Problem 31 - Calculus for Biology and Medicine

Question

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium point of $$ \begin{array}{l} x_{1}(t+1)=a x_{2}(t) \\ x_{2}(t+1)=x_{1}(t)-\cos \left(x_{2}(t)\right)+1 \end{array} $$ Assume that \(a>0\). For which values of \(a\) is \(\left.\right|_{0} ^{0}\) locally stable?

Step-by-Step Solution

Answer

The equilibrium point \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is stable for \(a < 1\).

Step 1: Define Equilibrium Point

An equilibrium point is a point \((x_1^*, x_2^*)\) where the system remains constant. To find the equilibrium point, set \(x_1(t+1) = x_1(t)\) and \(x_2(t+1) = x_2(t)\) and solve the system of equations.

Step 2: Apply Equilibrium Conditions

Given the system: \[x_1(t+1) = ax_2(t)\] \[x_2(t+1) = x_1(t) - \cos(x_2(t)) + 1\]Setting \(x_1(t+1) = x_1(t)\) and \(x_2(t+1) = x_2(t)\), the equilibrium conditions become:\[x_1 = ax_2\]\[x_2 = x_1 - \cos(x_2) + 1\]

Step 3: Solve for Equilibrium Points

Substitute \(x_1 = ax_2\) into the second equation:\[x_2 = ax_2 - \cos(x_2) + 1\]Solving for \(x_2\), we get:\[0 = (a-1)x_2 - \cos(x_2) + 1\]For \(x_2 = 0\), the equation \(a-1\)0 - \cos(0) + 1 = 0\ is satisfied if \cos(0) = 1\, confirming \(x_2 = 0\) and thus \(x_1 = 0\).

Step 4: Conclude Equilibrium Point

Since substituting \(x_2 = 0\) into \(x_1 = ax_2\) also gives \(x_1 = 0\), the equilibrium point is \(\left[\begin{array}{l}0 \ 0\end{array}\right]\).

Step 5: Analyze Stability Using Jacobian

To determine local stability, calculate the Jacobian matrix \(J\) at \(\left[\begin{array}{l}0 \ 0\end{array}\right]\).Partial derivatives:\[\frac{\partial f_1}{\partial x_1} = 0, \frac{\partial f_1}{\partial x_2} = a\]\[\frac{\partial f_2}{\partial x_1} = 1, \frac{\partial f_2}{\partial x_2} = \sin(x_2)\]At \(x_2 = 0\), \(\sin(0) = 0\).Jacobian becomes: \[J = \begin{bmatrix} 0 & a \ 1 & 0 \end{bmatrix}\]

Step 6: Determine Eigenvalues for Stability

Solve the characteristic equation: \[\text{det}(J - \lambda I) = \begin{vmatrix} -\lambda & a \ 1 & -\lambda \end{vmatrix} = 0\]This simplifies to:\[\lambda^2 - a = 0\].Thus, \(\lambda = \pm\sqrt{a}\).

Step 7: Discuss Stability Conditions

For local stability, both eigenvalues \(\lambda\) must have magnitude less than 1. Thus:\[\sqrt{a} < 1\].Solving gives \(a < 1\).

Key Concepts

Understanding Local Stability

Local stability refers to the behavior of a dynamical system in the vicinity of an equilibrium point. If an equilibrium point is locally stable, small disturbances or perturbations around that point will decay over time, and the system will return to equilibrium.

Think of it like a marble at the bottom of a bowl: if you nudge the marble slightly, it will roll back to the bottom (the equilibrium point). To determine local stability, we often use mathematical tools such as the Jacobian matrix and eigenvalues to study the linearized system around the equilibrium.

• An equilibrium point is called stable if, after a small perturbation, the system returns to it.
• Unstable if the system moves away after a disturbance.
• Semi-stable if some disturbances return it to the equilibrium and others do not.

Understanding local stability is crucial for analyzing how systems respond to minor changes, ensuring they perform as expected under slight conditions change.

The Role of the Jacobian Matrix

The Jacobian matrix is a fundamental concept in stability analysis of dynamical systems. It captures the local behavior near an equilibrium point.

In the context of a system of differential equations, the Jacobian is a matrix of first-order partial derivatives. It describes how each variable influences others around the equilibrium point. Suppose you have a set of equations representing the system:

• \(f_1(x_1, x_2, \ldots, x_n)\)
• \(f_2(x_1, x_2, \ldots, x_n)\)
• \(...\)

The Jacobian matrix, \(J\), consists of elements \(J_{ij} = \frac{\partial f_i}{\partial x_j}\)

The computed Jacobian for our system is:\[J = \begin{bmatrix} 0 & a \ 1 & 0 \end{bmatrix}\]
Analyzing such a matrix helps us determine the directions and strengths of different variables' effects on the system's state.

By evaluating the Jacobian matrix, we gain insights into the underlying dynamics of the system, helping us predict and control future states.

Deciphering Eigenvalues

Eigenvalues are critical in determining the stability of a system. They are derived from the Jacobian matrix and tell us how the system behaves near an equilibrium.

When you calculate the eigenvalues, you essentially look into how small perturbations to the system grow or decay over time. Here's how it works:

• Formulate the characteristic equation from the Jacobian matrix: \(\det(J - \lambda I) = 0\)
• Solve for the values of \(\lambda\) (eigenvalues).

For the given Jacobian:
\[\text{det}\left(\begin{bmatrix} -\lambda & a \ 1 & -\lambda \end{bmatrix}\right) = 0\]
This results in:
\[\lambda^2 - a = 0\]
Resulting eigenvalues are \(\lambda = \pm\sqrt{a}\).

These values provide essential insights:

• If all eigenvalues are less than one in magnitude, the system is stable.
• Eigenvalues greater than one can indicate instability.

Eigenvalues thus provide a numerical measure that aids in understanding the robustness of the system's equilibrium.

Exploring Stability Conditions

Stability conditions offer a way to classify the behavior of a system based on eigenvalues. As we've discovered, eigenvalues help tell if perturbations grow or shrink.

For our system, local stability hinges on whether the absolute values of eigenvalues \(\lambda\) are less than 1:
\[\sqrt{a} < 1\]
This inequality simplifies to:
\[a < 1\]
Subsequently,

• If \(a < 1\), the system returns to equilibrium, showing stability.
• When \(a \ge 1\), perturbations persist or amplify, indicating instability.

Stability conditions tell us the range of parameter values that ensure desired behavior. For systems like these, validating conditions helps engineers and scientists design and control systems efficiently. Stability analysis ultimately prevents systems from exhibiting unwanted behavior, ensuring reliability and safety.