Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
Question
Book Icon
Chapter 8.1, Problem 14E
Interpretation Introduction

Interpretation:

Consider the system equations are x˙1 = - x1+ F(I - bx2), x˙2= - x2 + F(I - bx1) where the gain function is given by F(x) = 11 + e-x, I is the strength of the input signal, and b is the strength of the mutual antagonism.

To sketch the phase plane for various values of I and b(both positive).

To show that the symmetric fixed point x1* = x2* = x* is always a unique solution.

To show that at a sufficiently large value of b, the symmetric solution loses stability at a pitchfork bifurcation and also find type of pitchfork bifurcation.

Concept Introduction:

A phase plane is defined as the graphical representation of the differential equation which represents the limit cycle of the defined system equation.

A phase portrait is defined as the geometrical representation of the trajectories of the dynamical system in the phase plane of the system equation. Every set of the original condition is signified by a different curve or point in the phase plane.

Blurred answer
Students have asked these similar questions
(b) Let I[y] be a functional of y(x) defined by [[y] = √(x²y' + 2xyy' + 2xy + y²) dr, subject to boundary conditions y(0) = 0, y(1) = 1. State the Euler-Lagrange equation for finding extreme values of I [y] for this prob- lem. Explain why the function y(x) = x is an extremal, and for this function, show that I = 2. Without doing further calculations, give the values of I for the functions y(x) = x² and y(x) = x³.
Please use mathematical induction to prove this
L sin 2x (1+ cos 3x) dx 59
Knowledge Booster
Background pattern image
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage