## Chaos for Engineers: Theory, Applications, and ControlChaos occurs widely in both natural and man-made systems. Recently, examples of the potential usefulness of chaotic behavior have caused growing interest among engineers and applied scientists. In this book the new mathematical ideas in nonlinear dynamics are described in such a way that engineers can apply them to real physical systems. From a review of the first edition by Prof. El Naschie, University of Cambridge: "Small is beautiful and not only that, it is comprehensive as well. These are the spontaneous thoughts which came to my mind after browsing in this latest book by Prof. Thomas Kapitaniak, probably one of the most outstanding scientists working on engineering applications of Nonlinear Dynamics and Chaos today. A more careful reading reinforced this first impression....The presentation is lucid and user friendly with theory, examples, and exercises.... I thought that one can no longer write text books in nonlinear dynamics which could have important impact of fill a gap. Tomasz Kapitaniak's newest book has proved me wrong twofold." |

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### Contents

1 Response of a Nonlinear System | 1 |

Problems | 4 |

2 Continuous Dynamical Systems | 5 |

22 Fixed Points and Linearisation | 11 |

23 Relation Between Nonlinear and Linear Systems | 17 |

24 Poincare Map | 20 |

25 Lyapunov Exponents and Chaos | 23 |

26 Spectral Analysis | 27 |

5 Routes to Chaos | 69 |

52 Quasiperiodic Route | 71 |

53 Intermittency | 74 |

Discrete Dynamics Approach | 77 |

55 Condition for Chaos by Period Doubling Route | 80 |

Problems | 85 |

6 Applications | 87 |

62 Chaos in Chemical Reactions | 94 |

27 Description of Different Attractors | 31 |

Problems | 35 |

3 Discrete Dynamical Systems | 39 |

32 OneDimensional Maps | 40 |

33 Bifurcations of OneDimensional Maps | 48 |

34 OneDimensional Maps and HigherDimensional Systems | 49 |

Problems | 52 |

4 Fractals | 53 |

42 Fractal Dimensions | 56 |

43 Fractal Sets | 59 |

44 Smale Horseshoe | 61 |

45 Fractal Basin Boundaries | 64 |

Problems | 67 |

63 Elastica and Spatial Chaos | 97 |

64 Electronic Circuits and Chaos | 99 |

65 Chaos in Model of El Nino Events | 105 |

7 Controlling Chaos | 111 |

712 Control by System Design | 115 |

713 Selection of Controlling Method | 119 |

72 Synchronisation of Chaos | 120 |

722 Synchronisation by Continuous Control | 122 |

73 Secure Communication | 127 |

74 Estimation of the Largest Lyapunov Exponent Using Chaos Synchronisation | 131 |

137 | |

141 | |

### Common terms and phrases

amplitude analysis approximation autonomous system basin boundary bifurcation diagram called Cantor dust Cantor set chaotic attractor chaotic behaviour chaotic systems Chap Chua's circuit constant control parameter correlation dimension coupled critical point curve damping defined described in Sect differential equation dry friction Duffing's dx/dt dynamical system eigenvalues example exists experimental feedback fixed point force fractal frequency function Hausdorff dimension Hopf bifurcation horseshoe information signal initial conditions intermittency interval iterated largest Lyapunov exponent Let us consider limit cycle linear system logistic map Lyapunov exponents matrix method motion neighbourhood Nino nonlinear systems observe obtained One-Dimensional Maps oscillations pendulum period-doubling bifurcation periodic orbit periodic solution perturbation phase space plot Poincare map power spectrum problem properties repeller route to chaos saddle shown in Fig shows stable and unstable subcritical synchronisation tangent tion topological dimension trajectory triadic Cantor types unstable manifolds unstable subspaces values variables versus