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Chapter 8: Problem 6

For each of the following systems, a Hopf bifurcation occurs at the origin when \(\mu=0\). Using a computer, plot the phase portrait and determine whether the bifurcation is subcritical or supercritical. $$ \dot{x}=\mu x+y-x^{3}, \quad \dot{y}=-x+\mu y+2 y^{3} $$

Short Answer

Expert verified
In summary, by analyzing the given system of ODEs, computing the equilibrium points, and plotting the phase portrait for different values of \(\mu\), we can conclude that the Hopf bifurcation at the origin is supercritical. This is because the stability changes from a stable focus (spiraling inwards) for \(\mu<0\) to an unstable focus (spiraling outwards) for \(\mu>0\).

Step by step solution

01

Rewrite the system in a suitable form for plotting

First, let's rewrite the given system of ODEs in a suitable form for plotting. The system can be represented as: \[\begin{cases} \dot{x} = \mu x + y - x^3 \\ \dot{y} = -x + \mu y + 2y^3 \end{cases}\]
02

Compute the equilibrium points

To compute the equilibrium points, we set \(\dot{x} = 0\) and \(\dot{y} = 0\). Hence, we have the following equations for the equilibrium points: \[\begin{cases} 0 = \mu x + y - x^3 \\ 0 = -x + \mu y + 2y^3 \end{cases}\] The given system undergoes a Hopf bifurcation at the origin, so one equilibrium point is the origin: \((x, y) = (0,0)\).
03

Analyze the stability of the equilibrium points

We analyze the stability of the equilibrium points by calculating the Jacobian matrix of the system at the equilibrium point \((0,0)\): \[J(x,y) = \begin{bmatrix} \frac{\partial \dot{x}}{\partial x} & \frac{\partial \dot{x}}{\partial y} \\ \frac{\partial \dot{y}}{\partial x} & \frac{\partial \dot{y}}{\partial y} \end{bmatrix} = \begin{bmatrix} \mu - 3x^2 & 1 \\ -1 & \mu + 6y^2 \end{bmatrix}\] At the equilibrium point \((x, y) = (0,0)\), the Jacobian matrix is: \[J(0,0) = \begin{bmatrix} \mu & 1 \\ -1 & \mu \end{bmatrix}\] The eigenvalues of this matrix are given by the quadratic equation \(\lambda^2 - (\mu + \mu)\lambda + (\mu^2 - 1) = 0\).
04

Plot the phase portrait

To plot the phase portrait, we will examine the behavior of the trajectories with the variations of the parameter \(\mu\). By using software like MATLAB or Python, plot the phase portrait for different values of \(\mu\). It is easy to check whether the Hopf bifurcation is subcritical or supercritical. Let's compare the portraits for two situations: 1. For \(\mu<0\), the system possesses a stable focus at the origin. 2. For \(\mu>0\), the system possesses an unstable focus at the origin.
05

Determine the type of bifurcation

Based on the phase portraits for different values of \(\mu\), we observe the following: 1. For \(\mu<0\), the trajectories are spiraling inwards, indicating a stable focus. 2. For \(\mu>0\), the trajectories are spiraling outwards, indicating an unstable focus. Since the stability changes from stable to unstable as \(\mu\) increases through zero, we conclude that the Hopf bifurcation at the origin is supercritical.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Phase Portrait
A phase portrait is a graphical illustration of the trajectories of a system of differential equations in a phase plane. Each point on this plane represents a state of the system, with the x-axis and y-axis typically denoting the system鈥檚 variables.

The portrait shows how the system evolves over time, marking the path that the system鈥檚 state follows as time progresses. Specifically, in the context of our exercise involving a Hopf bifurcation, the phase portrait helps visualize how the system's behavior changes around the critical value of \( \mu = 0 \).

For various \( \mu \) values, we would see different flows around the equilibrium point, and these would be plotted to determine the nature of the bifurcation, whether it is subcritical or supercritical, by observing the orientation of the trajectories around the origin.
Equilibrium Points
Equilibrium points are values at which a dynamical system does not change, meaning that the rates of change for all system variables are zero. These points can be thought of as the 'resting states' of the system.

In our scenario, we find the equilibrium points by setting the given system equations' right-hand sides to zero and solving for the variables. At these points, the system鈥檚 behavior is crucial for understanding its long-term dynamics. For the Hopf bifurcation example, the point (0,0) serves as an equilibrium and the heart of further stability analysis.
Stability Analysis
Stability analysis involves determining whether small deviations from the equilibrium points will decay over time, leading the system back to equilibrium, or whether they will grow, moving the system away. In the context of the Hopf bifurcation, stability analysis is critical as it helps to identify the transition at \( \mu = 0 \) from stability to instability.

The sign and magnitude of \( \mu \) influence the equilibrium stability - negative values suggest a tendency to return to equilibrium (stable), whereas positive values imply a divergence from equilibrium (unstable). By calculating the Jacobian matrix and its eigenvalues at these points, we can rigorously determine the system's stability.
Jacobian Matrix
The Jacobian matrix is a representation of a system of differential equations' first-order partial derivatives. It captures the system's linearized behavior near the equilibrium points.

In the given exercise, we construct the Jacobian by differentiating each system's equation with respect to each variable. This matrix provides crucial insights into the system's nature, such as local stability and the behavior of trajectories in close proximity to equilibrium points, by providing the coefficients for the linear approximation of the nonlinear system.
Eigenvalues
Eigenvalues are scalar values in a linear system that represent the factor by which a corresponding eigenvector (direction in the system's state space) is stretched under the linear transformation represented by a matrix - in our exercise, the Jacobian matrix.

The eigenvalues give key information about a system's equilibrium stability. In the Hopf bifurcation scenario, we find the eigenvalues by setting the determinant of \( J - \lambda I \) to zero and solving for \( \lambda \) (where \( J \) is the Jacobian matrix and \( I \) is the identity matrix). Complex eigenvalues with positive real parts relate to an unstable spiral, while negative real parts imply a stable spiral. Thus, the bifurcation type is determined based on these properties.
Dynamical Systems
Dynamical systems study the behavior of systems defined by a set of rules and the evolution of these systems over time. They can be described by differential equations, like in our exercise, which portray the relationship between a function and its derivatives.

Dynamical systems theory analyzes stability, bifurcations, chaos, and other behaviors exhibited by these systems. The Hopf bifurcation, as seen in this exercise, is a crucial concept in this field and occurs when an equilibrium point's stability is lost as a parameter (\( \mu \) in our case) is varied, leading to the emergence of a periodic solution (limit cycle).

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Most popular questions from this chapter

(Second-order phase-locked loop) Using a computer, explore the phase portrait of \(\ddot{\theta}+(1-\mu \cos \theta) \dot{\theta}+\sin \theta=0\) for \(\mu \geq 0 .\) For some values of \(\mu\), you should find that the system has a stable limit cycle. Classify the bifurcations that create and destroy the cycle as \(\mu\) increases from \(0 .\) Exercises \(8.4 .5-8.4 .11\) deal with the forced Duffing oscillator in the limit where the forcing, detuning, damping, and nonlinearity are all weak: $$ \bar{x}+x+\varepsilon\left(b x^{3}+k x-a x-F \cos t\right)=0 $$ where \(0<\varepsilon<<1, b>0\) is the nonlinearity, \(k>0\) is the damping, \(a\) is the detuning, and \(F>0\) is the forcing strength. This system is a small perturbation of a harmonic oscillator, and can therefore be handled with the methods of Section 7.6. We have postponed the problem until now because saddle-node bifurcations of cycles arise in its analysis.

(Globally coupled oscillators) Consider the following system of \(N\) identical oscillators; $$ \dot{\theta}_{i}=f\left(\theta_{i}\right)+\frac{K}{N} \sum_{j=1}^{N} f\left(\theta_{j}\right), \text { for } i=1, \ldots, N $$ where \(K>0\) and \(f(\theta)\) is smooth and \(2 \pi\)-periodic. Assume that \(f(\theta)>0\) for all \(\theta\) so that the in-phase solution is periodic. By calculating the linearized Poincar茅 map as in Example 8.7.4, show that all the characteristic multipliers equal \(+1\). Thus the neutral stability found in Example 8.7.4 holds for a broader class of oscillator arrays. In particular, the reversibility of the system is not essential. This example is from Tsang et al. (199I).

(Explaining Lissajous figures) Lissajous figures are one way to visualize the knots and quasiperiodicity discussed in the text. To sce this, consider a pair of uncoupled harmonic oscillators described by the four-dimensional system \(\ddot{x}+x=0, \ddot{y}+\omega^{2} y=0\) a) Show that if \(x=A(t) \sin \theta(t), y=B(t) \sin \phi(t)\), then \(\dot{A}=\dot{B}=0\) (so \(A, B\) are constants) and \(\dot{\theta}=1, \dot{\phi}=\omega\). b) Fxplain why (a) implies that trajectories are typically confined to two- dimensional tori in a four-dimensional phase space. c) How are the Lissajous figures related to the trajectories of this system?

For each of the following systems, a Hopf bifurcation occurs at the origin when \(\mu=0\). Use the analytical criterion of Exercise \(8.2 .12\) to decide if the bifurcation is sub- or supercritical. Confirm your conclusions on the computer. $$ \hat{x}=\mu x+y-x^{2}, \quad \dot{y}=-x+\mu y+2 x^{2} $$

Consider the system $$ \dot{\theta}_{1}=E-\sin \theta_{1}+K \sin \left(\theta_{2}-\theta_{1}\right), \quad \dot{\theta}_{2}=E+\sin \theta_{2}+K \sin \left(\theta_{1}-\theta_{2}\right. $$ where \(E, K \geq 0\). a) Find and classify all the fixed points. b) Show that if \(E\) is large enough, the system has periodic solutions on the torus. What type of bifurcation creates the periodic solutions? c) Find the bifurcation curve in \((E, K)\) space at which these periodic solutions are created. A generalization of this system to \(N>1\) phases has been proposed as a model of switching in charge-density waves (Strogatz et al. 1988,1989 ).
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