# Introduction to Dynamic Systems

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 ESE 210 Introduction to Dynamic Systems Catalog Description: This first course in systems modeling focuses on linear discrete-time systems. We draw on a set of examples used throughout the course as the necessary mathematical tools are developed. The examples demonstrate the breadth of systems models and are drawn from engineering, the biological sciences, and economics. MATLAB will be used extensively. Pre-requisite: MATH 240. Motivation (2 weeks) Systems Modeling Closed Equations that describe system are known Open Simulation needed for system modeling Complex Agent based modeling Review some simple models that are familiar Geometric Growth model Loan Repayment/Home mortgage National Income Model Exponential Growth/decay Cohort Population Model Introduction to use of MATLAB Modeling change with difference equations Life cycle of the modeling process Linear Difference Equations (3 weeks) Mathematical Properties Definitions Existence and uniqueness Initial conditions Existence and Uniqueness Theorem First-order linear equations General Solution Amortization example Cobweb Model MATLAB Linear Difference Equations Homogeneous Equation Fundamental Set of Solutions Linear Independence Solution of Nonhomogeneous Equation Linear Equations with Constant Coefficients Characteristic Equation Systems of First-order Equations Definitions Example – Leontief Economic Model Linear Systems Systems of Equations in Matrix Format (2 weeks) Linear Algebra Matrices and vectors Elementary Operations Determinants Inverses Linear transformations Eigenvalues/ eigenvectors Cohort Population Growth Model MATLAB Linear State Equations (3 weeks) Systems of First-order Difference Equations State space Conversion to State Form Block diagrams Homogeneous discrete time systems Natchez Social Structure Example General solution to linear discrete-time systems Superposition Time-invariant Systems National Income Model Revisited Equilibrium Points and Stability Cohort Population Growth Model Natchez Social Structure Example Revisited Concepts of Control (4 weeks) Z-transform Definition Sequences and convergence Rational polynomials, poles and zeros Transform Properties and Transform Pairs Transform Solution of Difference Equations Inversion by Partial Fractions State Space Methods ESE 210 John Keenan Page 4/7/2016

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