Problem 20 - Advanced Engineering Mathematics

Question

Use the Poincaré-Bendixson theorem to show that the plane autonomous system $$ \begin{aligned} &x^{\prime}=\epsilon x+y-x\left(x^{2}+y^{2}\right) \\ &y^{\prime}=-x+\epsilon y-y\left(x^{2}+y^{2}\right) \end{aligned} $$ has at least one periodic solution when \(\epsilon>0\). What occurs when \(\epsilon<0\) ?

Step-by-Step Solution

Answer

For \(\epsilon > 0\), a periodic solution exists. For \(\epsilon < 0\), no periodic solution exists.

Step 1: Identify System Type and Fixed Points

The given system is a planar autonomous system with equations \(x' = \epsilon x + y - x(x^2 + y^2)\) and \(y' = -x + \epsilon y - y(x^2 + y^2)\). First, identify the equilibrium points by setting \(x' = 0\) and \(y' = 0\). This results in the equations \(\epsilon x + y - x(x^2 + y^2) = 0\) and \(-x + \epsilon y - y(x^2 + y^2) = 0\). Testing \(x = 0\) and \(y = 0\), both equations hold true when \(\epsilon = 0\), indicating the point (0,0) is the only equilibrium for non-zero \(\epsilon\).

Step 2: Analyze System for \\(\epsilon > 0\\) Using Poincaré-Bendixson Theorem

For \(\epsilon > 0\), we need to determine the behavior of trajectories. Check for the existence of a closed bounded region where trajectories can be confined. Define the auxiliary function \(V(x, y) = \frac{1}{2}(x^2 + y^2)\). Calculate \(\frac{dV}{dt}\) which represents the time derivative of \(V\) along trajectories: \(\frac{dV}{dt} = x(x') + y(y') = \epsilon (x^2 + y^2) - (x^2 + y^2)^2\). This shows \(\frac{dV}{dt} > 0\) near the origin in the region where \(x^2 + y^2 < \epsilon\). According to the Poincaré-Bendixson theorem, since trajectories cannot escape and no fixed point lies in the bounded region, a periodic orbit must exist.

Step 3: Consider \\(\epsilon < 0\\) and Behavior Analysis

Under \(\epsilon < 0\), solve \(\frac{dV}{dt} = x(x') + y(y') = \epsilon(x^2 + y^2) - (x^2 + y^2)^2\). Here, \(\frac{dV}{dt} < 0\) suggests that any trajectory starting away from the origin spirals inward due to the dominant \(- (x^2 + y^2)^2\) term, suggesting no closed orbit exists. The trajectories will tend to the origin, indicating stability and the absence of periodic solutions.

Key Concepts

Autonomous Systems

Autonomous systems are a type of differential equation that do not explicitly depend on the independent variable, usually time. In simpler terms, the equations describe dynamic systems where the change depends only on the current state, not when they are observed. This feature makes them particularly suitable for modeling phenomena in the real world where time doesn't explicitly play a role, like certain ecological or physical processes.

The exercise provides a planar autonomous system, which means it can be visualized and analyzed in two dimensions. Autonomous systems often appear as a pair of differential equations. Here, we have:

• \(x' = \epsilon x + y - x(x^2 + y^2)\)
• \(y' = -x + \epsilon y - y(x^2 + y^2)\)

When analyzing these systems, especially using the Poincaré-Bendixson theorem, the autonomous nature simplifies the identification of potential periodic solutions and behaviors without the need to parameterize time.

Equilibrium Points

Equilibrium points in differential systems represent states where the system does not change. They are the solutions to the system's equations when all derivatives are set to zero, indicating a state of balance. For the given system:

• \(x' = \epsilon x + y - x(x^2 + y^2) = 0\)
• \(y' = -x + \epsilon y - y(x^2 + y^2) = 0\)

To find these points, you solve them simultaneously. In our particular exercise, the system simplifies under certain conditions, showing that \(x = 0\) and \(y = 0\) is an equilibrium point when \(\epsilon eq 0\).

This means when \(\epsilon > 0\), the nature of the system changes around this point, possibly leading to periodic behavior. However, when \(\epsilon < 0\), the trajectories tend to the origin, indicating stability without any periodic orbits.

Periodic Solutions

Periodic solutions are crucial in understanding the long-term behavior of dynamical systems. A periodic solution repeats itself after some period, creating a closed loop in the phase space—similar to a cycle. In the setup of the given challenge, the Poincaré-Bendixson theorem helps in predicting the existence of these periodic solutions.

Specifically, for \(\epsilon > 0\), the theorem assures that there is at least one periodic orbit within any closed bounded region devoid of equilibrium points. In our exercise, once we identify the closed region using an auxiliary function \(V(x, y) = \frac{1}{2}(x^2 + y^2)\), and a positive derivative \(\frac{dV}{dt}\) near the origin, we infer the presence of periodic orbits. Hence, the system demonstrates cyclic behavior in some regions under positive \(\epsilon\).

Stability Analysis

Stability analysis assesses whether an equilibrium point will exhibit resistance to small changes or perturbations. It's paramount in determining the response of a system to external influences. In the context outlined, we consider whether trajectories converge to or spiral away from equilibrium points.

For \(\epsilon < 0\), the calculated \(\frac{dV}{dt} < 0\), reveals that trajectories are invariably pulled towards the origin, creating an inward spiraling effect. This shows the system stabilizes around the equilibrium, ruling out periodic capture since the negative \(\epsilon\) suppresses oscillations leading to stabilization at the central point.

This implies that any disturbances are eventually negated, locking the system into its equilibrium, avoiding repetitive motions typically seen in periodic solutions.