Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 4

For each problem, determine equilibrium points, bifurcation points, and construct a bifurcation diagram. Discuss the different behaviors in each system. a. \(y^{\prime}=y-\mu y^{2}\) b. \(y^{\prime}=y(\mu-y)(\mu-2 y)\) c. \(x^{\prime}=\mu-x^{3}\) d. \(x^{\prime}=x-\frac{\mu x}{1+x^{2}}\).

Short Answer

Expert verified
a) Equilibrium points: \(y=0\), \(y=\mu\). Bifurcations occur at these points. b) Equilibrium points: \(y=0\), \(y=\mu\), \(y=\mu/2\). Bifurcations occur at these points. c) Equilibrium points: \(x=\sqrt[3]{\mu}\), \(x=-\sqrt[3]{\mu/2}\), \(x= \sqrt[3]{\mu/2}\). Bifurcations occur at these points. d) Equilibrium points: \(x=0\) and \(x=\sqrt{\mu-1}\),\(-\sqrt{\mu-1}\). Bifurcations occur at these points. Different behaviors observed in each system depend on the values of \(\mu\).

Step by step solution

01

Equilibrium Points for Equation a.

To find the equilibrium points, set the function equal to zero. Solve for y in the equation \(y^{\prime}=y-\mu y^{2}=0\). Doing this yields \(y=0\) and \(y= \mu\).
02

Bifurcation Points and Diagram for Equation a.

The system undergoes a bifurcation at \(y=0\) and \(y=\mu\). This is because these are the points where the stability of the system changes. Plot these points on a diagram with \(y\) on the y-axis and \(\mu\) on the x-axis.
03

Equilibrium Points for Equation b.

Solving the equation \(y^{\prime}=y(\mu-y)(\mu-2 y)=0\), gives three equilibrium points, \(y=0\), \(y=\mu\), and \(y=\mu/2\).
04

Bifurcation Points and Diagram for Equation b.

The system undergoes a bifurcation at \(y=0\), \(y=\mu\) and \(y=\mu/2\). Plot these points on a diagram with \(y\) on the y-axis and \(\mu\) on the x-axis.
05

Equilibrium Points for Equation c.

Solving the equation \(x^{\prime}=\mu-x^{3}=0\), we get the equilibrium points \(x=\sqrt[3]{\mu}\), \(x=-\sqrt[3]{\mu/2}\), and \(x= \sqrt[3]{\mu/2}\).
06

Bifurcation Points and Diagram for Equation c.

The system undergoes a bifurcation at \(x=\sqrt[3]{\mu}\), \(x=-\sqrt[3]{\mu/2}\), and \(x= \sqrt[3]{\mu/2}\). Plot these points on a diagram with \(x\) on the y-axis and \(\mu\) on the x-axis.
07

Equilibrium Points for Equation d.

Solving \(x^{\prime}=x-\frac{\mu x}{1+x^{2}}=0\) gives equilibrium points \(x=0\) and \(x=\sqrt{\mu-1}\),\(-\sqrt{\mu-1}\).
08

Bifurcation Points and Diagram for Equation d.

The system undergoes a bifurcation at \(x=0\), \(x=\sqrt{\mu-1}\), and \(x=-\sqrt{\mu-1}\). Plot these points on a diagram with \(x\) on the y-axis and \(\mu\) on the x-axis.
09

Discussion of system behavior

Each of the four system behaviors depends on the values of \(\mu\). For all, changes in stability occur at the bifurcation points which can be depicted in each bifurcation diagram. In the case of equations a and b, as \(\mu\) increases, the stable point transitions from one equilibrium point to another. For equations c and d, the systems show more complex bifurcations involving increases and decreases in stability at different equilibrium points.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

8\. Find the fixed points of the following systems. Linearize the system about each fixed point and determine the nature and stability in the neighborhood of each fixed point, when possible. Verify your findings by plotting phase portraits using a computer. a. $$ \begin{aligned} &x^{\prime}=x(100-x-2 y) \\ &y^{\prime}=y(150-x-6 y) \end{aligned} $$ b. $$ \begin{aligned} x^{\prime} &=x+x^{3} \\ y^{\prime} &=y+y^{3} \end{aligned} $$ C. $$ \begin{aligned} &x^{\prime}=x-x^{2}+x y \\ &y^{\prime}=2 y-x y-6 y^{2} \end{aligned} $$ d. $$ \begin{aligned} &x^{\prime}=-2 x y \\ &y^{\prime}=-x+y+x y-y^{3} \end{aligned} $$.

The Lorenz Model is a simple model for atmospheric convection developed by Edward Lorenz in \(1963 .\) The system is given by three equations: $$ \begin{aligned} &\frac{d x}{d t}=\sigma(y-x) \\ &\frac{d y}{d t}=x(\rho-z)-y \\ &\frac{d z}{d t}=x y-\beta z \end{aligned} $$ a. Find the equilibrium points of the system. b. Find the Jacobian matrix for the system and evaluate it at the equilibrium points. c. Determine any bifurcation points and describe what happens near bifurcation point(s). Consider \(\sigma=10, \beta=8 / 3\), and vary \(\rho\) d. This system is known to exhibit chaotic behavior. Lorenz found a so-called strange attractor for parameter values \(\sigma=10, \beta=8 / 3\), and \(\rho=28\). Using a computer, locate this strange attractor.

Another population model is one in which species compete for resources, such as a limited food supply. Such a model is given by $$ \begin{aligned} &x^{\prime}=a x-b x^{2}-c x y \\ &y^{\prime}=d y-e y^{2}-f x y \end{aligned} $$ In this case, assume that all constants are positive. a. Describe the effects/purpose of each terms. b. Find the fixed points of the model. c. Linearize the system about each fixed point and determine the stability. d. From the above, describe the types of solution behavior you might expect, in terms of the model.

Consider the family of differential equations \(x^{\prime}=x^{3}+\delta x^{2}-\mu x\) a. Sketch a bifurcation diagram in the \(x \mu\)-plane for \(\delta=0\). b. Sketch a bifurcation diagram in the \(x \mu\)-plane for \(\delta>0\). Hint: Pick a few values of \(\delta\) and \(\mu\) in order to get a feel for how this system behaves.

Plot phase portraits for the Lienard system $$ \begin{aligned} x^{\prime} &=y-\mu\left(x^{3}-x\right) \\ y^{\prime} &=-x \end{aligned} $$ for a small and a not so small value of \(\mu\). Describe what happens as one varies \(\mu\).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

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.

Sign-up for free