## Nonlinear SystemsThe theories of bifurcation, chaos and fractals as well as equilibrium, stability and nonlinear oscillations, are part of the theory of the evolution of solutions of nonlinear equations. A wide range of mathematical tools and ideas are drawn together in the study of these solutions, and the results applied to diverse and countless problems in the natural and social sciences, even philosophy. The text evolves from courses given by the author in the UK and the United States. It introduces the mathematical properties of nonlinear systems, mostly difference and differential equations, as an integrated theory, rather than presenting isolated fashionable topics. Topics are discussed in as concrete a way as possible and worked examples and problems are used to explain, motivate and illustrate the general principles. The essence of these principles, rather than proof or rigour, is emphasized. More advanced parts of the text are denoted by asterisks, and the mathematical prerequisites are limited to knowledge of linear algebra and advanced calculus, thus making it ideally suited to both senior undergraduates and postgraduates from physics, engineering, chemistry, meteorology et cetera as well as mathematics. |

### What people are saying - Write a review

User Review - Flag as inappropriate

در این کتاب دستگاه معادلات غیر خطی مورد بررسی غرار گرفته

### Contents

Introduction | 1 |

2 The origin of bifurcation theory | 3 |

3 A turning point | 6 |

4 A transcritical bifurcation | 10 |

5 A pitchfork bifurcation | 13 |

6 A Hopf bifurcation | 19 |

7 Nonlinear oscillations of a conservative system | 22 |

8 Difference equations | 28 |

Ordinary differential equations | 149 |

2 Hamiltonian systems | 153 |

3 The geometry of orbits | 155 |

4 The stability of a periodic solution | 157 |

Further reading | 161 |

Secondorder autonomous differential systems | 170 |

2 Linear systems | 172 |

3 The direct method of Liapounov | 178 |

9 An experiment on statics | 35 |

Further reading | 37 |

Problems | 38 |

Classification of bifurcations of equilibrium points | 48 |

2 Classification of bifurcations in one dimension | 49 |

3 Imperfections | 55 |

4 Classification of bifurcations in higher dimensions | 59 |

Further reading | 63 |

Difference equations | 68 |

2 Periodic solutions and their stability | 72 |

3 Attractors and volume | 74 |

32 Volume | 80 |

4 The logistic equation | 81 |

5 Numerical and computational methods | 94 |

6 Some twodimensional difference equations | 96 |

7 Iterated maps of the complex plane | 103 |

Further reading | 109 |

Some special topics | 125 |

2 Dimension and fractals | 127 |

3 Reoonnalization group theory | 132 |

32 Feigenbaums theory of scaling | 135 |

4 Liapounov exponents | 140 |

Further reading | 143 |

Problems | 144 |

4 The LindstedtPoincaré method | 181 |

5 Limit cycles | 186 |

6 Van der Pols equation | 190 |

Further reading | 197 |

Problems | 199 |

Forced oscillations | 214 |

regular perturbation theory | 215 |

3 Weakly nonlinear oscillations near resonance | 219 |

4 Subharmonics | 225 |

Further reading | 229 |

Problems | 230 |

Chaos | 233 |

2 Duffings equation with negative stiffness | 246 |

Melnikovs method | 251 |

4 Routes to chaos | 261 |

5 Analysis of time series | 264 |

Further reading | 277 |

Some partialdifferential problems | 283 |

Additional problems | 290 |

Answers and hints to selected problems | 305 |

316 | |

Motion picture and video index | 324 |

325 | |

### Common terms and phrases

approximation assume asymptotically attractor becomes bifurcation diagram called centre chaos chaotic Chapter closed coefficients complex condition Consider constant continuous cost curve Deduce defined definition depend describe difference equation differential equations dimension dt dt dx/dt eigenvalue equilibrium point example exists fact fixed point follows forcing function further given gives Hence increases initial integral interval length Liapounov limit cycle linearized system logistic manifold matrix method motion nonlinear nonlinear system Note occur orbit origin oscillations parameter period doubling periodic solution perturbation phase portrait plane Poincaré positive problem properties represents saddle point seen sequence Show shown simple Sketch solve space stable steady Suppose theory tion turning point two-cycle unstable values variable varies x)-plane Xn+1 zero