Question

a. Sketch the nullclines and describe how the critical points move as \(\alpha\) increases. b. Find the critical points. c. Let \(\alpha=2\). Classify each critical point by investigating the corresponding approximate linear system. Draw a phase portrait in a rectangle containing the critical points. d. Find the bifurcation point \(\alpha_{0}\) at which the critical points coincide. Locate this critical point, and find the eigenvalues of the approximate linear system. Draw a phase portrait. e. For \(\alpha>\alpha_{0},\) there are no critical points. Choose such a value of \(\alpha\) and draw a phase portrait. $$x^{\prime}=-4 x+y+x^{2}, \quad y^{\prime}=-\alpha-x+y$$

Short answer

Short Answer: To analyze the given two-dimensional nonlinear system, first, we find the nullclines and the critical points by solving for x and y when \(x^{\prime}=0\) and \(y^{\prime}=0\). The behavior of the critical points can be determined by calculating the eigenvalues of the Jacobian matrix. We can then draw the phase portrait for the system with a chosen value of \(\alpha\), such as 2. To find the bifurcation point \(\alpha_0\) where the critical points coincide, we solve for \(\alpha\) in terms of the critical points and analyze the system's behavior at this value. Finally, we draw the phase portrait for a chosen value of \(\alpha>\alpha_0\) to analyze the system's behavior.

Step-by-step solution

Find the Nullclines

To find the nullclines, set each equation in the system equal to 0 separately and solve for x and/or y. $$ x^{\prime}=-4x+y+x^{2}=0 \quad \Rightarrow \quad y=(4-x)x $$ $$ y^{\prime}=-\alpha-x+y=0 \quad \Rightarrow \quad y=\alpha+x $$

Sketch the Nullclines

Now sketch the nullclines on a graph with x-axis as x and y-axis as y. You should see that the nullclines intersect at the critical points of the system, and the behavior of the critical points changes as \(\alpha\) increases. #b. Critical Points#

Find the Critical Points

To find the critical points, set both \(x^{\prime}\) and \(y^{\prime}\) equal to 0 simultaneously and solve for x and y. Equate the nullclines which will give: $$ (4-x)x = \alpha+x $$ Now, solve this equation for x and substitute back into one of the nullclines to find y. There may be multiple critical points depending on the value of \(\alpha\). #c. Classify Critical Points and Phase Portrait for \(\alpha=2\)#

Find the Approximate Linear System

Let \(\alpha=2\). Before classifying the critical points, find the Jacobian matrix evaluated at the critical points (\(x_0\), \(y_0\)): $$ J(x_0, y_0) = \begin{pmatrix} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} \\ \frac{\partial y'}{\partial x} & \frac{\partial y'}{\partial y} \end{pmatrix} = \begin{pmatrix} 1-2x_0 & 1 \\ -1 & 1 \end{pmatrix} $$

Classify the Critical Points

Now, we can classify each critical point by finding the eigenvalues of the Jacobian matrix. The behavior of the critical point will depend on the eigenvalues.

Draw the Phase Portrait

Draw the phase portrait for the system with \(\alpha=2\), showing the classified critical points and the behavior of the corresponding approximate linear system in a rectangle containing all critical points. #d. Bifurcation Point and Phase Portrait#

Find the Bifurcation Point \(\alpha_0\)

To find the bifurcation point \(\alpha_0\) at which the critical points coincide, we need to find the value of \(\alpha\) for which the system has multiple solutions for the critical points. Solve the equation for x in terms of \(\alpha\) and find the bifurcation point.

Analyze the System at the Bifurcation Point

Now, analyze the behavior of the system at the bifurcation point by finding the eigenvalues of the Jacobian matrix evaluated at the coinciding critical points.

Draw the Phase Portrait

Draw the phase portrait for the system at the bifurcation point, showing the coinciding critical points and the behavior of the corresponding approximate linear system. #e. Phase Portrait for \(\alpha>\alpha_0\)#

Choose a Value of \(\alpha>\alpha_0\)

Now, choose a value of \(\alpha\) that is greater than the bifurcation point \(\alpha_0\). Since there are no critical points for these values of \(\alpha\), we can analyze the behavior of the system by looking at the phase portrait.

Draw the Phase Portrait

Draw the phase portrait for the chosen value of \(\alpha>\alpha_0\), showing the behavior of the nonlinear system.

Key concepts

Nullclines

Nullclines are special curves in the phase space of dynamical systems where either the derivative of the variable x or y equals zero. In other words, they help us find the steady states of the system where the change in one variable temporarily halts.
To find nullclines, we need to solve the given system of equations by setting each derivative to zero separately. For example, with the equations:

• \(x^{\prime}=-4x+y+x^{2}=0\) leads to \(y=(4-x)x\)
• \(y^{\prime}=-\alpha-x+y=0\) results in \(y=\alpha+x\)

The intersection points of these nullclines indicate potential critical points. As \(\alpha\) changes, the appearance and location of nullclines—and hence the behavior of critical points—might shift, giving insights into the system's dynamics.

Critical Points

Critical points, sometimes referred to as equilibrium points, are locations within the phase space where both derivatives of the system are zero. This means the system is in a state of balance, with no net change over time.
To determine these points, we set both equations for the nullclines equal to each other:

• \( (4-x)x = \alpha +x \)

Solving this equation yields the critical points for various values of \(\alpha\). These points play a crucial role in understanding how the dynamical system behaves at specific parameters. Different \(\alpha\) values may result in multiple, one, or no critical points.
Uncovering these critical points allows us to deeply analyze how the system transitions between different states of equilibrium as parameters like \(\alpha\) vary.

Phase Portrait

The phase portrait is a visual representation of a dynamical system's trajectories in the phase space. It provides a map of all possible behaviors and states the system can exhibit over time.
When \(\alpha=2\), critical points can first be classified through linear approximation by examining the Jacobian matrix. This involves:

• Finding eigenvalues of the matrix \(J(x_0, y_0) = \begin{pmatrix} 1-2x_0 & 1 \ -1 & 1 \end{pmatrix}\)

By evaluating these eigenvalues, we get insights into the stability of the critical points—whether a point is stable, a saddle, or otherwise. Drawing a phase portrait involves plotting trajectories that align with these calculations and classification.
This step-by-step analysis allows us to visualize how the system evolves with time, giving clarity on how variables like \(\alpha\) affect system dynamics.

Bifurcation Analysis

Bifurcation analysis helps us understand how small changes in system parameters can lead to significant changes in the behavior of the system. It identifies crucial points, known as bifurcation points, where the qualitative nature of the system's equilibrium changes.
Finding the bifurcation point \(\alpha_0\) involves recognizing when critical points coincide. We solve the critical point condition \((4-x)x = \alpha +x\) to find specific values of \(\alpha\). At these points, the behavior of the system dramatically shifts, often leading to the appearance or disappearance of critical points.
Once the bifurcation point is identified, the system can be further analyzed by finding the eigenvalues at these critical points. This determines the stability of coinciding critical points and can be represented in a phase portrait.
Exploring beyond \(\alpha_0\), for \(\alpha>\alpha_0\), reveals configurations where no critical points exist, helping to visualize the system's behavior in new parameter realms. Understanding bifurcation enables predicting and explaining shifts in dynamical system structures.