Chapter 9.3, Problem 1E

Interpretation Introduction

Interpretation:

To find why isn’t the quasi periodic system ${\dot{\text{θ}}}_{\text{1}}{\text{ = ω}}_{\text{1}}$, ${\dot{\text{θ}}}_2{\text{ = ω}}_2$ considered chaotic, if the trajectories of the system are not periodic. To find the largest Liapunov exponent for the system, without using a computer.

Concept Introduction:

• ➢ A chaotic system should be sensitive to the initial conditions.

• ➢ Two trajectories in phase space with initial separation ${\overrightarrow{\text{δ}}}_0$ diverge at a rate given by $\left|\overrightarrow{\text{δ}}\left(\text{t}\right)\right|\text{ = }\left|{\overrightarrow{\text{δ}}}_{\text{0}}\right|{\text{ e}}^{\text{- λt}}$

  Here, $\text{λ}$ is the Liapunov exponent.

Answer to Problem 1E

Solution:

• a) The given quasi periodic system is not chaotic because it’s not sensitive to initial conditions.

• b) The largest Liapunov exponent is $\text{0}$.

Explanation of Solution

• a) The quasi periodic system is given as

  ${\dot{\text{θ}}}_{\text{1}}{\text{ = ω}}_{\text{1}}$, ${\dot{\text{θ}}}_2{\text{ = ω}}_2$

  This system is periodic, but it is not sensitive to the initial conditions.

  One of the fundamental requirements of chaos is that it should be sensitive to the initial conditions.

  Thus, it can be concluded that the given quasi periodic system is not chaotic.

• b) Consider ${\dot{\text{θ}}}_{\text{1}}{\text{ = ω}}_{\text{1}}$,

  The equation can be rewritten as

  $\frac{{\text{dθ}}_{\text{1}}}{\text{dt}}{\text{ = ω}}_{\text{1}}$

  ${\text{dθ}}_{\text{1}}{\text{ = ω}}_{\text{1}}\text{ dt}$

  Integrating both sides,

  ${\displaystyle \int {\text{dθ}}_{\text{1}}}\text{ = }{\displaystyle \int {\text{ω}}_{\text{1}}\text{ dt}}$

  ${\text{θ}}_{\text{1}}{\text{ = ω}}_{\text{1}}{\text{t + C}}_1$

  Similarly, consider ${\dot{\text{θ}}}_2{\text{ = ω}}_2$,

  The equation can be rewritten as

  $\frac{{\text{dθ}}_2}{\text{dt}}{\text{ = ω}}_2$

  ${\text{dθ}}_2{\text{ = ω}}_2\text{ dt}$

  Integrating both sides,

  ${\displaystyle \int {\text{dθ}}_2}\text{ = }{\displaystyle \int {\text{ω}}_2\text{ dt}}$

  ${\text{θ}}_2{\text{ = ω}}_2{\text{t + C}}_2$

  For the two trajectories in phase space with initial separation ${\overrightarrow{\text{δ}}}_0$, diverge at a rate given by

  $\left|\overrightarrow{\text{δ}}\left(\text{t}\right)\right|\text{ = }\left|{\overrightarrow{\text{δ}}}_{\text{0}}\right|{\text{ e}}^{\text{- λt}}$

  Here, $\text{λ}$ is the Liapunov exponent.

  Introducing the error $\overrightarrow{\text{δ}}\left(\text{t}\right)$ to the initial conditions makes a blob around the initial conditions, but the blob moves together and doesn’t distort.

  The system can be interpreted as movement of a dot on a sphere. ${\text{ω}}_{\text{1}}$ controls how fast the dot moves in the latitudinal direction and ${\text{ω}}_2$ controls how fast the dot moves in the longitudinal direction.

  $\left|\overrightarrow{\text{θ}}\left(\text{t}\right)\text{ + }\overrightarrow{\text{δ}}\left(\text{t}\right)\text{ - }\overrightarrow{\text{θ}}\left(\text{t}\right)\right|\text{ = }\left|\overrightarrow{\text{δ}}\left(\text{t}\right)\right|$

  $\text{ = }\left|\left({\overrightarrow{\text{δ}}}_{\text{1}}{\text{, }\overrightarrow{\text{δ}}}_{\text{2}}\right)\right|$

  $\text{ = }\left|\left({\overrightarrow{\text{δ}}}_{\text{1}}{\text{, }\overrightarrow{\text{δ}}}_{\text{2}}\right)\right|{\text{ e}}^{\text{0 t}}$

  Hence $\text{λ = 0}$.

  Thus, the largest Liapunov exponent is $\text{0}$.