Question

The system of differential equations: $$ \begin{aligned} &x^{\prime}=\mu x-y-x^{3}, \\ &y^{\prime}=x, \end{aligned} $$ is called the van der Pol system. It arises in the study of non-linear semiconductor circuits, where \(y\) represents a voltage and \(x\) the current. It is in the Gallery menu. a) Find the equilibrium points for the system. Use pplane6 only to check your computations. b) For various values of \(\mu\) in the range \(0<\mu<5\), find the equilibrium points, and find the type of each, i.e, is it a nodal sink, a saddle point, ...? You should find that there are at least two cases depending on the value of \(\mu\). Don't worry too much about non-generic cases. Use pplane6 only to check your computations. c) Use pplane6 to illustrate the behavior of solutions to the system in each of the cases found in b). Plot enough solutions to illustrate the phenomena you discover. Be sure to start some orbits very close to \((0,0)\), and some near the edge of the display window. Put arrows on the solution curves (by hand after you have printed them out) to indicate the direction of the motion. (The display window \((-5,5,-5,5)\) will allow you to see the interesting phenomena.) d) For \(\mu=1\) plot the solutions to the system with initial conditions \(x(0)=0\), and \(y(0)=0.2\). Plot both components of the solution versus \(t\). Describe what happens to the solution curves as \(t \rightarrow \infty\).

Short answer

Equilibrium point is at (0,0). Type depends on \(\mu\): saddle for \(\mu^2 > 4\), spiral for \(\mu^2 < 4\).

Step-by-step solution

Set Differential Equations to Zero

To find the equilibrium points of the van der Pol system, set both differential equations to zero: \[ x' = \mu x - y - x^3 = 0 \] \[ y' = x = 0 \] These conditions indicate that at equilibrium, \( x = 0 \).

Solve for y

Substitute \( x = 0 \) into the first equation: \[ \mu(0) - y - (0)^3 = 0 \] Thus, \( -y = 0 \), leading to \( y = 0 \).

Identify Equilibrium Point

The result from Steps 1 and 2 gives us the equilibrium point: \((x, y) = (0, 0)\).

Linearize System Around Equilibrium

We linearize the system by evaluating the Jacobian matrix at the equilibrium point. The system is: \[ x' = \mu x - y - x^3, \quad y' = x \] The Jacobian matrix is: \[ J = \begin{bmatrix} \frac{\partial (\mu x - y - x^3)}{\partial x} & \frac{\partial (\mu x - y - x^3)}{\partial y} \ \frac{\partial x}{\partial x} & \frac{\partial x}{\partial y} \end{bmatrix} \] Evaluating the partial derivatives, we get:\[ J = \begin{bmatrix} \mu - 3x^2 & -1 \ 1 & 0 \end{bmatrix} \] At \((0,0)\), the Jacobian matrix is:\[ J = \begin{bmatrix} \mu & -1 \ 1 & 0 \end{bmatrix} \]

Determine Eigenvalues of Jacobian

Calculate the eigenvalues of the Jacobian for different values of \( \mu \). The characteristic equation of the Jacobian is:\[ \det(\begin{bmatrix} \mu - \lambda & -1 \ 1 & -\lambda \end{bmatrix}) = 0 \] Resulting in the quadratic equation:\[ \lambda^2 - \mu\lambda + 1 = 0 \] Finding roots using the quadratic formula gives:\[ \lambda = \frac{\mu \pm \sqrt{\mu^2 - 4}}{2} \]

Determine Stability and Type of Equilibrium

Analyze the eigenvalues for different \( \mu \):- If \( \mu^2 > 4 \), both eigenvalues are real, indicating a saddle point.- If \( \mu^2 < 4 \), eigenvalues are complex with non-zero real parts, indicating a spiral (either sink or source depending on the sign of \( \mu \)).

Illustrate Solution Behavior with Computational Tools

Use computational tools like pplane6 to simulate the system for various \( \mu \) values:\- Illustrate persistent cycles or paths for tested \((x(0), y(0))\) values.- Ensure the illustration clearly shows the directional behavior and types of orbits around equilibrium.- Use window \((-5,5,-5,5)\) to see the orbit patterns.

Plot and Analyze for Specific \(\mu=1\)

Using initial conditions \(x(0) = 0, y(0) = 0.2\): \- Plot solutions against time \(t\)\.- Observe the behaviors as \(t \to \infty\), typically implying convergence to a limit cycle or steady state.

Key concepts

van der Pol system

The van der Pol system is a type of ordinary differential equation that is particularly famous for its role in modeling certain electrical circuits, biological systems, and mechanical oscillations. It is a non-linear system defined by the equations \( x' = \mu x - y - x^3 \) and \( y' = x \). Here, \( x \) often represents current and \( y \) represents voltage in circuits. The system is characterized by the presence of the non-linear term \(-x^3\), which introduces complexity to the behavior of the system, often leading to phenomena like limit cycles and bifurcation depending on the value of the parameter \( \mu \). Understanding the van der Pol system is crucial as it provides insights into how subtle changes in system parameters can drastically change system behavior. Always check that the system equations are correctly interpreted and analyzed using both analytical and computational tools.

equilibrium points

Equilibrium points in a differential equation system are crucial as they represent the states where the system experiences no change—meaning \( x' = 0 \) and \( y' = 0 \). For the van der Pol system defined by \( x' = \mu x - y - x^3 \) and \( y' = x \), setting these to zero leads to the system of equations: \( \mu x - y - x^3 = 0 \) and \( x = 0 \). By substituting \( x = 0 \) into the first equation, it becomes \( -y = 0 \), which gives the equilibrium point \( (0, 0) \). This is the only equilibrium point for the system. Discovering equilibrium points helps in understanding the behavior of solutions around them, which is essential when determining the stability and type of the point.

Jacobian matrix

To assess the local behavior of a system near an equilibrium point, we use the Jacobian matrix. It provides a linear approximation of the system near these points. For our van der Pol system, the Jacobian matrix \( J \) is obtained by calculating the partial derivatives of the system's functions. This results in the matrix:\[J = \begin{bmatrix} \mu - 3x^2 & -1 \ 1 & 0 \end{bmatrix}\]Evaluating \( J \) at the equilibrium point \( (0, 0) \), we substitute \( x = 0 \), yielding:\[J = \begin{bmatrix} \mu & -1 \ 1 & 0 \end{bmatrix}\]The Jacobian matrix helps identify how slight perturbations around the equilibrium point affect the system’s dynamics. Calculating it properly is key in advancing to the analysis of eigenvalues which determines stability.

eigenvalues

Eigenvalues are paramount for understanding the stability of equilibrium points. They are derived from the characteristic equation of the Jacobian matrix associated with a system. For the van der Pol system’s Jacobian:\[\det(\begin{bmatrix} \mu - \lambda & -1 \ 1 & -\lambda \end{bmatrix}) = 0\]This results in the quadratic equation:\[\lambda^2 - \mu\lambda + 1 = 0\]Solving using the quadratic formula, we find:\[\lambda = \frac{\mu \pm \sqrt{\mu^2 - 4}}{2}\]By analyzing these results:

• If \( \mu^2 > 4 \), the system results in real eigenvalues, indicating a saddle point.
• If \( \mu^2 < 4 \), the eigenvalues are complex, suggesting a spiral behavior, pointing towards either a sink or source.

The sign and magnitude of the real parts of the eigenvalues determine the nature (stable/unstable) of the equilibrium.

pplane6 tool

The pplane6 tool is a powerful computational application for analyzing systems of ordinary differential equations, such as the van der Pol system. It was designed to allow interactive exploration of these systems by providing a graphical interface where solutions can be plotted. This tool helps students and researchers visualize phase portraits:

• Illustrate the trajectories of solutions given different initial conditions.
• Observe behavior as parameters like \( \mu \) are varied.
• See how changes affect stability and the nature of equilibrium points.

Using pplane6, you can confirm theoretical findings about equilibrium points and eigenvalues by plotting the orbits around these points, highlighting stable and unstable behaviors visually. This visual analysis is invaluable in grasping the concepts intuitively.