hw5_solution

hw5_solution - ME201 Advanced Dynamics(Fall 2007 HW-5...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
ME201 Advanced Dynamics (Fall 2007) HW-5 Solution ( 100 pts ) 1. 30 pts Consider the standard map studied in class. I 0 = I + δ sin(2 πφ ) φ 0 = φ + I + δsin (2 πφ ) Study the change of the phase space of the standard map as δ changes from 0 to 2 π . Specifically, plot the cases of δ = { 0 . 2 / 2 π, 0 . 6 / 2 π, 1 / 2 π, 2 π } Describe your investigation. In the following figure we have iterations of the standard map with varying δ . We find that as δ increases, the existence of new resonant orbits arise. Orbits of different period arise which are surrounded by islands and KAM curves. In between these curves and the islands exist zones of chaos or instability. Note that the figures look different based on initial conditions chosen (i.e. a rational or irrational grid of points). Here we have chosen random initial conditions. Figure 1: Standard map with increasing δ 2. 40 pts Find the criterion for existence of 1/2 period orbit of the standard map (hint: study the second iterate of the map). Compare your analytical criterion with what you see in the numerical phase portrait for small δ < 1 / 2 π . What happens close to that trajectory as δ is increased? ME201 Advanced Dynamics 1 HW5 Solution
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
You may consider the standard map as a function of the form: I ( k + 1) = f ( I ( k ) ( k )) φ ( k + 1) = g ( I ( k ) ( k )) We are looking for conditions in which I ( k + 2) = I ( k ), φ ( k + 2) = φ ( k ), this is the 1/2 period orbit. Note also that this will occur at all points which are the first period orbits: I ( k + 1) = I ( k ), φ ( k +1) = φ ( k ) so we need to identify these first (it is assumed in the problem that we are interested in the orbits that are strictly 1/2 periodic). The conditions for a single period orbit are:
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 4

hw5_solution - ME201 Advanced Dynamics(Fall 2007 HW-5...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online