/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 For the following prototypical e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 1

For the following prototypical examples, plot the phase portraits as \(\mu\) varies: a) \(\dot{x}=\mu x-x^{2}, \quad \dot{y}=-y\) (transcritical bifurcation) b) \(\dot{x}=\mu x+x^{3}, \quad \dot{y}=\cdots y\) (subcritical pitchfork bifurcation) For each of the following systems, find the eigenvalues at the stable fixed point as a function of \(\mu\), and show that one of the eigenvalues tends to zero as \(\mu \rightarrow 0\).

Short Answer

Expert verified
For system (a) (transcritical bifurcation), we have found that the fixed points are at \((0, 0)\) and \((\mu, 0)\), and both fixed points are neutrally stable. For system (b) (subcritical pitchfork bifurcation), the fixed points are at \((0, 0)\), \((\sqrt{-\mu}, 0)\), and \((-\sqrt{-\mu}, 0)\). When \(\mu \ge 0\), only the fixed point \(x = 0\) exists. For \(\mu < 0\), we have demonstrated that the fixed points at \(\pm\sqrt{-\mu}\) are stable. Moreover, when studying the eigenvalues of the Jacobian for system (b) at the stable fixed point \(\sqrt{-\mu}\), we found \(\lambda_1 = -1\) and \(\lambda_2 = 2\mu\), with \(\lambda_2 \rightarrow 0\) as \(\mu \rightarrow 0\).

Step by step solution

01

Find the fixed points

To find the fixed points, we'll set the right-hand side of each equation equal to zero: \(\mu x - x^2 = 0, -y = 0\) The fixed points are at \(y = 0\), and we solve for \(x\) from the first equation: \(x(\mu - x) = 0\). Thus, the fixed points are \((0, 0)\) and \((\mu, 0)\).
02

Analyze stability at the fixed points

Since the \(\dot{y} = -y\), the y-component of the fixed points is always stable, and the dynamics at the fixed points will depend only on the x-component. Observing that a fixed point is stable if \(\dot{x} < 0\) when \(x\) is slightly perturbed. When \(x = 0\), \(\dot{x} = 0\). When \(x = \mu\), \(\dot{x} = (\mu - x)^2 = 0\). Since the x-component is always 0, both fixed points are neutrally stable. The transcritical bifurcation occurs at \(\mu = 0\), where the fixed points collide and exchange stability. For system (b):
03

Find the fixed points

Similar to system (a), we set the right-hand side of each equation equal to zero: \(\mu x + x^3 = 0, \cdots y = 0\) Without loss of generality, let the second equation be \(\dot{y} = - y\). The fixed points are at \(y = 0\), and we solve for \(x\) from the first equation: \(x(\mu + x^2) = 0\). Thus, the fixed points are \((0, 0)\), \((\sqrt{-\mu}, 0)\), and \((-\sqrt{-\mu}, 0)\). When \(\mu \ge 0\) only the fixed point \(x = 0\) exists.
04

Analyze stability at the fixed points

The stability analysis for the y-component of the fixed points is the same as system (a). For the x-component, a fixed point is stable if \(\dot{x} < 0\) when \(x\) is slightly perturbed. When \(x = 0\), \(\dot{x} = 0\) if \(\mu \ge 0\). When \(x = \pm\sqrt{-\mu}\) and \(\mu < 0\), \(\dot{x} = -\mu x + x^3 = x^3 (1 + \frac{x^2}{\mu}) < 0\) since \(\mu < 0\) and \(x \ne 0\). Thus, there is a subcritical pitchfork bifurcation at \(\mu = 0\). We won't be performing Steps 3 for both systems as it requires graphical illustrations. However, one can always use a plotting software to visualize the phase portraits for different values of \(\mu\). For system (b):
05

Find the eigenvalues at the stable fixed point as a function of \(\mu\)

The Jacobian matrix of system (b) at a fixed point \((x_0, 0)\) is given by: \[J(x_0, 0) = \begin{bmatrix} \mu + 3x_0^2 & 0 \\ 0 & -1 \end{bmatrix}\] At the stable fixed point \(x_0 = \sqrt{-\mu}\) in the case when \(\mu < 0\), the eigenvalues of the Jacobian are: \[\lambda_1 = -1 \quad \text{and} \quad \lambda_2 = \mu + 3(-\mu) = 2\mu\]
06

Show that one of the eigenvalues tends to zero as \(\mu \rightarrow 0\)

As we can see, when \(\mu \rightarrow 0\), the eigenvalue \(\lambda_2 \rightarrow 0\). Therefore, one of the eigenvalues tends to zero when \(\mu \rightarrow 0\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Phase Portraits
Phase portraits are graphical representations that help us understand the behavior of dynamical systems over time. They plot the trajectory of a system's state variables in a phase space, which illustrates how the system evolves. In a two-dimensional system, like those in our exercise, the phase space is made up of axes representing variables such as \(x\) and \(y\). By analyzing these portraits, we can visually identify dynamical features such as fixed points and trajectories.
For example, in the problem given, the phase portraits change as the parameter \(\mu\) varies, indicating a bifurcation. Bifurcation refers to a qualitative change in the behavior of a system, which can be observed in phase portraits when the number or stability of fixed points change.
When constructing phase portraits, key features to look for include:
  • Fixed points, where trajectories start or end.
  • Stable and unstable points, depicted by attracting or repelling motions.
  • Invariant sets where trajectories remain enclosed.
If you need to visualize these for a better grasp, plotting them using software tools like Python or MATLAB can be immensely helpful.
Stability Analysis
Stability analysis helps us understand how a system responds to small perturbations in its state. Determining the stability of fixed points is vital in predicting the long-term behavior of dynamical systems.
In simplest terms, a fixed point is stable if, after a slight disturbance, the system returns to that point. Conversely, if the system diverges away, the fixed point is unstable. This analysis often involves looking at the sign and magnitude of derivatives or using tools like the Jacobian matrix.
In the original exercise, consider system \(\dot{y} = -y\), where any perturbation in \(y\) decays over time because the trajectories move towards the fixed point \(y = 0\). Therefore, it's stable. Similarly, analyzing the \(x\) component's dynamics in the given systems reveals complex behaviors linked to the parameter \(\mu\).
Remember that the nuances of stability depend on various factors, such as the nature of the bifurcation (like transcritical or pitchfork) which can further be understood through phase portraits.
Fixed Points
Fixed points are critical elements in the study of dynamical systems. They are the points where the system's variables do not change over time. In mathematical terms, these are solutions to the equations where all derivatives are zero, as shown in the exercise's setting of equations' right-hand side to zero.
In our examples, fixed points are determined by solving equations like \(\mu x - x^2 = 0\) and \(\mu x + x^3 = 0\). This gives us specific points like \( (0,0) \) and \( (\mu, 0) \) among others.
Here's why fixed points matter:
  • They provide a framework for predicting the system's behavior and possible states.
  • They pave the way for conducting stability analysis, which tells us whether these states are achievable or temporary under perturbations.
Grasping the concept of fixed points and their changes as parameter \(\mu\) varies is central to understanding bifurcations in the systems discussed.
Eigenvalues
Eigenvalues are fundamental in analyzing the stability of linear dynamical systems. By calculating eigenvalues of a system鈥檚 Jacobian matrix at its fixed points, we understand how small perturbations will behave over time.
In the context of the exercise, when we insert the fixed point into the Jacobian matrix, we calculate eigenvalues to determine stability. For instance, if all eigenvalues have negative real parts, the fixed point is stable.
Consider the example provided: the eigenvalues calculated show a scenario where one of them approaches zero as \(\mu\) tends to zero. This phenomenon often indicates a change in stability, frequently linked to bifurcations.
Eigenvalues help us:
  • Assess the stability of fixed points efficiently.
  • Detect changes in the behavior pattern as parameters in the system are altered.
  • Predict potential bifurcations and transitions in system dynamics.
Understanding how to capture and interpret these values forms a backbone in studying dynamical systems, particularly in the frameworks of bifurcation theory and stability analysis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the biased van der Pol oscillator \(\ddot{x}+\mu\left(x^{2}-1\right) \dot{x}+x=a\). Find the curves in \((\mu, a)\) space at which Hopf bifurcations occur. The next three exercises deal with the system \(\dot{x}=-y+\mu x+x y^{2}\), \(\dot{y}=x+\mu y-x^{2}\).

(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).

(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.)

(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?

(Laser model) In Fxercise \(3.3 .1\) we introduced the laser model $$ \begin{aligned} &\dot{n}=G n N-k n \\ &\dot{N}=-G n N-f N+p \end{aligned} $$ where \(N(t)\) is the number of excited atoms and \(n(t)\) is the number of photons in the laser field. The parameter \(G\) is the gain coefficient for stimulated emission, \(k\) is the decay rate due to loss of photons by mirror transmission, scattering, etc., \(f\) is the decay rate for spontaneous emission, and \(p\) is the pump strength. All parameters are positive, except \(p\), which can have either sign. For more information, see Milonni and Eberly (1988). a) Nondimensionalize the system. b) Find and classify all the fixed points. c) Sketch all the qualitatively different phase portraits that occur as the dimensionless parameters are varied. d) Plot the stability diagram for the system. What types of bifurcation occur?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.