## Dynamical Systems with Applications using Mathematica®This book provides an introduction to the theory of dynamical systems with the ® aid of the Mathematica computer algebra system. It is written for both senior undergraduates and graduate students. The ?rst part of the book deals with c- tinuous systems using ordinary differential equations (Chapters 1–10), the second part is devoted to the study of discrete dynamical systems (Chapters 11–15), and Chapters 16 and 17 deal with both continuous and discrete systems. It should be pointedoutthatdynamicalsystemstheoryisnotlimitedtothesetopicsbutalso- compassespartialdifferentialequations,integralandintegrodifferentialequations, stochastic systems, and time-delay systems, for instance. References [1]–[4] given at the end of the Preface provide more information for the interested reader. The author has gone for breadth of coverage rather than ?ne detail and theorems with proofs are kept at a minimum. The material is not clouded by functional analytic and group theoretical de?nitions, and so is intelligible to readers with a general mathematical background. Some of the topics covered are scarcely covered el- where. Most of the material in Chapters 9, 10, 14, 16, and 17 is at a postgraduate levelandhasbeenin?uencedbytheauthor’sownresearchinterests. Thereismore theory in these chapters than in the rest of the book since it is not easily accessed anywhere else. It has been found that these chapters are especially useful as ref- ence material for senior undergraduate project work. The theory in other chapters of the book is dealt with more comprehensively in other texts, some of which may be found in the references section of the corresponding chapter. |

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### Contents

0 | 1 |

Mathematica Program Index | 2 |

Differential Equations | 16 |

Planar Systems | 41 |

1 | 69 |

4 | 84 |

6 | 126 |

8 | 170 |

Fractals and Multifractals | 331 |

Chaos Control and Synchronization | 363 |

6 | 369 |

17 | 386 |

4 | 408 |

6 | 415 |

18 | 421 |

2 | 424 |

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### Common terms and phrases

age class algorithm applied asymptotically stable behavior bifurcation diagram bistable region box-counting chaos control chaotic attractor Chapter circuit Consider constructed defined depicted in Figure determine differential equations dynamical systems eigenvalues eigenvectors Example feedback fiber fixed point fractal dimension graph Hamiltonian systems Hence Hénon map Hopfield network infinite number infinity initial conditions input Julia sets Koch curve Liénard systems limit cycle linear stability analysis logistic map loop Lyapunov exponents Lyapunov function Mathematica Mathematica Commands Mathematica program mathematical multifractal neural networks neuron nonlinear Notebook optical orbits oscillations output perturbations phase portrait Phys Poincaré maps points of period polynomial population reader saddle point second iterative method Section segments SFR resonator shown in Figure simple solution curves solving species stable focus stage synaptic weights synchronization theorem theory trajectories unstable manifolds unstable node vector field xn+1 yn+1 zero zn+1