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.

Key concepts

Phase Plane

In the study of differential equations, the phase plane is a valuable tool for visualizing the behavior of dynamic systems. For a system of first-order equations like the one given in the problem, plotting the variables on a two-dimensional plane allows us to see their trajectories over time.

• The horizontal axis usually represents one variable (e.g., \(x\)), while the vertical axis represents the other (e.g., \(y\)).
• Each point in the phase plane corresponds to a particular state of the system, defined by those values of \(x\) and \(y\).
• This helps us identify fixed points, limit cycles, and the overall behavior of the system as time progresses.

Visualizing trajectories can give insights into aspects such as equilibrium points and stability, which are crucial for understanding complex systems.

Oscillation

Oscillation refers to a repetitive variation, typically in time, of some measure about a central value or between two or more different states. In differential equations, oscillations often arise as solutions to systems involving periodic or cyclical behavior.

• The system in the exercise demonstrates oscillation through periodic behavior of the variables \(x\) and \(y\).
• The period of the oscillation is the time it takes to complete one full cycle.
• In the given system, changing parameters \(a\) and \(b\) affects the speed and characteristics of the oscillations.

Understanding oscillations helps in predicting how systems behave over time, particularly one's that involve cycles, such as biological rhythms or mechanical vibrations.

Trajectory Analysis

Trajectory analysis involves studying the paths that solutions of a differential equation take in the phase plane over time. It provides insights into the dynamic nature of the system.

• In our context, a trajectory is a curve in the phase plane that shows the change in both \(x\) and \(y\) as time progresses, starting from the given initial conditions \(x(0)=5\) and \(y(0)=2\).
• The shape of these trajectories can indicate stability, spirals, or closed loops, which correspond to different dynamic behaviors.
• By examining how trajectories change with differing parameters \(a\) and \(b\), we can observe how they influence system dynamics, providing a visual grasp of complex behavior.

Trajectory analysis not only aids in comprehending current system dynamics but also in forecasting future behavior under various initial conditions or parameter changes.

Numerical Solver

When it comes to differential equations, especially nonlinear ones, finding analytical solutions can be difficult. That's where numerical solvers come in useful. These computational tools allow us to approximate the solutions to complicated systems, making analysis feasible.

• Numerical solvers use algorithms to calculate the approximate path of a trajectory based on the given differential equations and initial conditions.
• For the problem at hand, tools like Euler's method, Runge-Kutta methods, or software like MATLAB and Python can be employed to compute trajectories.
• Using numerical solvers in plotting helps visualize how the system evolves, offering a practical approach to understand oscillations and trajectory characteristics.

By harnessing numerical solvers, students can tackle real-world problems where analytical solutions are difficult or impossible, allowing for exploration and deeper understanding of dynamic systems.