Question

Consider the system $$ d x / d t=a x[1-(y / 2)], \quad d y / d t=b y[-1+(x / 3)] $$ where \(a\) and \(b\) are positive constants. Observe that this system is the same as in the example in the text if \(a=1\) and \(b=0.75 .\) Suppose the initial conditions are \(x(0)=5\) and \(y(0)=2\) (a) Let \(a=1\) and \(b=1 .\) Plot the trajectory in the phase plane and determine (or cstimate) the period of the oscillation. (b) Repeat part (a) for \(a=3\) and \(a=1 / 3,\) with \(b=1\) (c) Repeat part (a) for \(b=3\) and \(b=1 / 3,\) with \(a=1\) (d) Describe how the period and the shape of the trajectory depend on \(a\) and \(b\).

Short answer

Answer: The period and shape of the trajectory depend on the values of a and b such that varying these parameters results in different oscillation patterns. By plotting the trajectories for different values of a and b, we can observe how the period of oscillation and shape of the trajectory are affected. In general, changing the values of a and b will lead to changes in the period and trajectory shape, highlighting the critical role these parameters play in the oscillating system's behavior.

Step-by-step solution

Part (a) - Plot trajectories and estimate the period of oscillation for a=1 and b=1

To plot the trajectory, we can use a numerical solver to obtain the solution to the system of differential equations for the given initial conditions \(x(0) = 5\) and \(y(0) = 2\) and a =1, b=1. After obtaining the trajectory, we can visualize the result to analyze the period of oscillation.

Part (b) - Plot trajectories with a=3 and a=1/3, keeping b=1

To understand the effect of varying \(a\), we need to plot the trajectories in the phase plane for a=3 and a=1/3, making sure we keep b=1. We can repeat the same process as in part (a), but this time with different values of \(a\).

Part (c) - Plot trajectories with b=3 and b=1/3, keeping a=1

Now we want to analyze the effect of varying \(b\). To do this, we need to plot trajectories for b=3 and b=1/3, while keeping a=1 constant. We can repeat the same process as in previous parts, just changing the values of \(b\) instead.

Part (d) - Describe the dependence of period and trajectory shape on a and b

After plotting the trajectories for different values of \(a\) and \(b\), we can describe how the period and shape of the trajectory depend on these parameters. By analyzing the graphs, it is possible to draw conclusions about how varying \(a\) and \(b\) affect the period of oscillation and the shape of the trajectories.