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Chapter 8: Problem 2

Discuss the bifurcations of the system \(\dot{r}=r(\mu-\sin r), \dot{\theta}=1\) as \(\mu\) varies.

Short Answer

Expert verified
In summary, the given system has equilibrium points at \((0,\theta)\) for all \(\theta\) and other equilibrium points satisfying \(\mu = \sin r\). A bifurcation occurs as \(\mu\) passes through 0, with the origin changing its stability from stable for \(\mu < 0\) to unstable for \(\mu > 0\). Additionally, as \(\mu\) increases, more equilibrium points appear, characterized by the relation \(\mu = \sin r\).

Step by step solution

01

1. Find equilibrium points

To find the equilibrium points, we need to solve \(\dot{r} = 0\) together with \(\dot{\theta}=1\). The angular equation cannot be zero, so let's focus on the radial equation: Setting \(\dot{r} = 0\), we get: \(r(\mu - \sin r) = 0\) There are two possible solutions for this equation: 1. \(r = 0\) 2. \(\mu - \sin r = 0 \implies \mu = \sin r\) The equilibrium points are given by \((r,\theta)\), where \(r\) satisfies one of the conditions above.
02

2. Determine the stability

To determine the stability of the equilibrium points, we'll linearize the system near these points. To do this, we will find the Jacobian matrix and then analyze the sign of its eigenvalues. The Jacobian matrix of the system is given by: \[J(r,\theta) = \begin{bmatrix} \frac{\partial}{\partial r}(r(\mu - \sin r)) & \frac{\partial}{\partial \theta}(r(\mu - \sin r)) \\ \frac{\partial}{\partial r}(1) & \frac{\partial}{\partial \theta}(1) \end{bmatrix} =\begin{bmatrix} \mu - \sin r - r \cos r & 0 \\ 0 & 0 \end{bmatrix}\] Now, let's look at the two cases from the equilibrium points: 1. For \(r = 0\), $J(0,\theta) = \begin{bmatrix} \mu & 0 \\ 0 & 0 \end{bmatrix}\(. The eigenvalues are \)\mu\( and 0. Thus, the equilibrium point \)(0,\theta)\( is stable if \)\mu < 0\( and unstable if \)\mu > 0$. 2. For \(\mu = \sin r\), $J(r,\theta) = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. The eigenvalues are both 0. The dynamics near this equilibrium point will depend on higher-order terms.
03

3. Analyze bifurcations as \(\mu\) varies

As \(\mu\) varies, the stability of the equilibrium points changes: 1. For \(\mu < 0\), the origin \((0,\theta)\) is stable. There are no other equilibrium points. 2. For \(\mu = 0\), there is a bifurcation at the origin, as the equilibrium point changes its stability. Additionally, there is another equilibrium point for any \(r\) satisfying \(\mu = \sin r\), which means \(r = 0\) (again). Therefore, there are infinitely many equilibrium points on the circle with radius \(r = 0\), but they will be of higher-order dynamics. 3. For \(\mu > 0\), the origin \((0,\theta)\) becomes unstable. There are additional equilibrium points characterized by \(\mu = \sin r\). Since \(\sin r\) is bounded between \(-1\) and \(1\), there can be multiple equilibrium points depending on \(\mu\). In summary, a bifurcation occurs when \(\mu\) passes through 0, as the stability of the origin changes. As \(\mu\) increases, additional equilibrium points appear characterized by the relation \(\mu = \sin r\).

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Key Concepts

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

Equilibrium Points
When analyzing nonlinear dynamic systems like the one given by \(\dot{r}=r(\mu-\sin r), \dot{\theta}=1\), identifying equilibrium points is a fundamental step. Equilibrium points, also known as fixed points or critical points, are found where the rate of change of the system's variables is zero. In our case, this means solving \(\dot{r} = 0\) while \(\dot{\theta}=1\) is a nonzero constant, signifying continuous rotation.

There are two scenarios in which the radial rate of change \(\dot{r}\) becomes zero: when \(r = 0\) or when \(\mu = \sin r\). These points are significant because they represent the states where the system will not evolve further unless perturbed. The collection of these points creates a landscape of possible behaviors the system might exhibit as it evolves over time. A deep understanding of equilibrium points is essential as they form the foundation for subsequent stability analysis and bifurcation examination.

Knowing that \(\dot{\theta}\) is always positive implies that the system will keep rotating, however the character of that motion鈥攚hether spiraling out, in, or maintaining a consistent loop鈥攄epends on the radial equilibrium around which this motion unfolds.
Jacobian Matrix
The Jacobian matrix is a powerful tool in the stability analysis of nonlinear dynamic systems. For the system at hand, the Jacobian matrix \(J(r,\theta)\) provides a local linear approximation of the changes around an equilibrium point

Components of the Jacobian

For the given system, the off-diagonal elements of the Jacobian matrix are zero because the \(\dot{r}\) equation does not depend on \(\theta\), and \(\dot{\theta}\) does not depend on \(r\) or \(\theta\). This simplification reflects the decoupled nature of the rotation from the radial dynamics.

Eigenvalues and Dynamics

The eigenvalues of the Jacobian matrix, calculated from the diagonal elements, are instrumental in understanding the behavior near equilibrium points. If the real parts of all eigenvalues are negative, the point is stable; if any have positive real parts, the point is unstable. In the context of our exercise, the Jacobian informs us about the stability of the radial component of motion while the system keeps rotating with a constant angular velocity.
Stability Analysis
Stability analysis evaluates how a system responds to small disturbances around an equilibrium point. A point is stable if the system returns to it after a slight disturbance. The Jacobian matrix derived in the previous section plays a vital role in this analysis for our nonlinear dynamic system.

By calculating the eigenvalues of the Jacobian at the equilibrium points, we can infer their stability. Depending on the sign of these eigenvalues, an equilibrium point will be classified as stable or unstable. In our case:
  • For \(r = 0\), the point's stability hinges on the value of \(\mu\). If \(\mu < 0\), the radial motion's equilibrium point is a stable one since it attracts nearby trajectories. On the contrary, if \(\mu > 0\), it is unstable and repels nearby trajectories.
  • When \(\mu = \sin r\), we find ourselves dealing with a degenerate case where the Jacobian's eigenvalues are zero; this necessitates a more nuanced exploration beyond linear approximations to determine stability.
As \(\mu\) varies, equilibrium points' stability can change, leading to bifurcations鈥攕hifts in the qualitative structure of the system dynamics鈥攔evealing the complex interplay between stability, system parameters, and the geometry of the trajectories in the phase space.

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Most popular questions from this chapter

(Bead on rotating hoop, revisited) In Section \(3.5\), we derived the following dimensionless equation for the motion of a bead on a rotating hoop: $$ \varepsilon \frac{d^{2} \phi}{d \tau^{2}}=-\frac{d \phi}{d \tau}-\sin \phi+\gamma \sin \phi \cos \phi $$ Here \(\varepsilon>0\) is proportional to the mass of the bead, and \(\gamma>0\) is related to the spin rate of the hoop. Previously we restricted our attention to the overdamped limit \(\varepsilon \rightarrow 0\). a) Now allow any \(\varepsilon>0\). Find and classify all bifurcations that occur as \(\varepsilon\) and \(\gamma\) vary. b) Plot the stability diagram in the positive quadrant of the \(\varepsilon, \gamma\) plane.

(Degenerate bifurcation, not Hopf) Consider the damped Duffing oscillator \(\ddot{x}+\mu \hat{x}+x-x^{3}=0\) a) Show that the origin changes from a stable to an unstable spiral as \(\mu\) decreases though zero. b) Plot the phase portraits for \(\mu>0, \mu=0\), and \(\mu<0\), and show that the bifurcation at \(\mu=0\) is a degenerate version of the Hopf bifurcation.

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. $$ \dot{x}=\mu x+y-x^{3}, \quad \dot{y}=-x+\mu y+2 y^{3} $$

(Logistic equation with periodically varying carrying capacity) Consider the logistic equation \(\dot{N}=r N(1-N / K(t))\), where the carrying capacity is positive, smooth, and \(T\)-periodic in \(t .\) a) Using a Poincar茅 map argument like that in the text, show that the system has at least one stable limit cycle of period \(T\), contained in the strip \(K_{\min } \leq N \leq K_{\max ^{\prime}}\) b) Is the cycle necessarily unique?

(Mechanical example of quasiperiodicity) The equations $$ m \ddot{r}=\frac{h^{2}}{m r^{3}}-k, \quad \dot{\theta}=\frac{h}{m r^{2}} $$ govern the motion of a mass \(m\) subject to a central force of constant strength \(k>0\). Here \(r, \theta\) are polar coordinates and \(h>0\) is a constant (the angular momentum of the particle). a) Show that the system has a solution \(r=r_{0}, \dot{\theta}=\omega_{\theta}\), corresponding to uniform circular motion at a radius \(r_{e}\) and frequency \(\omega_{\theta} .\) Find formulas for \(r_{0}\) and \(\omega_{\theta}\). b) Find the frequency \(\omega\), of small radial oscillations about the circular orbit. c) Show that these small radial oscillations correspond to quasiperiodic motion by calculating the winding number \(\omega_{r} / \omega_{\theta}\). d) Show by a geometric argument that the motion is either periodic or quasiperiodic for any amplitude of radial oscillation. (To say it in a more interesting way, the motion is never chaotic.) e) Can you think of a mechanical realization of this system?
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