Problem 1 - Advanced Engineering Mathematics

Question

Show that \((0,0)\) is an asymptotically stable critical point of the nonlinear autonomous system $$ \begin{aligned} &x^{\prime}=\alpha x-\beta y+y^{2} \\ &y^{\prime}=\beta x+\alpha y-x y \end{aligned} $$ when \(\alpha<0\) and an unstable critical point when \(\alpha>0\). [Hint: Switch to polar coordinates.]

Step-by-Step Solution

Answer

The point \((0,0)\) is asymptotically stable if \(\alpha < 0\) and unstable if \(\alpha > 0\).

Step 1: Identify Critical Points

Start by identifying the critical points of the given system of differential equations. A critical point occurs where both derivatives are zero: \(x' = 0\) and \(y' = 0\). This simplifies the system:\[\begin{aligned} 0 &= \alpha x - \beta y + y^2 \ 0 &= \beta x + \alpha y - xy \end{aligned}\]

Step 2: Convert to Polar Coordinates

To analyze the stability of the point \((0,0)\), switch to polar coordinates with \(x = r\cos\theta\) and \(y = r\sin\theta\), where \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan(y/x)\). Substitute these into the differential equations to get:\[ \begin{aligned} r' &= \left(\alpha r\cos\theta - \beta r\sin\theta + (r\sin\theta)^2 \right)\cos\theta + \left(\beta r\cos\theta + \alpha r\sin\theta - r^2\cos\theta\sin\theta \right)\sin\theta\ \theta' &= \frac{d}{dt}\arctan(y/x) \end{aligned} \] Solve the equations to express \(r'\) and \(\theta'\).

Step 3: Simplifying Expressions

Simplify the expressions obtained from substituting polar coordinates and focus on terms involving \(r\). The primary interest is in how \(r\) changes as this will determine stability:\[ r' = r(\alpha \cos^2\theta + \alpha \sin^2\theta - r\sin(2\theta)/2) \ln(\alpha r) \pm \text{other terms} \] Focus on the dominant terms for small \(r\).

Step 4: Analyze Stability Conditions

For small \(r\), all terms with higher powers of \(r\) become negligible. Focus on the linear term \(\alpha r\). If \(\alpha < 0\), \(r' < 0\) indicating that \(r\) decreases, making the system stable. Conversely, if \(\alpha > 0\), \(r' > 0\) causing \(r\) to increase and implying instability at the critical point.

Step 5: Conclude Stability Analysis

Conclude that the point \((0,0)\) is asymptotically stable when \(\alpha < 0\), since \(r\) decreases to zero. On the other hand, the system is unstable when \(\alpha > 0\) as \(r\) increases. This analysis in polar coordinates confirms the variation in stability based on the sign of \(\alpha\).

Key Concepts

Nonlinear Autonomous Systems

In mathematics, nonlinear autonomous systems are a type of differential equations where the system doesn't explicitly depend on the independent variable, often time. This means that the equations rely entirely on the variables themselves, making them autonomous. Here, we consider a system of equations that describe how variables evolve based on their nonlinear relationships to each other.

Understanding nonlinear autonomous systems helps us analyze complex dynamical behaviors like oscillations or chaotic behaviors. These systems can describe real-world phenomena where the change rates depend on the current state of the system instead of an external input, such as population dynamics or certain mechanical systems.

• An example is the given system: \[ \begin{aligned} x^{\prime} &= \alpha x - \beta y + y^{2} \ y^{\prime} &= \beta x + \alpha y - xy \end{aligned} \]
• Here, \(x'\) and \(y'\) represent how \(x\) and \(y\) change over time.
• The relationship includes products and squares of variables, making it nonlinear.

Analyzing such systems requires advanced techniques, like switching to polar coordinates, to determine the behavior around points of interest, known as critical points.

Polar Coordinates

Polar coordinates provide an alternative way to represent points in a plane, particularly suitable for problems with rotational or radial symmetry. In this system, instead of using Cartesian coordinates \(x\), \(y\), each point is represented by its distance from the origin, \(r\), and the angle relative to a fixed direction, \(\theta\).

For the nonlinear system, converting to polar coordinates is beneficial to analyze the stability of the origin, as damping or amplifying effects might become more evident in this format. The key conversion equations are:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \arctan(y/x)\)

By substituting these into the original differential equations, one can derive expressions for \(r'\) and \(\theta'\) to examine how these quantities change. This conversion transforms the problem and often simplifies the analysis of the system's stability.

Critical Points

Critical points are specific locations in a dynamical system where the derivative (rate of change) of each variable is zero. At these points, the system might rest or change behavior, such as stabilizing or becoming chaotic. Finding critical points provides insight into the potential constant states or equilibrium conditions of the system.

For our system, a critical point occurs where \(x' = 0\) and \(y' = 0\). Solving these equations:

• \(0 = \alpha x - \beta y + y^2\)
• \(0 = \beta x + \alpha y - xy\)

reveals possible conditions the system can satisfy for the time rate of change of each variable to be zero. These critical points are then analyzed to determine their behavior, which results in understanding their stability.

Stability Analysis

Stability analysis involves assessing how small perturbations near critical points behave over time. This determines whether a system will return to equilibrium after a disturbance or move away, possibly escalating to new behavior patterns.

Using polar coordinates in the stability analysis of our system, we focus on the radius \(r\), representing distance from origin. Observing \(r'\) tells us how this distance changes. Simplifying gives:

• For \(\alpha < 0\), the term \(\alpha r\) dominates, indicating \(r' < 0\) and that \(r\) decreases, showing attraction to the origin.
• Conversely, for \(\alpha > 0\), \(r' > 0\), meaning \(r\) increases, showing repulsion from the origin.

Thus, the origin \(0,0\) is asymptotically stable when \(\alpha < 0\) as any small disturbance fades away, restoring equilibrium. When \(\alpha > 0\), it is unstable, as disturbances grow, illustrating the system's susceptibility to changes.