Problem 5 - Advanced Engineering Mathematics

Question

In Problems, write the given nonlinear second-order differential equation as a plane autonomous system. Find all critical points of the resulting system. $$ x^{\prime \prime}+x=\epsilon x^{3} \text { for } \epsilon>0 $$

Step-by-Step Solution

Answer

The critical points are \((0, 0), \left(\pm \frac{1}{\sqrt{\epsilon}}, 0\right)\).

Step 1: Define New Variables

First, we need to convert the second-order differential equation into a first-order system. Define the variables: Let \( y_1 = x \) and \( y_2 = x' \).

Step 2: Rewrite Differential Equation

Using the new variables, rewrite the original equation. The second derivative \( x'' \) becomes the derivative of \( y_2 \), so we have \( y_2' = x'' = \epsilon x^3 - x \). In terms of \( y_1 \) and \( y_2 \), this becomes \( y_2' = \,\epsilon y_1^3 - y_1 \).

Step 3: Formulate Autonomous System

Create a system using the equations from earlier steps. The system is:\[ \begin{align*} y_1' &= y_2, \ y_2' &= \epsilon y_1^3 - y_1. \end{align*} \] This represents our system of first-order differential equations.

Step 4: Find Critical Points

To find the critical points, set \( y_1' = 0 \) and \( y_2' = 0 \): - \( y_1' = y_2 = 0 \) - \( y_2' = \epsilon y_1^3 - y_1 = 0 \). Solving \( \epsilon y_1^3 - y_1 = 0 \) gives \( y_1(\epsilon y_1^2 - 1) = 0 \). This results in the critical points: \( y_1 = 0 \), \( y_2 = 0 \); and \( y_1 = \pm \frac{1}{\sqrt{\epsilon}} \), \( y_2 = 0 \).

Key Concepts

Plane Autonomous System

A plane autonomous system is a type of system of differential equations that does not explicitly depend on the independent variable, often time, making it 'autonomous'. In the context of our exercise, we transformed the given nonlinear second-order differential equation into such a system.

To do this, we redefined the original second-order equation into a set of first-order equations in terms of new variables. This is done by introducing new variables to represent the original function and its derivative, converting the problem into the plane of these variables.

For example:

• Let \( y_1 = x \): this variable represents the original function.
• Let \( y_2 = x' \): this stands for the first derivative, a new direction in the system.

These redefined equations \( y_1' = y_2 \) and \( y_2' = \epsilon y_1^3 - y_1 \) form the plane autonomous system, describing how each variable changes independently in time without explicit time dependence. This approach simplifies the analysis of nonlinear dynamics.

Critical Points

Critical points in a differential equation system are where the system doesn't change—it's in a state of equilibrium. These points indicate the steady states that the system tends to stabilize at, or away from which disturbances can cause new behaviors. To find these equilibrium points, we usually set all derivatives in the system to zero and solve for the variables.

In our transformed autonomous system:

• We set \( y_1' = 0 \) which implies \( y_2 = 0 \).
• Simultaneously, \( y_2' = 0 \) becomes \( \epsilon y_1^3 - y_1 = 0 \). This simplifies to solving \( y_1(\epsilon y_1^2 - 1) = 0 \).

This gives the critical points of the system: \( y_1 = 0, y_2 = 0 \) indicating the origin, and \( y_1 = \pm \frac{1}{\sqrt{\epsilon}}, y_2 = 0 \), indicating additional points on the \( y_1 \)-axis. These points help us understand the system dynamics—whether it is stable, and how it behaves near these points.

Second-Order Differential Equations

Second-order differential equations involve the second derivative of a function and are common in modeling physical systems. Our original equation, \( x^{\prime \prime} + x = \epsilon x^{3} \), represents a nonlinear second-order differential equation due to the presence of the \( x^3 \) term.

The characteristic of these equations:

• They often describe systems with parallels in the real world, such as motion or wave phenomena.
• Nonlinearity, like the \( x^3 \) term, implies that solutions to these equations can exhibit complex behaviors like chaos or multiple equilibrium positions.

Converting a second-order differential equation into a first-order system, as we did, allows for easier graphical and analytical study. This makes understanding the evolution of systems in terms of its variables more tangible. Whether for mechanical systems, electromagnetic fields, or any context relying on change rates, second-order equations are essential.