Dynamical Systems. 1.0 Ordinary Differential Equations. 2.0 Dynamical Systems


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1 . Ordinary Differential Equations. An ordinary differential equation (ODE, for short) is an equation involves a dependent variable and its derivatives with respect to an independent variable. When we speak of a dynamical systems generally we mean that the independent variable is time. Thus, the derivatives refer to rates of change, i.e. velocity and acceleration, of the dependent variable.. Eamples a < A linear differential equation a 3 < A nonlinear differential equation. Dynamical Systems Here, means the rate of change of over time, t.. A dynamical system has two parts: ) A state, which completely describes the state of the system at a given time ) A function (or rule), which tells us, given the current state of the system, what the state will be at the net instant in time. This is also called the system s dynamics. In both eamples above, the state is simply the dependent variable,, and the functions are given by the differential equations themselves.. Ordinary differential equations are one type of dynamical system; there are other types as well..3 One way to look at the behavior of a dynamical system is as a time series. A time series shows the value of the dependent variable(s) as a function of the independent variable. The following figure illustrates three important facts about dynamical systems ) the system has an initial condition, the value of at t ; ) the system has a solution which is a function of time, (t) 3) the system has a stationary state, a value of for which there is no longer any change. Page of 5
2 Solution of a time.4 Initial conditions must always be specified. The initial condition is usually written as () or..5 Solutions, the trajectory over time, can be easy or hard to come by:.5. It is always possible to find an analytical solution for a linear ODE. In other words, the trajectory over time, (t), can be written in closed form. For the linear differential equation above, the solution is: t () e at..5. Sometimes nonlinear ODEs can also be solved analytically. For eample, the solution to the nonlinear ODE above is ,. ke t k ( ) More often, however, the solution to a nonlinear ODE cannot be written down in this concise fashion. Usually, nonlinear dynamical systems must be solved numerically. This involves programming the functions into a computer and iterating over time to trace out the trajectory stepbystep. Since the time steps must usually be very small for this method to work, considerable computation is generally required..6 Sometimes it is possible to learn important facts about the behavior of a dynamical system without solving it, either analytically or numerically. The single most important fact to know about a dynamical system is: What are the stationary states?.6. The simplest type of stationary state, and the only one that we will look at here, is the fied point. Page of 5
3 3. Stability.5.6. A fied point is defined as a point at which. Substituting this definition into the original equation allows us to find the fied points, if we are clever enough. For the linear equation above, we have a. Therefore, if a, the fied point is. (Why is the condition a necessary? What happens of a?) For our eample nonlinear equation, we have r( r ), yielding the fied points r and r ± a. (Be sure you can find these yourself!) 3. Once we know the fied points, the net step is to understand their stability. A fied point is called stable when initial conditions away from the fied point lead to trajectories that approach the fied point. A fied point is called unstable when initial conditions near the fied point lead to trajectories that move away from the fied point..5 stable unstable a <.5 fied point: In the above figure the fied point,, is stable whenever a <. But when, the same fied point is unstable. 3.3 A linear system, such as the on above, has a single fied point, and it is either stable or unstable depending on parameter values. For nonlinear systems, the situation is more comple. There may be more than one fied point, and even other kinds attractors, such as limit cycles. Page 3 of 5
4 ± stable, a < unstable, In our eample nonlinear system there are three fied points,,, and. is stable when <, and in this case the other two fied points do not eist (Why?). When >, becomes unstable; the other fied points come into eistence and are stable. 4. Bifurcations 4. As the last eample shows, the structure of a nonlinear dynamical system can be much richer than that of a linear dynamical system. A variety of qualitatively different behaviors can be ehibited by the system, depending upon its parameter values. 4. A diagram that shows the different behaviors (i.e. stable states) of a dynamical system as a function of its parameter value(s) is called a bifurcation diagram. For our nonlinear eample, the associated diagram looks like this: < > bifurcation point Page 4 of 5
5 4.3 The key point about nonlinear dynamical systems is that small changes in parameter values can cause qualitative changes in behavior. Mathematicians call this phenomenon a bifurcation, physicists call it a non equilibrium phase transition. 4.4 This is the aspect of nonlinear dynamics that makes it especially useful as a modeling tool. The same system can ehibit many different kinds of behavior; the bifurcation diagram maps out these behaviors systematically. 4.5 The main issue in modeling is to map the behavior under study onto the behavior of a dynamical system. Page 5 of 5
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