/*! 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 9 The next exercises are designed ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 9

The next exercises are designed to test your ability to distinguish among the various types of bifurcations - it's easy to confuse them! In each case, find the values of \(r\) at which bifurcations occur, and classify those as saddle- node, transcritical, supercritical pitchfork, or subcritical pitchfork. Finally, sketch the bifureation diagram of fixed points \(x^{*}\) vs. \(r\). $$ \dot{x}=x+\tanh (r x) $$

Short Answer

Expert verified
In summary, we analyzed the dynamical system \(\dot{x}=x+\tanh(rx)\) and found that there is a supercritical pitchfork bifurcation happening at \(r=\pm1\). The bifurcation diagram has three regions forming a "V" shape, with the vertex at the fixed point \(x^*=0\) and the bifurcation points located at \(r=\pm1\).

Step by step solution

01

Find the fixed points

To find the fixed points, we need to set the right-hand side of the given equation, \(x+\tanh(rx)\), equal to zero, and solve for \(x^*\): \(0 = x^*+\tanh(rx^*)\) This equation doesn't have a closed-form solution, so we must proceed by understanding its behavior.
02

Analyze the equation and find possible bifurcation points

Notice that \(\tanh(rx^*)\) is an odd function, meaning \(\tanh(-rx^*)=-\tanh(rx^*)\). Now let's analyze the behavior of the equation for different values of \(r\): 1. Case \(r=0\): \(0=x^*\), this is a trivial solution. 2. Case \(r>0\): - as \(x^*>0\), then \(\tanh(rx^*)>0\), and \(x^*>-\tanh(rx^*)\) - as \(x^*<0\), then \(\tanh(rx^*)<0\), and \(x^*>-\tanh(rx^*)\) - as \(x^*=0\), then \(x^*=-\tanh(0)=0\) 3. Case \(r<0\): - as \(x^*>0\), then \(\tanh(rx^*)<0\), and \(x^*<-\tanh(rx^*)\) - as \(x^*<0\), then \(\tanh(rx^*)>0\), and \(x^*<-\tanh(rx^*)\) - as \(x^*=0\), then \(x^*=-\tanh(0)=0\) Remember that we are looking for the points where the fixed points change their stability, which occurs when \(\frac{d}{dx}(x^*+\tanh(rx^*))\) crosses zero. To determine the stability, we will analyze the derivative of the equation.
03

Calculate the derivative

Differentiate the equation with respect to \(x\): \(\frac{d}{dx}(x^*+\tanh(rx^*)) = 1 + r\cdot(1-\tanh^2(rx^*))\) Now, we will analyze the stability for the different cases we previously established. 1. Case \(r=0\): - at \(x^*=0\), the derivative is \(\frac{d}{dx}(x^*+\tanh(rx^*)) = 1\), meaning the fixed point is stable. 2. Case \(r>0\): - at \(x^*=0\), the derivative is \(\frac{d}{dx}(x^*+\tanh(rx^*)) = 1 + r\), meaning both fixed points (positive and negative) are stable given r > 1. 3. Case \(r<0\): - at \(x^*=0\), the derivative is \(\frac{d}{dx}(x^*+\tanh(rx^*)) = 1 + r\), meaning both fixed points (positive and negative) are stable given r < -1.
04

Identify and classify bifurcations

From the previous analysis, we can identify two bifurcation points, \(r_1=0\), and \(r_2=\pm1\). Now we need to classify the type of bifurcations: 1. At \(r=0\), the fixed point is stable (i.e., it's a sink). Given that the stability of this fixed point does not change as the parameter \(r\) changes, this is not a bifurcation point after all. 2. At \(r=\pm1\), the behavior of fixed points changes: - For \(r>1\), there are two stable fixed points (positive and negative). - For \(r<-1\), there are also two stable fixed points (positive and negative). - For \(-1
05

Sketch the bifurcation diagram

To sketch the bifurcation diagram, we will plot the fixed points \(x^*\) against the parameter \(r\): 1. For \(r<-1\), there are two stable fixed points. 2. For \(-11\), there are two stable fixed points. The bifurcation diagram will have three different regions, forming a "V" shape, with the bifurcation points located at \(r=\pm1\) and the fixed point \(x^*=0\) being the "vertex" of the "V".

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.

Fixed Points
Understanding fixed points is crucial when studying dynamical systems. These are essentially the values at which the system does not change over time, serving as potential points of equilibrium. For example, when we have the differential equation \( \dot{x} = x + \tanh(rx) \), fixed points are found by setting \( \dot{x} = 0 \) and solving for \( x^* \). In this particular equation, fixed points are solutions to the equation \( x^* = -\tanh(rx^*) \).

Analyze the function behavior to predict such points. If \( r = 0 \) we have a trivial solution, while non-zero \( r \) introduces the hyperbolic tangent function into play, indicating the presence of non-trivial fixed points. The number, location, and nature (stable or unstable) of these fixed points can change when the parameter \( r \) varies, leading to bifurcations鈥攁 concept we'll explore further in the other sections.
Stability of Dynamical Systems
Investigating the stability of these fixed points gives us insights into the dynamical system's long-term behavior. When a fixed point is stable, slight deviations from this point will eventually return to it, implying a sense of equilibrium. To ascertain stability, we differentiate the right-hand side of our system's equation with respect to \( x \) and appraise the derivative at the fixed point.

For the given system \( \dot{x} = x + \tanh(rx) \), stability is determined by the sign of the derivative \(1 + r\cdot(1-\tanh^2(rx^*))\). When this derivative is positive, it indicates instability; conversely, when negative, the fixed point is stable. At the juncture where this derivative equals zero, a change in stability may occur; this marks a potential bifurcation point鈥攖he moment where qualitative changes in system dynamics are expected to occur.
Pitchfork Bifurcation
The term pitchfork bifurcation describes a scenario wherein a system exhibits a change in the stability of fixed points as parameters vary. Consider our equation from earlier; when the parameter \( r \) passes through critical values of \( \pm1 \), the system undergoes such a bifurcation. There are two primary types: supercritical and subcritical.

In a supercritical pitchfork bifurcation, as \( r \) increases past a critical value, a stable fixed point becomes unstable while two new stable fixed points appear. The bifurcation diagram depicting this will resemble an uppercase 'Y.' It's analogous to a balanced state becoming unstable and splitting into two alternative stable states. Meanwhile, a subcritical pitchfork bifurcation presents an inverse scenario, with an unstable fixed point giving way to stability. Visualize our system's behavior using a diagram, showing how these transitions occur at \( r=\pm1 \), catering to the labeled bifurcation as supercritical due to the preservation and bifurcation of stable fixed points.

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 system \(\dot{x}=r x+x^{3}-x^{5}\), which exhibits a subcritical pitchfork bifurcation. a) Find algebraic expressions for all the fixed points as \(r\) varies. b) Sketch the vector fields as \(r\) varies. Be sure to indicate all the fixed points and their stability. c) Calculate \(r_{y}\), the parameter value at which the nonzero fixed points are born in a saddle-node bifurcation.

In parts (a)-(c), let \(V(x)\) be the potential, in the sense that \(\dot{x}=-d V / d x .\) Sketch the potential as a function of \(r .\) Be sure to show all the qualitatively different cases, including bifurcation values of \(r\). a) (Saddle-node) \(\dot{x}=r-x^{2}\) b) (Transcritical) \(\hat{x}=r x-x^{2}\) c) (Subcritical pitchfork) \(\dot{x}=r x+x^{3}-x^{5}\)

The first-order system \(\dot{u}=a u+b u^{3}-c u^{5}\), where \(b, c>0\), has a subcritical pitchfork bifurcation at \(a=0\). Show that this equation can be rewritten as $$ \frac{d x}{d \tau}=r x+x^{3}-x^{3} $$ where \(x=u / U, \tau=t / T\), and \(U, T\), and \(r\) are to be determined in terms of \(a, b\), and \(c\).

The equation \(\dot{N}=r N\left(1-\frac{N}{k}\right)-H\) provides an extremely simple model of a fishery. In the absence of fishing, the population is assumed to grow logistically. The effects of fishing are modeled by the term \(-H\), which says that fish are caught or "harvested" at a constant rate \(H>0\), independent of their population \(N\). (This assumes that the fishermen aren't worried about fishing the population dry-they simply catch the same number of fish every day.) a) Show that the system can be rewritten in dimensionless form as $$ \frac{d x}{d \tau}=x(1-x)-h $$ for suitably defined dimensionless quantities \(x, \tau\), and \(h\). b) Plot the vector field for different values of \(h\). c) Show that a bifurcation occurs at a certain value \(h_{c}\), and classify this bifurcation. d) Discuss the long-term behavior of the fish population for \(hh_{c}\), and give the biological interpretation in each case. There's something silly about this model-the population can become negative! A better model would have a fixed point at zero population for all values of \(H .\) See the next exercise for such an improvement.

In the following exercises, sketch all the qualitatively different vector fields that occur as \(r\) is varied. Show that a pitchfork bifurcation occurs at a critical value of \(r\) (to be determined) and classify the bifurcation as supercritical or subcritical. Finally, sketch the bifurcation diagram of \(x^{*}\) vs. \(r\). $$ \dot{x}=r x-4 x^{3} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.