Problem 11 - Classical Mechanics

Question

Show that the origin is the only critical point for the system $$ \begin{aligned} &\dot{x}=-y+\alpha x\left(\beta-x^{2}-y^{2}\right) \\ &\dot{y}=x+\alpha y\left(\beta-x^{2}-y^{2}\right) \end{aligned} $$ where \(\alpha, \beta\) are real parameters, with \(\alpha\) fixed and positive and \(\beta\) allowed to take different values. Show that the character of the critical point and the existence of a limit cycle depend on the parameter \(\beta\), so that the system undergoes a supercritical Hopf bifurcation at \(\beta=0\). (Hint: Make the change from Cartesian co-ordinates to plane polars.)

Step-by-Step Solution

Answer

Question: Describe the process of determining the existence of a supercritical Hopf bifurcation in the given system of differential equations and explain how the parameter β affects the presence and stability of a limit cycle. Answer: To determine the existence of a supercritical Hopf bifurcation, we first find the critical points by setting the given equations for ẋ and ẏ to zero. The origin (0, 0) is the only critical point we find in this case. We then change from Cartesian coordinates to polar coordinates (x, y) to (r, θ), and express ẋ and ẏ in terms of r and θ using the chain rule. We solve for ṙ and θ̇ by solving the system of two equations we obtain. Analyzing the character of the critical point using the equations for ṙ and θ̇, we see that β determines the presence and stability of a limit cycle. If β > 0, we have a stable limit cycle; if β < 0, we have an unstable limit cycle. When β = 0, there is no limit cycle and the system undergoes a supercritical Hopf bifurcation.

Step 1: Finding the critical points

To find the critical points, we need to set the equations for \(\dot{x}\) and \(\dot{y}\) to zero: $$ \begin{aligned} & 0 = -y + \alpha x (\beta - x^2 - y^2) \\ & 0 = x + \alpha y (\beta - x^2 - y^2) \end{aligned} $$ Now we'll solve this system of equations for \(x\) and \(y\). Since both equations are equal to zero, let's add the two equations together: $$ 0 = x - y + \alpha(x+y)(\beta-x^2-y^2) $$ We can notice that \((x-y)\) and \((x+y)\) are factors. Factorizing and simplifying the equation, we get: $$ 0=(x-y)(1+\alpha \beta - \alpha x^{2}- \alpha y^{2}) $$ Solving for \(x\) and \(y\), we find that \(x=y\) or \(1+\alpha \beta - \alpha x^{2}- \alpha y^{2}=0\). Since \(x=y\), we plug this back into either one of our original equations, and we get \(0 = x - x\). This shows that the origin (0, 0) is the only critical point.

Step 2: Changing to polar coordinates

To analyze the character of the critical point and the existence of a limit cycle, we'll change from Cartesian coordinates to polar coordinates, with \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, we'll use the chain rule to express \(\dot{r}\) and \(\dot{\theta}\): $$ \begin{aligned} \dot{r} \cos \theta - r \sin \theta \dot{\theta} &= - r \sin \theta + \alpha r \cos \theta (\beta - r^2) \\ \dot{r} \sin \theta + r \cos \theta \dot{\theta} &= r \cos \theta + \alpha r \sin \theta (\beta - r^2) \end{aligned} $$

Step 3: Solve for \(\dot{r}\) and \(\dot{\theta}\)

Now we have a system of two equations: $$ \begin{aligned} & \dot{r} \cos \theta - r \sin \theta \dot{\theta} + r \sin \theta - \alpha r \cos \theta (\beta - r^2) = 0 \\ & \dot{r} \sin \theta + r \cos \theta \dot{\theta} - r \cos \theta - \alpha r \sin \theta (\beta - r^2) = 0 \end{aligned} $$ Solving this system for \(\dot{r}\) and \(\dot{\theta}\) gives us: $$ \begin{aligned} & \dot{r} = \alpha r (\beta - r^2) \\ & \dot{\theta} = 1 \end{aligned} $$

Step 4: Analyzing the character of the critical point and the existence of a limit cycle

The equations we found in the previous step are: $$ \begin{aligned} & \dot{r} = \alpha r (\beta - r^2) \\ & \dot{\theta} = 1 \end{aligned} $$ These equations show that the position of the limit cycle depends on the parameter \(\beta\). When \(\beta > 0\), the value of \(\dot{r}\) is positive, meaning that the system has a stable limit cycle. When \(\beta < 0\), the value of \(\dot{r}\) is negative, meaning that the system has an unstable limit cycle. When \(\beta = 0\), there is no limit cycle, and the system undergoes a supercritical Hopf bifurcation.

Key Concepts

Polar Coordinates

Polar coordinates are a useful tool for solving problems involving circular or spiral patterns, particularly in planar systems. By transforming the standard Cartesian coordinates \(x, y\) into polar coordinates \(r, \theta\), where \(r\) is the radius and \(\theta\) is the angle from a reference direction, we can simplify our analysis of rotational dynamics. In the system given by the problem, by changing variables into polar coordinates using the relations \(x = r \cos \theta\) and \(y = r \sin \theta\), the problem becomes easier to solve.
By applying the chain rule, we find that \(\dot{r}\) and \(\dot{\theta}\) capture the system's behavior in terms of radial and angular movements. This change allows us to look at the characteristics of the system, such as the stability of solutions like the limit cycle, without the complication of x and y interactions. Polar coordinates thus help in simplifying and breaking down complex dynamical systems into more manageable subcomponents.

Limit Cycle

A limit cycle is a closed trajectory in the phase space of a dynamical system where solutions can converge, or, in some cases, diverge depending on their stability. In realistic systems, limit cycles can describe oscillating phenomena such as biological rhythms, mechanical vibrations, or electrical circuits.
In the exercise, the system's limit cycle behavior depends on the parameter \(\beta\). When \(\beta > 0\), the system exhibits a stable limit cycle, meaning that the trajectories of the system eventually settle into oscillations around this closed path. However, when \(\beta < 0\), these cycles are unstable, and the system's paths either move away from or fail to consistently encircle the cycle. At the critical point where \(\beta = 0\), the limit cycle disappears, resulting in a Hopf bifurcation—a critical situation where the system abruptly changes behavior, transitioning from simple stability to stable oscillatory motion as \(\beta\) passes through zero.

Critical Points

Critical points, also known as equilibrium points, are where the system doesn’t change, meaning \(\dot{x} = 0\) and \(\dot{y} = 0\). In this exercise, finding critical points involves setting the system's velocity equations to zero and solving for \(x\) and \(y\).
The origin, or \(x = 0, y = 0\), is determined to be the only critical point in the system. Understanding the nature of these points, especially within the context of Hopf bifurcations, is essential. As \(\beta\) varies, this critical point's stability changes, influencing whether nearby trajectories converge or diverge from it. For values of \(\beta > 0\), the system may stabilize around the origin before expanding into a limit cycle. At \(\beta = 0\), the system transitions through a supercritical Hopf bifurcation, changing from no cycles to exhibiting limit cycles, hence marking a critical transition in the system's dynamics.