Problem 1 - Advanced Engineering Mathematics

Question

In Problems, show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ \begin{aligned} &x^{\prime}=2+x y \\ &y^{\prime}=x-y \end{aligned} $$

Step-by-Step Solution

Answer

The system has no periodic solutions due to the absence of critical points and the Bendixson-Dulac criterion.

Step 1: Identify the System of Differential Equations

We are given the system \(\begin{aligned} x' &= 2 + xy \ y' &= x - y \end{aligned}\). This is a plane autonomous system expressed in terms of \(x'\) and \(y'\).

Step 2: Determine the Jacobian Matrix

To analyze the critical points, we first find the Jacobian matrix of the system. The Jacobian matrix \(J\) for the system is calculated from the partial derivatives of \(x'\) and \(y'\): \[ J = \begin{bmatrix} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} \ \frac{\partial y'}{\partial x} & \frac{\partial y'}{\partial y} \end{bmatrix} = \begin{bmatrix} y & x \ 1 & -1 \end{bmatrix} \]

Step 3: Identify Critical Points

Set \(x' = 0\) and \(y' = 0\) to find critical points: 1. \(2 + xy = 0\) implies \(xy = -2\).2. \(x - y = 0\) implies \(x = y\).Combining these, the equations imply that there are no real solutions for \(x\) and \(y\) because \(x^2 eq -2\).

Step 4: Examine Absence of Critical Points

Since there are no real critical points, the system cannot have periodic solutions passing through any critical points because no critical points exist to form closed trajectories.

Step 5: Use Bendixson-Dulac Criterion

According to the Bendixson-Dulac criterion, if the divergence of the vector field is never zero in a simply connected region, the system has no periodic solutions. Calculating the divergence: \[ \frac{\partial}{\partial x}(2 + xy) + \frac{\partial}{\partial y}(x-y) = y + (-1) = y - 1 \]This means we can choose test regions where \(y - 1 eq 0\) such as any region where \(y eq 1\). Hence, there are no periodic solutions based on this criterion.

Key Concepts

Jacobian matrix

The Jacobian matrix is a crucial tool in studying dynamical systems. It describes the linear approximation of a function near a given point. In the context of a plane autonomous system, we derive the Jacobian matrix from the partial derivatives of the differential equations involved. By focusing on the system \( x' = 2 + xy \) and \( y' = x - y \), the Jacobian matrix \( J \) is composed of: \[J = \begin{bmatrix} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} \frac{\partial y'}{\partial x} & \frac{\partial y'}{\partial y} \end{bmatrix} = \begin{bmatrix} y & x \ 1 & -1 \end{bmatrix}\]

• The element in the first row and first column is the partial derivative of \( x' \) with respect to \( x \).
• The element in the first row and second column is the partial derivative of \( x' \) with respect to \( y \).
• The element in the second row and first column is the partial derivative of \( y' \) with respect to \( x \).
• The element in the second row and second column is the partial derivative of \( y' \) with respect to \( y \).

The Jacobian plays a vital role in understanding the nature of critical points and system behavior near these points.

Critical points

Critical points are points where the system does not change, which means the derivatives are zero. To find them, you solve for \( x' = 0 \) and \( y' = 0 \). In our given system:

• First equation: \( 2 + xy = 0 \) implies that \( xy = -2 \).
• Second equation: \( x - y = 0 \) implies that \( x = y \).

To find a solution, substitute \( x = y \) into \( xy = -2 \), leading to \( x^2 = -2 \), which has no real solutions. Hence, there are no real critical points for this system.Finding critical points is essential because they often correspond to equilibrium states of a system. However, in this case, the absence of real critical points indicates that the system does not stabilize in any real point on the plane.

Bendixson-Dulac criterion

The Bendixson-Dulac criterion is a mathematical method to determine whether an autonomous system can have periodic solutions. The criterion states that if the divergence of a vector field can be observed, and it is never zero in a simply connected region, then the system cannot have any periodic solutions.For our system, calculate the divergence as follows:\[ \frac{\partial}{\partial x}(2 + xy) + \frac{\partial}{\partial y}(x-y) = y - 1 \]

• If \( y - 1 e 0 \) in any region, no periodic solutions exist.
• This implies that for simply connected regions where \( y e 1 \), periodic solutions are not possible.

This criterion helps in ruling out closed trajectories, which are indicative of periodic solutions, ensuring the system's behavior is predictable and non-cyclic in the regions considered.

Periodic solutions

Periodic solutions are solutions of a differential system that repeat after a fixed interval, forming closed loops in the phase space. These solutions indicate that the system returns to its initial state periodically.In examining the given system, the focus is to prove it has no periodic solutions:

• The Jacobian matrix did not reveal any real critical points, eliminating potential centers for periodic orbits.
• The Bendixson-Dulac criterion further confirmed no periodic solutions by showing that the divergence \( y - 1 \) is non-zero in simply connected areas that do not include \( y = 1 \).

In practical terms, this means the system does not loop back on itself, ensuring all trajectories will eventually move away from each point rather than cycling back, providing insight into the nature and predictability of the autonomous system.