Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 3.7, Problem 1E
Interpretation Introduction

Interpretation:

To show there is always unstable fixed point at x* = 0 for the given equation

Concept Introduction:

To check the stability at any point, differentiate the equation and put the value in it.

Quotient Rule for differentiation (fg)'=f'gg'fg2 and other basic rules of differentiation.

For small value of x predation is very weak, thus the budworm population increases exponentially, when x0

Expert Solution & Answer
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Answer to Problem 1E

Solution:

Instability at x* = 0 is proved below.

Explanation of Solution

Given information:

dx = rx (1- xk) - x21 + x2

The given expression is dimensionless for budworm population growth

dx = rx (1- xk) - x21 + x2

Where τ is the dimensionless time, r is the growth rate, and k is the carrying capacity

To check the stability of the equation at x* = 0, differentiate the equation with respect to x and put x = 0

ddx(rx (1- xk) - x21 + x2)

ddx(rx - rx2k - x21 + x2)=r2rxk2x(1+x2)2

By substituting x = 0 in the above equation

ddx(rx (1- xk) - x21 + x2)=r

Given condition is  r > 0, therefore x = 0 is always an unstable fixed point.

This result is also verified by plotting graph for equation

x˙ = rx (1- xk) - x21 + x2

Nonlinear Dynamics and Chaos, Chapter 3.7, Problem 1E , additional homework tip  1

Nonlinear Dynamics and Chaos, Chapter 3.7, Problem 1E , additional homework tip  2Nonlinear Dynamics and Chaos, Chapter 3.7, Problem 1E , additional homework tip  3

From the above graphs, it is clear that for r > 0, there is always an unstable point at x = 0

Conclusion

For the given type of system, there is always an unstable fixed point at x = 0

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