## An Introduction to Difference EquationsThis book integrates both classical and modern treatments of difference equations. It contains the most updated and comprehensive material on stability, Z-transform, discrete control theory, and asymptotic theory, continued fractions and orthogonal polynomials. Yet the presentation is simple enough for the book to be used by advanced undergraduate and beginning graduate students in mathematics, engineering science, and economics. Moreover, scientists and engineers who are interested in discrete mathematical models will find it useful as a reference. The book contains a large set of applications in a variety of disciplines, including neural networks, feedback control, Markov chains, trade models, heat transfer, propagation of plants, epidemic models and host-parasitoid systems. Each section ends with an extensive and highly selected set of exercises. |

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### Contents

Linear Difference Equations of Higher Order | 47 |

Systems of Difference Equations | 105 |

Stability Theory | 154 |

Copyright | |

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2-cycle a k x k assume Ax(n bounded solution Bu(n Casoratian Chapter characteristic equation characteristic polynomial characteristic roots coefﬁcients completely controllable completely observable complex numbers Consider the difference continued fraction converges Corollary corresponding deﬁned difference equation x(n differential equation eigenvalues eigenvectors equilibrium point Example Exercises 4.5 exists Figure ﬁnd ﬁrst ﬁxed point Floquet multipliers formula fundamental matrix fundamental set Hence inﬁnite Jordan blocks Jordan form k x k matrix Lemma Let x(n Liapunov function linear difference equations linearly independent Markov chain Mathematics method minimal solution n e Z+ nonlinear nonsingular nonsingular matrix obtain orthogonal polynomials oscillates oscillatory period positive deﬁnite positive integer Problem 13 Proof Prove radius of convergence real number satisﬁes second-order difference equation sequence set of solutions Show solution of 7.3.2 solution x(n Suppose system x(n Theorem theory transform uniformly stable unstable vector Z-transform zero solution