Dynamic Systems

Dyamical System

A dynamical system is a system that changes in time. A discrete dynamical system is a system that changes at discrete points in time. A dynamical system model gives us a good, but not completely accurate, discription of how real world quantities changes over time. Dynamical systems model quantities such as the population size of a species in an ecology, interest on loans or savings accounts, the price of an economic good, the number of people contracting a disease, how much pollutants are in a lake or river, or how much drug is in a person's bloodstream, etc. The state of a dynamical system is the value of it at a given time period. A first-order discrete dynamical system is a system in which its current state depends on its previous state. Likewise, a second-order dynamical system is a system in which its current state depends on the previous two states. A first-order discrete dynamical system has the form a(n+1)=f(a(n)). First-order discrete dynamical systems may be linear, affine, or nonlinear. Linear first-order discrete dynamical systems have the form, a(n+1)=ra(n), n=0,1,2,... Dynamical systems are linear if the graph of its function, y=f(x), is a straight line through the origin. Affine first-order discrete dynamical systems have the form, a(n+1)=ra(n)+b, n=0,1,2,... Dynamical systems are affine if the graph of its function, y=f(x), is a straight line with y-intercept not equal to 0. If the graph of the function of the dynamical system, y=f(x), isn't a straight line, then the dynamical system is nonlinear. Dynamical systems sometimes have equilibrium states, which are states at which the dynamical system doesn't change. The equilibrium state of a linear first-order discrete dynamical system is equal to 0. The equilibrium state of an affine first-order discrete dynamical system is a=b/(1-r) if r isn't equal to 1. If r=1 and b=0, then every number is an equilibrium state. If r=1 and b doesn't equal 0, then there is no equilibrium state.