Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780429972195
Author: Steven H. Strogatz
Publisher: Taylor & Francis
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Chapter 9.2, Problem 5E
Interpretation Introduction

Interpretation:

To give a partial sketch of the stability diagram, using the analytical results obtained about bifurcations in the Lorenz equations. To plot the pitchfork and Hopf bifurcation curves in the (σ, r) parameter plane, by assuming b = 1, and σ, r0.

Concept Introduction:

  • ➢ The Lorenz equations are given as

    x˙=σ(y - x),

    y˙=rx - xz - y,

    z˙=xy - bz

    Here, σ, r, b > 0.

  • ➢ The Jacobian matrix is given by

    A = (x˙xx˙yx˙zy˙xy˙yy˙zz˙xz˙yz˙z)

  • ➢ The Eigen value λ can be calculated using the characteristic equation

    |(A - λI)| = 0

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