## Discrete Dynamical ModelsThis book provides an introduction to the analysis of discrete dynamical systems. The content is presented by an unitary approach that blends the perspective of mathematical modeling together with the ones of several discipline as Mathematical Analysis, Linear Algebra, Numerical Analysis, Systems Theory and Probability. After a preliminary discussion of several models, the main tools for the study of linear and non-linear scalar dynamical systems are presented, paying particular attention to the stability analysis. Linear difference equations are studied in detail and an elementary introduction of Z and Discrete Fourier Transform is presented. A whole chapter is devoted to the study of bifurcations and chaotic dynamics. One-step vector-valued dynamical systems are the subject of three chapters, where the reader can find the applications to positive systems, Markov chains, networks and search engines. The book is addressed mainly to students in Mathematics, Engineering, Physics, Chemistry, Biology and Economics. The exposition is self-contained: some appendices present prerequisites, algorithms and suggestions for computer simulations. The analysis of several examples is enriched by the proposition of many related exercises of increasing difficulty; in the last chapter the detailed solution is given for most of them. |

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### Contents

1 | |

2 Linear difference equations | 25 |

onestep scalar equations | 84 |

bifurcations and chaos | 125 |

onestep vector equations | 179 |

6 Markovchains | 227 |

7 Positive matrices and graphs | 256 |

8 Solutions of the exercises | 285 |

Appendix C Basic probability | 351 |

Appendix D Linear Algebra | 352 |

Appendix E Topology | 363 |

Appendix F Fractal dimension | 365 |

Appendix G Tables of Ztransforms | 371 |

Appendix H Algorithms and hints for numerical experiments | 375 |

381 | |

383 | |

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### Common terms and phrases

algebraic Algorithm alleles analysis associated attraction basin behavior called Cantor set coefﬁcients complex number components compute consider constant convergence corresponding DDS I,f deduce deﬁned Deﬁnition denotes difference equation discrete dynamical system dominant eigenvalue eigenvalue eigenvector Example Exercise exists finite ﬁrst ﬁxed points follows formula function f given graph homogeneous initial conditions integer interval invariant probability distribution irreducible ISBN iterations Lemma limk linear locally asymptotically stable logistic logistic map Markov chain Matematica method modulus monotonicity Moreover multiplicity nodes nonlinear Notice obtain parameter periodic orbits permutation permutation matrix Pk+1 polynomial population probability distribution problem Proof prove recursive Remark roots Salinelli sequence solution solve square matrix stable and attractive stochastic matrix strictly positive strongly connected Theorem topological conjugacy topologically conjugate trajectories transform transition matrix vector verify Xk+1 Yk+1 Z-transform zero