ 158
8.4.3 LAG SYNCHRONIZATION FOR X 2 X\ 158
8.5 INVERSE SYNCHRONIZATIONS: MACKEY-GLASS AND IKEDA SYSTEMS 161
REFERENCES 163
9 INTERMITTENCY TRANSITION TO GENERALIZED SYNCHRONIZATION 165 9.1
INTRODUCTION 165
9.2 BROAD RANGE (SLOW/DELAYED) INTERMITTENCY TRANSITION TO GS FOR LINEAR
ERROR FEEDBACK COUPLING OF THE FORM (XI (F ) - X 2 (T)) * * 166 9.3
STABILITY CONDITION 167
9.4 APPROXIMATE (INTERMITTENT) GENERALIZED SYNCHRONIZATION 168 9.5
CHARACTERIZATION OF IGS 171
9.6 NARROW RANGE (IMMEDIATE) INTERMITTENCY TRANSITION TO GS FOR LINEAR
DIRECT FEEDBACK COUPLING OF THE FORM X\{T) 173 9.7 BROAD RANGE
INTERMITTENCY TRANSITION TO GS FOR NONLINEAR ERROR FEEDBACK COUPLING OF
THE FORM {F{X\(T - X 2 )) - F(X 2 (T - R 2 ))) . 178 9.8 NARROW RANGE
INTERMITTENCY TRANSITION TO GS FOR NONLINEAR
DIRECT FEEDBACK COUPLING OF THE FORM F(X\ (T - X 2 )) 181
9.9 INTERMITTENCY TRANSITION TO GENERALIZED SYNCHRONIZATION:
MACKEY-GLASS & IKEDA SYSTEMS 185
9.9.1 BROAD RANGE INTERMITTENCY TRANSITION TO GS 186 9.9.2 NARROW RANGE
INTERMITTENCY TRANSITION TO GS 190 REFERENCES 1 99
10 TRANSITION FROM PHASE TO GENERALIZED SYNCHRONIZATION 201 10.1
INTRODUCTION 201
10.2 PHASE-COHERENT AND NON-PHASE-COHERENT ATTRACTORS 202 10.3 CPS IN
CHAOTIC SYSTEMS 203
10.4 CPS AND TIME-DELAY SYSTEMS 205
10.5 CPS FROM POINCARE SURFACE OF SECTION OF THE TRANSFORMED ATTRACTOR
207
IMAGE 5
CONTENTS
10.6 CPS FROM RECURRENCE QUANTIFICATION ANALYSIS 210
10.7 CPS FROM THE LYAPUNOV EXPONENTS 213
10.8 CONCEPT OF LOCALIZED SETS 214
10.9 TRANSITION FROM PHASE TO GENERALIZED SYNCHRONIZATION: MACKEY-GLASS
& IKEDA SYSTEMS 215
10.9.1 CPS FROM POINCARE SECTION OF THE TRANSFORMED ATTRACTOR . 217
10.9.2 CPS FROM RECURRENCE QUANTIFICATION ANALYSIS 218 10.9.3 CPS FROM
THE LYAPUNOV EXPONENTS 219
10.9.4 CPS IN COUPLED IKEDA SYSTEMS 221
10.10 SUMMARY 225
REFERENCES 225
11 DTM INDUCED OSCILLATING SYNCHRONIZATION 227
11.1 INTRODUCTION 227
11.2 ESTIMATION OF THE EFFECT OF DELAY TIME MODULATION 228 11.2.1
FILLING FACTOR 228
11.2.2 LENGTH OF POLYGON LINE 230
11.2.3 AVERAGE MUTUAL INFORMATION 230
11.3 COUPLED SYSTEM AND STABILITY CONDITION IN THE PRESENCE OF DELAY
TIME MODULATION 233
11.4 OSCILLATING SYNCHRONIZATION 235
11.5 INTERMITTENT ANTICIPATORY SYNCHRONIZATION 238
11.6 COMPLETE SYNCHRONIZATION 241
11.7 INTERMITTENT LAG SYNCHRONIZATION 242
11.8 COMPLEX OSCILLATING SYNCHRONIZATION 245
11.9 DTM INDUCED OSCILLATING SYNCHRONIZATION: MACKEY-GLASS & IKEDA
SYSTEMS 245
11.9.1 COUPLED MACKEY-GLASS SYSTEMS 245
11.9.2 COUPLED IKEDA SYSTEMS 246
11.10 SUMMARY 248
REFERENCES 249
12 EXACT SOLUTIONS OF CERTAIN TIME DELAY SYSTEMS: THE CAR-FOLLOWING
MODELS 251
12.1 INTRODUCTION 251
12.2 THE CAR-FOLLOWING MODELS 251
12.3 THE NEWELL MODEL 252
12.4 THE TANH CAR-FOLLOWING MODEL 255
12.5 OTHER DEVELOPMENTS 257
REFERENCES 258
IMAGE 6
XVI CONTENTS
APPENDIX
A COMPUTING LYAPUNOV EXPONENTS FOR TIME-DELAY SYSTEMS 259 A. 1
INTRODUCTION 259
A.2 LYAPUNOV EXPONENTS OF AN N-DIMENSIONAL DYNAMICAL SYSTEM . . . 259
A.2.1 COMPUTATION OF LYAPUNOV EXPONENTS 260
A.3 LYAPUNOV EXPONENTS OF A DDE 261
REFERENCES 262
B A BRIEF INTRODUCTION TO SYNCHRONIZATION IN CHAOTIC DYNAMICAL SYSTEMS
263
B.I INTRODUCTION 263
B.2 CHARACTERIZATION OF SYNCHRONIZATION 265
B.2.1 COMPLETE SYNCHRONIZATION 268
B.2.2 PHASE SYNCHRONIZATION 269
B.2.3 LAG SYNCHRONIZATION 271
B.2.4 ANTICIPATORY SYNCHRONIZATION 272
B.2.5 GENERALIZED SYNCHRONIZATION 273
REFERENCES 275
C RECURRENCE ANALYSIS 279
C.I INTRODUCTION 279
C.2 RECURRENCE PLOTS AND THEIR VARIANTS 280
C.2.1 RECURRENCE PLOTS 280
C.2.2 CROSS RECURRENCE PLOTS (CRP) 282
C.2.3 JOINT RECURRENCE PLOTS (JRP) 283
C.3 RECURRENCE QUANTIFICATION ANALYSIS (RQA) 284
C.3.1 GENERALIZED AUTOCORRELATION FUNCTION, P(T) 284 C.3.2 CORRELATION
OF PROBABILITY OF RECURRENCE (CPR) 285 C.3.3 JOINT PROBABILITY OF
RECURRENCE (JPR) 285
C.3.4 SIMILARITY OF PROBABILITY OF RECURRENCE (SPR) 286 C.4
SYNCHRONIZATION AND RECURRENCES 286
C.4.1 PS IN MUTUALLY COUPLED ROESSLER SYSTEMS 286
C.4.2 PHASE TO LAG SYNCHRONIZATION 288
REFERENCES 291
D SOME MORE EXAMPLES OF DDES 293
D. 1 INTRODUCTION 293
D.2 DDES WITH CONSTANT DELAY 293
D.2.1 HUTCHINSON'S EQUATION/DELAYED LOGISTIC EQUATION 293 D.2.2
GOPALSAMY AND LADAS POPULATION MODEL 293
D.2.3 STEM-CELL MODEL 294
D.2.4 PUPIL CYCLING MODEL 294
IMAGE 7
CONTENTS XVII
D.3 DDES WITH DISCRETE DELAYS 295
D.3.1 AUSTRALIAN BLOWFLY MODEL 295
D.3.2 WILSON AND COWAN MODEL 295
D.3.3 HUMAN RESPIRATORY MODEL 295
D.4 DDES WITH DISTRIBUTED DELAY 296
D.4.1 VOLTERRA'S LOGISTIC EQUATION 296
D.4.2 NEURAL NETWORK WITH DISTRIBUTED DELAY 296
D.4.3 CHEMOSTAT MODEL 297
D.5 DDES WITH STATE-DEPENDENT DELAY 297
D.5.1 POPULATION MODEL 297
D.5.2 LOGISTIC MODEL WITH STATE DEPENDENT DELAY 297 D.5.3 MECHANICAL
MODEL FOR MACHINE TOOL CHATTER 298 D.6 DDES WITH TIME-DEPENDENT DELAY
298
D.6.1 STEM-CELL EQUATION 298
D.6.2 NEURAL NETWORK MODEL 298
REFERENCES 299
GLOSSARY 301
INDEX 309 |
| any_adam_object | 1 |
| author | Lakshmanan, Muthuswamy 1946- Senthilkumar, Dharmapuri Vijayan |
| author_GND | (DE-588)112070434 |
| author_facet | Lakshmanan, Muthuswamy 1946- Senthilkumar, Dharmapuri Vijayan |
| author_role | aut aut |
| author_sort | Lakshmanan, Muthuswamy 1946- |
| author_variant | m l ml d v s dv dvs |
| building | Verbundindex |
| bvnumber | BV037245386 |
| classification_rvk | UG 3900 |
| ctrlnum | (OCoLC)701477243 (DE-599)DNB1004244274 |
| dewey-full | 515.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.39 |
| dewey-search | 515.39 |
| dewey-sort | 3515.39 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV037245386 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T10:59:53Z |
| institution | BVB |
| isbn | 9783642149375 9783642149382 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021158808 |
| oclc_num | 701477243 |
| open_access_boolean | |
| owner | DE-11 DE-20 DE-634 DE-703 |
| owner_facet | DE-11 DE-20 DE-634 DE-703 |
| physical | XVII, 313 S. graph. Darst. 24 cm |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in synergetics Springer complexity |
| spelling | Lakshmanan, Muthuswamy 1946- Verfasser (DE-588)112070434 aut Dynamics of nonlinear time-delay systems M. Lakshmanan ; D. V. Senthilkumar Berlin [u.a.] Springer 2010 XVII, 313 S. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Springer series in synergetics Springer complexity Literaturangaben Nichtlineare Dynamik (DE-588)4126141-0 gnd rswk-swf Differentialgleichung mit nacheilendem Argument (DE-588)4199298-2 gnd rswk-swf Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf Synchronisierung (DE-588)4130847-5 gnd rswk-swf Zeitverzögertes System (DE-588)4780926-7 gnd rswk-swf Differentialgleichung mit nacheilendem Argument (DE-588)4199298-2 s Zeitverzögertes System (DE-588)4780926-7 s Nichtlineare Dynamik (DE-588)4126141-0 s Synchronisierung (DE-588)4130847-5 s DE-604 Nichtlineare Optimierung (DE-588)4128192-5 s 1\p DE-604 Senthilkumar, Dharmapuri Vijayan Verfasser aut X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3500791&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021158808&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lakshmanan, Muthuswamy 1946- Senthilkumar, Dharmapuri Vijayan Dynamics of nonlinear time-delay systems Nichtlineare Dynamik (DE-588)4126141-0 gnd Differentialgleichung mit nacheilendem Argument (DE-588)4199298-2 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd Synchronisierung (DE-588)4130847-5 gnd Zeitverzögertes System (DE-588)4780926-7 gnd |
| subject_GND | (DE-588)4126141-0 (DE-588)4199298-2 (DE-588)4128192-5 (DE-588)4130847-5 (DE-588)4780926-7 |
| title | Dynamics of nonlinear time-delay systems |
| title_auth | Dynamics of nonlinear time-delay systems |
| title_exact_search | Dynamics of nonlinear time-delay systems |
| title_full | Dynamics of nonlinear time-delay systems M. Lakshmanan ; D. V. Senthilkumar |
| title_fullStr | Dynamics of nonlinear time-delay systems M. Lakshmanan ; D. V. Senthilkumar |
| title_full_unstemmed | Dynamics of nonlinear time-delay systems M. Lakshmanan ; D. V. Senthilkumar |
| title_short | Dynamics of nonlinear time-delay systems |
| title_sort | dynamics of nonlinear time delay systems |
| topic | Nichtlineare Dynamik (DE-588)4126141-0 gnd Differentialgleichung mit nacheilendem Argument (DE-588)4199298-2 gnd Nichtlineare Optimierung (DE-588)4128192-5 gnd Synchronisierung (DE-588)4130847-5 gnd Zeitverzögertes System (DE-588)4780926-7 gnd |
| topic_facet | Nichtlineare Dynamik Differentialgleichung mit nacheilendem Argument Nichtlineare Optimierung Synchronisierung Zeitverzögertes System |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3500791&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021158808&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT lakshmananmuthuswamy dynamicsofnonlineartimedelaysystems AT senthilkumardharmapurivijayan dynamicsofnonlineartimedelaysystems |