Chapter 9.5, Problem 6E

Interpretation Introduction

Interpretation:

Using numerical integration, find three different initial conditions, for which there is transient chaos and three, for which there is no transient chaos. State the rule that predicts whether initial condition produces transient and non-transient chaos.

Concept Introduction:

• ➢ Lorenz equations

  $\begin{array}{l}\dot{\text{x}}=\sigma (\text{y}-\text{x})\\ \dot{\text{y}}=\text{rx}-\text{y}-\text{xz}\\ \dot{\text{z}}=\text{xy}-\text{bz}\\ \text{Here σ, r, b > 0}\end{array}$

  The solution of Lorenz equations oscillates irregularly for wide range of parameters, never exactly repeating, but always remaining in the bounded region of phase space.

• ➢ Chaos: It is an aperiodic long-term behavior in a deterministic system that sensitively depends upon the initial conditions.

• ➢ Transient Chaos: It is the characteristic behavior of nonlinear systems; it can be identified by the nature of trajectories. In the phase space, the trajectories behave chaotically for some time before meeting an external attractor; then it is said that the system has transient chaos. It occurs due to the presence of the chaotic saddle point in the phase space.

• ➢ In transient chaos, the trajectory is seen as it leaves the strange attractor, but after spiraling in the left region, it comes to the right region and spirals down towards the stable fixed point C⁺.

• ➢ The chaotic saddle can be created or destroyed by varying the parameter.

• ➢ Transient chaos is also called metastable chaos or pre-turbulence.