Problem 1 - Biocalculus Calculus for the Life Sciences

Question

Each of the nonlinear systems has an equilibrium at \(\left(\hat{x}_{1}, \hat{x}_{2}\right)=(0,0) .\) Find the linearization near this point. $$\begin{array}{l}{1-6 \text { Each of the nonlinear systems has an equilibrium at }} \\ {\left(\hat{x}_{1}, \hat{x}_{2}\right)=(0,0) \text { . Find the linearization near this point. }}\end{array}$$ $$\begin{array}{l}{\frac{d x_{1}}{d t}=4 x_{1}-2 x_{1} x_{2}} \\ {\frac{d x_{2}}{d t}=-2 x_{2}+8 x_{1} x_{2}}\end{array}$$

Step-by-Step Solution

Answer

Linearization yields a Jacobian matrix with eigenvalues 4 and -2 at (0,0).

Step 1: Identify the System of Equations

The given system is composed of two equations: 1. \( \frac{dx_1}{dt} = 4x_1 - 2x_1x_2 \)2. \( \frac{dx_2}{dt} = -2x_2 + 8x_1x_2 \).

Step 2: Evaluate Partial Derivatives at Equilibrium

To linearize the system, we calculate the Jacobian matrix at \((0,0)\). First, compute the partial derivatives: \( \frac{\partial f_1}{\partial x_1} = 4 - 2x_2 \) and \( \frac{\partial f_1}{\partial x_2} = -2x_1 \).\( \frac{\partial f_2}{\partial x_1} = 8x_2 \) and \( \frac{\partial f_2}{\partial x_2} = -2 + 8x_1 \).

Step 3: Form the Jacobian Matrix

Substitute the equilibrium point \((0,0)\) into the partial derivatives to get the Jacobian matrix:\[ J = \begin{bmatrix} 4 & 0 \ 0 & -2 \end{bmatrix} \]

Step 4: Interpret the Jacobian Matrix

This Jacobian matrix \( J = \begin{bmatrix} 4 & 0 \ 0 & -2 \end{bmatrix} \) shows that at the point \((0,0)\), the system behaves linearly with solutions that can be described using the eigenvalues of the matrix. The eigenvalues are \( \lambda_1 = 4 \) and \( \lambda_2 = -2 \), indicating saddle-type behavior at the equilibrium.

Key Concepts

Equilibrium Point

An equilibrium point in the context of differential equations is where the system does not change over time. For the given system of equations, the equilibrium point is \((\hat{x}_{1}, \hat{x}_{2})=(0,0)\). At this point, all derivatives with respect to time are zero. This means that if the system is initialized at this point, it will remain there indefinitely, assuming no external perturbations.

Finding the equilibrium is typically the first step in analyzing the system's dynamics. It gives us a snapshot to understand the system's inherent stability or instability.
In real-world terms, imagine a ball at the bottom of a bowl (stable equilibrium) or perched at the top of a hill (unstable equilibrium). The system will stay at these points until an external force acts on it.

Jacobian Matrix

The Jacobian matrix is a fundamental tool in systems of differential equations. It consists of all the first-order partial derivatives of a vector-valued function. For our nonlinear system, the Jacobian matrix contains the slopes given by these partial derivatives. It is constructed by evaluating these derivatives at

When computed at the equilibrium point, the Jacobian provides a linear approximation of the system.

• The first row contains \(\frac{\partial f_1}{\partial x_1}\) and \(\frac{\partial f_1}{\partial x_2}\).
• The second row includes \(\frac{\partial f_2}{\partial x_1}\) and \(\frac{\partial f_2}{\partial x_2}\).

For the example given, the Jacobian matrix was computed at the equilibrium point (0,0) and results in:\[ J = \begin{bmatrix} 4 & 0 \ 0 & -2 \end{bmatrix} \].
This matrix helps identify how the system behaves near the equilibrium point.

Eigenvalues

Eigenvalues are often associated with the stability of systems described by differential equations. Once the Jacobian matrix is obtained, eigenvalues are derived to understand the local dynamics near the equilibrium point.

For the given matrix\[ J = \begin{bmatrix} 4 & 0 \ 0 & -2 \end{bmatrix} \], the eigenvalues can be easily seen from the diagonal entries:

• \(\lambda_1 = 4\)
• \(\lambda_2 = -2\)

These eigenvalues indicate the type of stability at the equilibrium.

• Positive eigenvalue (\(\lambda_1 = 4\)) suggests the system moves away from the equilibrium, implying instability.
• Negative eigenvalue (\(\lambda_2 = -2\)) suggests that the system tends toward the equilibrium, implying stability, typically in one axis or dimension.

Therefore, the mixed signs indicate a saddle point, where some motions stabilize, while others destabilize.

Nonlinear Systems

Nonlinear systems are mathematical models where the change in a variable is not directly proportional to the size of the variable. Such systems can exhibit complex behavior including oscillations, bifurcations, and chaos.

In the example, the system\( \frac{dx_1}{dt} = 4x_1 - 2x_1x_2 \) and \( \frac{dx_2}{dt} = -2x_2 + 8x_1x_2 \) are nonlinear due to the product terms \(x_1x_2\). This nonlinearity means solutions are not simply additive, and small changes in initial conditions could lead to vastly different outcomes.

• Nonlinear terms often result in systems that cannot be solved analytically, requiring approximation techniques such as linearization.
• The linearization process involves approximating the nonlinear system by a linear one near the equilibrium points. This is where the Jacobian matrix comes in handy.

Through linearization, the complex nonlinear behaviors can be studied in local, simpler linear forms around equilibrium points, giving insights into potential system behaviors.