A Geometric Way of Thinking

Pictures are often more helpful than formulas for analyzing nonlinear systems. Here we illustrate this point by a simple example. Along the way we will introduce oneof the most basic techniques of dynamics interpreting a differential equation as a vector field. 

To emphasize our point about formulas versus pictures, we have chosen one of the few nonlinear equations that can be solved in closed form. We separate the variables and then integrate  

This result is exact, but a headache to interpret. For example, can you answer the following questions  

We think of t as time, x as the position of an imaginary particle moving along the real line, and x as the velocity of that particle. Then the differential equation x = sin x represents a vector field on the line it dictates the velocity vector X at each x. To sketch the vector field, it is convenient to plot x versus x, and then draw arrows o n the x-axis to indicate the corresponding velocity vector at each X. The arrows point to the right when x 0 and to the left when x 0. 

Armed with this picture, we can now easily understand the solutions to the differential equation x = sin. r. We just start our imaginary particle at Xp and watch how it is carried along by the flow. 

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