Introduction to the Modeling and Analysis of Complex Systems

4.2. CLASSIFICATIONS OF MODEL EQUATIONS 37Nonlinear system Anything else (e.g., equation involving squares, cubes, radicals,trigonometric functions, etc., of state variables).First-order system A difference equation whose rules involve state variables of theimmediate past (at time t − 1) only a .Higher-order system Anything else.a Note that the meaning of “order” in this context is different from the order of terms in polynomials.Autonomous system A dynamical equation whose rules don’t explicitly include timet or any other external variables.Non-autonomous system A dynamical equation whose rules do include time t orother external variables explicitly.Exercise 4.3 Decide whether each of the following examples is (1) linear or nonlinear,(2) first-order or higher-order, and (3) autonomous or non-autonomous.1. x t = ax t−1 + b2. x t = ax t−1 + bx t−2 + cx t−33. x t = ax t−1 (1 − x t−1 )4. x t = ax t−1 + bxt − 2 2 + c √ x t−1 x t−35. x t = ax t−1 x t−2 + bx t−3 + sin t6. x t = ax t−1 + by t−1 , y t = cx t−1 + dy t−1Also, there are some useful things that you should know about these classifications:Non-autonomous, higher-order difference equations can always be converted intoautonomous, first-order forms, by introducing additional state variables.For example, the second-order difference equationx t = x t−1 + x t−2 (4.5)