1Difference Equations and Chaos3/5/20082Opening Discussion■What did we talk about last class?■Do you have any questions about the project?3Difference Equations■Similar to differential equations are difference equations. These are discrete equations where we calculate the next value of the system from the previous one.■These systems are sometimes referred to as mappings.■Our numerical solutions to differential equations actually convert them to difference equations. Symplectic integrators are often called symplectic maps.4The Logistic Map■A common example of a map is the logistic map. The formula for the logistic map is extremely simple: xn+1=rxn(1-xn).■Iterating a simple 1-D map like this can be viewed by drawing a “cobweb diagram”. The formula is quadratic in x, but opens downward.■Fixed points are places where the curve crosses the y=x line. Depending on the slope at that the intersection the fixed points might be stable or unstable.■Let's see if we can write code to draw a cobweb diagram.5Bifurcation■If we vary the value of r, interesting things happen to the behavior of this system. At small values there is a single, stable fixed point. At larger values, that fixed point becomes unstable and we get a period-2 cycle instead. This split is called a bifurcation.■Increasing the value a bit more produces another bifurcation to a period-4 cycle.■The separation between bifurcations gets smaller and the system actually becomes chaotic.■Let's write code to draw a bifurcation diagram.6Reminders■The midterm will be a week from today. It is open book and open
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