Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 3

Use Bendixson's negative criterion to show that if \(P\) is an isolated saddle point there cannot be a limit cycle in the neighborhood of \(P\) that contains only \(P\).

Short Answer

Expert verified
Bendixson's criterion shows no limit cycle exists around an isolated saddle point P since divergence does not change sign.

Step by step solution

01

Understand Bendixson's Criterion

Bendixson's criterion states that a planar autonomous system cannot have a limit cycle in a region where the divergence of its vector field does not change sign. If we can show that this is the case for a neighborhood around the saddle point P, we will conclude there can be no limit cycle.
02

Identify the System's Divergence

For a planar system described by \(\frac{dx}{dt} = f(x,y)\) and \(\frac{dy}{dt} = g(x,y)\), the divergence of the vector field \((f,g)\) is given by \(div(F) = \frac{abla \bullet F}{abla} = \frac{abla \bullet (f, g)}{abla} = \frac{\frac{d f}{d x} + \frac{d g}{d y}}{abla}\). Calculate this for the given system.
03

Check the Divergence Near Saddle Point P

Evaluate the divergence \( div(F) = \frac{d f}{d x} + \frac{d g}{d y} \) at points near P. If this divergence does not change sign in the region surrounding P, proceed to the next step. Otherwise, re-evaluate the region around P.
04

Apply Bendixson's Negative Criterion

Given that P is an isolated saddle point, evaluate the nature of divergence around P. For a saddle point, typically, the divergence does not change sign clearly around the point. Use this to apply the criterion.
05

Conclude No Limit Cycle

Since the divergence of the vector field around the saddle point P does not change sign, Bendixson's criterion implies that there can be no limit cycle in this neighborhood surrounding P. Therefore, there is no limit cycle that contains only P.

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!

Key Concepts

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

Limit Cycle
A limit cycle is a closed trajectory in a dynamical system. It represents a periodic solution that an autonomous system of differential equations may exhibit. Imagine tracing a closed loop repeatedly. That loop is the limit cycle.
Limit cycles are crucial in various scientific fields because they depict sustained oscillations. For example, biological rhythms, like the heartbeat, are modeled using limit cycles.
Bendixson's negative criterion helps determine if such cycles are absent in certain regions. If the divergence of the vector field doesn't change sign around a critical point, no limit cycle exists there.
Saddle Point
A saddle point is a type of equilibrium in a dynamical system found at a critical point where trajectories are repelled along some directions and attracted in others. Visualize a mountain pass between two peaks. Moving in one direction feels like climbing, while moving sideways feels like descending.
Saddle points are important in understanding the system's stability. They indicate that the system has both stable and unstable behaviors depending on the direction of approach.
For an isolated saddle point, Bendixson's criterion can ensure no limit cycles exist in its neighborhood. Here, the divergence typically shows no sign change around such points.
Divergence of Vector Field
The divergence of a vector field measures the net rate at which 'density' exits a point. For a vector field \(F = (f, g)\), its divergence \(div(F)\) is given by \(\frac{ \partial f}{ \partial x} + \frac{ \partial g}{ \partial y}\).
Think of it as gauging the flow's 'outwardness.' If the divergence is positive, there's a net flow outwards; if negative, net flow inwards.
In applying Bendixson's criterion, we look at how the divergence behaves. If it stays positive or negative without switching signs, the area cannot house a limit cycle. This is key to analyzing systems near critical points like saddle points.
Planar Autonomous System
A planar autonomous system consists of differential equations that model the rate of change in the system's state. This system is defined by two equations: \(\frac{d x}{d t} = f(x, y)\) and \(\frac{d y}{d t} = g(x, y)\).
'Autonomous' means the system's rules don't change over time. The right-hand side functions \(f\) and \(g\) are purely dependent on state variables \(x\) and \(y\), not time \(t\).
These systems are visualized as vector fields in the plane. By studying their flow and critical points, we understand the system's long-term behavior such as stability or presence of cycles. Bendixson's criterion aids in analyzing these features, specifically ruling out limit cycles in certain areas based on divergence behavior.

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

Lefschetz (1977) discusses the following system of equations $$ \begin{aligned} &\frac{d x}{d t}=-y+x f\left(x^{2}+y^{2}\right) \\ &\frac{d y}{d t}=x+y f\left(x^{2}+y^{2}\right) \end{aligned} $$ Show that this system is equivalent to the polar equation $$ \frac{d r}{d \theta}=r f\left(r^{2}\right) \text {. } $$

A number of modifications of the Lotka-Volterra predator-prey model that have been suggested over the years are given in the last box in Section \(6.2\). By considering several combinations of prey density-dependent growth and predator density-dependent attack rate, detemine whether such modifications might lead to limit-cycle oscillations. You may wish to do the following: (a) Check to see whether the Kolmogorov conditions are satisfied. (b) Plot nullclines by hand (or by writing a simple computer program and check to see whether the Poincaré-Bendixson conditions are satisfied. (c) Determine the stability properties of steady states.

Find the polar form of the following equations and determine whether periodic trajectories exist. If so, find their stability. (a) \(\quad \frac{d x}{d t}=\frac{2 \pi y}{1+\left(x^{2}+y^{2}\right)^{1 / 2}}, \quad \frac{d y}{d t}=\frac{2 \pi x}{1+\left(x^{2}+y^{2}\right)^{1 / 2}}\). (b) $$ \begin{aligned} &\frac{d x}{d t}=y+\frac{x}{\left(x^{2}+y^{2}\right)^{1 / 2}}\left[1-\left(x^{2}+y^{2}\right)\right] \\ &\frac{d y}{d t}=-x+\frac{y}{\left(x^{2}+y^{2}\right)^{1 / 2}}\left[1-\left(x^{2}+y^{2}\right)\right] \end{aligned} $$

Morris and Lecar (1981) describe a semiqualitative model for voltage oscillations in the giant muscle fiber of the barnacle. In this system the important ions are potassium and calcium (not sodium). The equations they suggest are the following: $$ \begin{aligned} I &=C \frac{d v}{d t}+g_{\mathrm{L}}\left(v-v_{\mathrm{L}}\right)+g \mathrm{ca} M\left(v-v_{\mathrm{Ca}}\right)+g_{\mathrm{K}} N\left(v-v_{\mathrm{K}}\right) \\ \frac{d M}{d t} &=\lambda_{M}(v)\left[M_{=}(v)-M\right] \\ \frac{d N}{d t} &=\lambda_{v}(v)\left[N_{=}(v)-N\right] \end{aligned} $$ where $$ \begin{aligned} v &=\text { voltage, } \\ M &=\text { fraction of open } \mathrm{Ca}^{2+} \text { channels, } \\ N &=\text { fraction of open } \mathrm{K}^{+} \text {channels. } \end{aligned} $$ (a) Interpret these three equations. For reasons detailed in their papers Morris and Lecar define the functions \(M_{\infty}, \lambda_{s}, N_{*}\), and \(\lambda_{N}\) as follows: $$ \begin{array}{ll} M_{\infty}(v)=\frac{1}{2}\left(1+\tanh \frac{v-v_{1}}{v_{2}}\right), & N_{\infty}(v)=\frac{1}{2}\left(1+\tanh \frac{v-v_{3}}{v_{4}}\right), \\ \lambda_{M}(v)=\lambda_{M} \cosh \frac{v-v_{1}}{2 v_{2}}, & \lambda_{N}(v)=\bar{\lambda}_{N} \cosh \frac{v-v_{3}}{2 v_{4}} \end{array} $$ (b) Sketch or describe the voltage dependence of these functions. $$ \text { Note: } \begin{aligned} \cosh x &=\frac{e^{x}+e^{-x}}{2}, \quad \sinh x=\frac{e^{x}-e^{-x}}{2} \\ \tanh x &=\frac{\sinh x}{\cosh x} \end{aligned} $$ (c) Morris and Lecar consider the reduced \(v N\) system to be an approximation to the whole model. What assumption underlies this approximation? (d) Show that in the reduced \(\mathrm{vN}\) system the variables are constrained to satisfy the following inequalities: $$ \begin{aligned} 0 &

Consider the equations $$ \frac{d x}{d t}=y, \quad \frac{d y}{d t}=-x . $$ (b) By transforming variables, obtain $$ \frac{d r^{2}}{d t}=0, \quad \frac{d \theta}{d t}=-1 . $$ (b) Conclude that there are circular solutions. What is the direction of rotation? Are these cycles stable?
See all solutions

Recommended explanations on Physics Textbooks

Solid State Physics

Read Explanation

Physics of Motion

Read Explanation

Medical Physics

Read Explanation

Scientific Method Physics

Read Explanation

Astrophysics

Read Explanation

Translational Dynamics

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