Problem 5 - Advanced Engineering Mathematics

Question

In Problems, show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ \begin{aligned} &x^{\prime}=-\mu x-y \\ &y^{\prime}=x+y^{3} \\ &\text { for } \mu<0 \end{aligned} $$

Step-by-Step Solution

Answer

The system has no periodic solutions because the divergence is always positive or zero, never allowing negative values.

Step 1: Analyze the System of Equations

We have the system of differential equations: \( x' = -\mu x - y \) and \( y' = x + y^3 \). We need to determine if this system has any periodic solutions. A periodic solution is one where trajectories in the phase plane repeat after some period \( T > 0 \).

Step 2: Calculate the Divergence

The divergence of a vector field \( F(x, y) = (P, Q) \) in a plane is given by \( abla \cdot F = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \). Here, \( P = -\mu x - y \) and \( Q = x + y^3 \). So, the divergence is: \[ abla \cdot F = \frac{\partial (-\mu x - y)}{\partial x} + \frac{\partial (x + y^3)}{\partial y} = -\mu + 3y^2. \]

Step 3: Examine Conditions for Periodic Solutions

For a plane autonomous system to have periodic solutions, a necessary condition is that the divergence should change sign or be zero throughout the entire plane, implying zero net divergence. However, the divergence here is \( -\mu + 3y^2 \). Since \( \mu < 0 \), this means the term \( -\mu \) is positive, making \( -\mu + 3y^2 \) always positive or zero (never negative).

Step 4: Conclude No Periodic Solutions

Since the divergence is strictly non-negative (or never negative) throughout the plane and can never be negative, the system does not allow trajectories that loop back on themselves to form repeated cycles. Thus, this plane autonomous system has no periodic solutions.

Key Concepts

Periodic Solutions

In differential equations, a periodic solution refers to a solution that repeats over a fixed interval. Practically, this means the system's trajectory in the phase plane retraces its path after some interval, known as the period. Understanding periodic solutions is crucial as they indicate oscillating behaviors found in many natural phenomena, such as waves or cycles in population dynamics.
In autonomous systems, periodic solutions are indicative of nontrivial cyclic behavior, which can be critically important for understanding long-term behavior in biological, physical, and engineering systems. These solutions would form closed loops in the phase plane. However, for their existence, certain mathematical conditions must be met. For the given system, the divergence criterion is not met, indicating that periodic solutions cannot exist. This analysis helps in comprehending the system's limitations regarding oscillatory motion.

Divergence Theorem

The divergence theorem is a fundamental result in vector calculus connecting the flow (or divergence) of a vector field through a surface to a flux over the boundary. When applied to autonomous differential equations, divergence plays a key role in analyzing the possible behavior of solutions, such as the existence of periodic solutions.

• The divergence of a vector field \( F(x, y) = (P, Q) \) is calculated as \( abla \cdot F = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \).
• In our analysis, the divergence result, \( -\mu + 3y^2 \), does not change sign but stays non-negative.

This uniform non-negative divergence indicates that solutions tend not to close upon themselves, establishing the absence of periodic trajectories. A zero or sign-changing divergence often hints at potential region oscillations—missing here.

Phase Plane Analysis

Phase plane analysis is a powerful visualization tool for studying dynamical systems described by differential equations. It helps understand how solutions evolve over time by plotting trajectories in a coordinate space defined by state variables like \( x \) and \( y \).

• In our system, each point represents the system's current state, and the entire path corresponds to its evolution.
• A closed path indicates periodic behavior, but for our exercise, the divergence does not allow for such paths.

By critically examining this visual representation and utilizing associated mathematical tools (like divergence), we can draw qualitative conclusions, such as the non-periodicity of solutions in this context. This analytical approach is crucial for engineers and scientists aiming to predict system behavior over extended periods.