Damped oscillator

Damped oscillator

We consider a damped oscillator (a pendulum in the linear approximation), ruled by the differential equation

We naively discretized this equation (using a sampling time of one second) with

Choosing omega=1/8 and various values for k, and choosing initial condition theta_p=8 and theta_v=0, we have the following system oscillator_i0.lts.

• The analysis results can be seen here.

  l

  #e

  k=3/4

  k=1/4

  k=1/8

  k=1/16

  k=1/64

  k=0

  running time

  file

  thetap

  thetav

  thetap

  thetav

  thetap

  thetav

  thetap

  thetav

  thetap

  thetav

  thetap

  thetav

  1

  4

  [0,8.0]

  [-0.125,0]

  [0,8.264]

  [-0.248,0]

  [0,10.217]

  [-0.402,0]

  [-1.818,8.262]

  [-0.625,0.128]

  [-6.580,8]

  [-0.956,0.785]

  [-infty,infty]

  [-infty,infty]

  oscillator_i0_l1.ps

  2

  8

  [0,8.0]

  [-0.125,0]

  [0,8.184]

  [-0.233,0]

  [0,9.286]

  [-0.394,0]

  [-1.680,8.090]

  [-0.600,0.123]

  [-6.580,8]

  [-0.924,0.760]

  [-infty,infty]

  [-infty,infty]

  oscillator_i0_l2.ps

  4

  32

  [0,8.0]

  [-0.125,0]

  [0,8.0]

  [-0.231,0]

  [0,8.08]

  [-0.393,0]

  [-1.648,8.0]

  [-0.592,0.122]

  [-6.580,8]

  [-0.915,0.751]

  [-infty,infty]

  [-infty,infty]

  oscillator_i0_l4.ps

  l	:	corresponds to "-log l" option
  #e	:	number of template expressions on the coefficients of Jordan normal form
  		(related to the number "l" and the number of different coefficient of the Jordan normal form)
  k	:	damping factor
  thetap	:	minimum and maximum angular position in discrete trajectories
  thetav	:	minimum and maximum angular speed in discrete trajectories

• If we now choose the initial condition theta_p=8 and theta_v=8, we have these results:

  l

  #e

  k=3/4

  k=1/4

  k=1/8

  k=1/16

  k=1/64

  k=0

  running time

  file

  thetap

  thetav

  thetap

  thetav

  thetap

  thetav

  thetap

  thetav

  thetap

  thetav

  thetap

  thetav

  4

  32

  [0,16.0]

  [-4.125,8]

  [0,16.099]

  [-0.25,8]

  [0,31.416]

  [-1.560,8]

  [-8.742,42.555]

  [-3.144,8]

  [-49.598,60.421]

  [-6.901,8]

  [-infty,infty]

  [-infty,infty]

  oscillator_i1_l4.ps

Some comments:

• If k=0, the discretized version diverges. Indeed the Jordan matrix have an eigenvalue with a norm strictly greater than 1.
• When k is larger, the oscillations become smaller both in magnitude and maximal speed.
• When the initial speed is zero, the analysis discovers that when k>omega, the position cannot be negative.

Damped oscillator