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Chapter 8: Problem 2

Consider the driven pendulum \(\phi^{\prime \prime}+\alpha \phi^{\prime}+\sin \phi=I\). By numerical computation of the phase portrait, verify that if \(\alpha\) is fixed and sufficiently small, the system's stable limit cycle is destroyed in a homoclinic bifurcation as \(I\) decreases. Show that if \(\alpha\) is too large, the bifurcation is an infinite-period bifurcation instead.

Short Answer

Expert verified
By converting the driven pendulum equation (\(\phi^{\prime\prime}+\alpha \phi^{\prime}+\sin\phi=I\)) into a 2D autonomous system, we can analyze the system's behavior by drawing phase portraits for different combinations of \(\alpha\) and \(I\). Numerically simulating and plotting the system for different initial conditions allows us to verify the following bifurcation behaviors: 1. When \(\alpha\) is fixed and small, the stable limit cycle is destroyed in a homoclinic bifurcation as \(I\) decreases. 2. If \(\alpha\) is too large, the system experiences an infinite-period bifurcation instead of a homoclinic bifurcation.

Step by step solution

01

Convert the ODE into a 2D autonomous system

To analyze this second-order ODE, we will first need to convert it into a system of two first-order ODEs. We can achieve this by defining \(x(t) = \phi(t)\) and \(v(t) = \phi^{\prime}(t)\), where \(x(t)\) represents the angle of the pendulum and \(v(t)\) represents the angular velocity. Now, the given ODE can be rewritten as: \[ \begin{cases} \frac{dx}{dt} = v(t) \\ \frac{dv}{dt} = -\alpha v(t) - \sin(x(t)) + I \end{cases} \]
02

Compute equilibrium points and study their stability

Now that we have the system of first-order ODEs, we need to compute the equilibrium points to determine the possible stable limit cycle behavior. Equilibrium points are points in the phase space where the system is stationary, which means that the derivative of both \(x(t)\) and \(v(t)\) are zero. Thus, we need to solve the following simultaneous equations: \[ \begin{cases} v = 0 \\ -\alpha v - \sin(x) + I = 0 \end{cases} \] By substituting \(v=0\) into the second equation, we have: \[ -\sin(x)+I = 0 \] The equilibrium points can be found by solving this equation for \(x\). The stability of these points will be determined by analyzing the Jacobian matrix of the system, which contains the partial derivatives of the functions defining the system.
03

Draw the phase portrait for different combinations of \(\alpha\) and \(I\)

At this point, we will need to rely on numerical methods to simulate the system and obtain the phase portrait. This can be achieved by employing numerical ODE solvers like Runge-Kutta or ode45 (if using MATLAB) to calculate the time evolution of the system for different starting conditions. Once we have the time series for several starting conditions, we can plot the phase portrait, showing the angle and angular velocity on the axes, with different colors or arrows indicating the direction field. We should verify the following behaviors for different values of \(\alpha\) and \(I\): 1. For fixed and small \(\alpha\), as \(I\) decreases, we should observe that the stable limit cycle is destroyed in a homoclinic bifurcation. 2. For larger values of \(\alpha\), we should observe an infinite-period bifurcation instead of a homoclinic bifurcation. These bifurcations in the phase portrait will help us verify the behaviors described in the problem statement.

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Key Concepts

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

Driven Pendulum
A driven pendulum is a type of pendulum that experiences external periodic forcing. Unlike a simple pendulum, it doesn鈥檛 just move back and forth under gravity, but also has an extra force, represented by the term \(I\) in our equation. This external force can change the motion significantly, leading to complex and interesting behavior.
  • The equation \(\phi^{\prime\prime} + \alpha \phi^{\prime} + \sin(\phi) = I\) describes this dynamic system.
  • Here, \(\alpha\) is a damping factor that resists the swing of the pendulum. If \(\alpha\) is small, the pendulum can swing freely, resembling a more classic trajectory.
  • The sinusoidal term \(\sin(\phi)\) contributes to the nonlinearity, causing varying responses depending on \(\phi\)'s angle.
Analyzing a driven pendulum is fascinating due to its relationship with chaotic dynamics and its unpredictable outcomes.
Bifurcation Theory
Bifurcation theory is a powerful mathematical tool used to study changes in the qualitative or topological structure of a given family of dynamical systems. In essence, it helps in understanding how different parameters in a system can lead to sudden changes, like the emergence or disappearance of equilibria or limit cycles.
  • A "homoclinic bifurcation" in the driven pendulum context happens when a stable limit cycle collides with a saddle point as parameter \(I\) changes.
  • An "infinite-period bifurcation" occurs when the system's oscillation period grows forever long as another parameter, like \(\alpha\), is adjusted.
These bifurcations can drastically alter the behavior of the pendulum system, signaling the potential for more complex, or less predictable, motion patterns.
Phase Portrait
A phase portrait is a graphical representation of a dynamical system's trajectories in the phase plane, with each set of initial conditions represented by different trajectories. For the driven pendulum, this provides a comprehensive snapshot of how the system evolves over time.
  • Axes in the phase portrait represent the angle of the pendulum \(x(t)\) and its angular velocity \(v(t)\).
  • Stable limit cycles appear as loops in this portrait, representing steady-state oscillations.
  • By observing changes in the phase portrait under varying \(\alpha\) and \(I\), we can visually detect bifurcations and other system behaviors.
Exploring phase portraits is invaluable for predicting and verifying outcomes in nonlinear dynamics, offering both a qualitative and quantitative understanding.
Equilibrium Points
Equilibrium points in the context of the driven pendulum are configurations where the pendulum does not change its state 鈥� essentially, where all forces balance out. These points are crucial for understanding the stability of the system and predicting future behavior.
  • To find equilibrium, set \(v = 0\) and solve \(-\sin(x) + I = 0\) which indicates the points where the pendulum ceases to swing.
  • The stability of these points can be determined by analyzing the Jacobian matrix derived from the system's equations.
  • Stable equilibria reflect a tendency for the system to return to these points after small disturbances, while unstable ones will repel small perturbations.
Knowing the equilibrium points helps in mapping out phase portraits and understanding potential bifurcations, making them central to nonlinear dynamics studies.

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Most popular questions from this chapter

(Bacterial respiration) Fair茅n and Velarde (1979) considered a model for respiration in a bacterial culture. The equations are $$ \dot{x}=B-x-\frac{x y}{1+q x^{2}}, \quad \dot{y}=A-\frac{x y}{1+q x^{2}} $$ where \(x\) and \(y\) are the levels of nutrient and oxygen, respectively, and \(A, B, q>0\) are parameters. Investigate the dynamics of this model. As a start, find all the fixed points and classify them. Then consider the nullclines and try to construct a trapping region. Can you find conditions on \(A, B, q\) under which the system has a stable limit cycle? Use numerical integration, the Poincar茅-Bendixson theorem, results about Hopf bifurcations, or whatever else seems useful. (This question is deliber- ately open-ended and could serve as a class project; see how far you can go.)

(Interacting bar magnets) Consider the system $$ \begin{aligned} &\dot{\theta}_{1}=K \sin \left(\theta_{1}-\theta_{2}\right)-\sin \theta_{1} \\\ &\dot{\theta}_{2}=K \sin \left(\theta_{2}-\theta_{1}\right)-\sin \theta_{2} \end{aligned} $$ where \(K \geq 0\). For a rough physical interpretation, suppose that two bar magnets are confined to a plane, but are free to rotate about a common pin joint, as shown in Figure \(1 .\) Let \(\theta_{1}, \theta_{2}\) denote the angular orientations of the north poles of the magnets. Then the term \(K \sin \left(\theta_{2}-\theta_{1}\right)\) represents a repulsive force that tries to keep the two north poles \(180^{\circ}\) apart. This repulsion is opposed by the \(\sin \theta\) terms, which model external magnets that pull the north poles of both bar magnets to the east. If the inertia of the magnets is negligible compared to viscous damping, then the equations above are a decent approximation to the true dynamics. a) Find and classify all the fixed points of the system. b) Show that a bifurcation occurs at \(K=\frac{1}{2} .\) What type of bifurcation is it? (Hint: Recall that \(\sin (a-b)=\cos b \sin a-\sin b \cos a .)\) c) Show that the system is a "gradient" system, in the sense that \(\dot{\theta}_{i}=-\partial V / \partial \theta_{1}\) for some potential function \(V\left(\theta_{1}, \theta_{2}\right)\), to be determined. d) Use part (c) to prove that the system has no periodic orbits. e) Sketch the phase portrait for \(0\frac{1}{2}\).

Consider the predator-prey model, $$ \dot{x}=x\left(b-x-\frac{y}{1+x}\right), \quad \dot{y}=y\left(\frac{x}{1+x}-a y\right) $$ where \(x, y \geq 0\) are the populations and \(a, b>0\) are parameters. a) Sketch the nullclines and discuss the bifurcations that occur as \(b\) varies. b) Show that a positive fixed point \(x^{3}>0, y^{*}>0\) exists for all \(a, b>0\). (Don't try to find the fixed point explicitly; use a graphical argument instead.) c) Show that a Hopf bifurcation occurs at the positive fixed point if $$ a=a_{c}=\frac{4(b-2)}{b^{2}(b+2)} $$ and \(b>2\). (Hint: A necessary condition for a Hopf bifurcation to occur is \(\tau=0\), where \(\tau\) is the trace of the Jacobian matrix at the fixed point. Show that \(\tau=0\) if and only if \(2 x^{*}=b-2 .\) Then use the fixed point conditions to express \(a_{e}\) in terms of \(x^{*}\). Finally, substitute \(x^{*}=(b-2) / 2\) into the expression for \(a_{c}\) and you're done.) d) Using a computer, check the validity of the expression in (c) and determine whether the bifurcation is subcritical or supercritical. Plot typical phase portraits above and below the Hopf bifurcation.

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. In Example 8.2.I, we argued that the system \(\dot{x}=\mu x-y+x y^{2}\), \(\dot{y}=x+\mu y+y^{3}\) undergoes a subcritical Hopf bifurcation at \(\mu=0\). Use the analytical criterion to confirm that the bifurcation is subcritical.

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