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Re: Poincare section for double pendulum

  • To: mathgroup at smc.vnet.net
  • Subject: [mg120596] Re: Poincare section for double pendulum
  • From: JUN <noeckel at gmail.com>
  • Date: Sat, 30 Jul 2011 06:02:05 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <j0u7jr$fbg$1@smc.vnet.net>

On Jul 29, 5:04 am, gal bevc <gal.b... at gmail.com> wrote:
> Hello,
>
> I'm a relatively new user of Mathematica, who doesn't have much of
> programming skills. For my undergraduate assingment I must analyze chaotic
> motion of double pendulum.
>
> Until now i have got system of differential equations for equations of
> motion for double pendulum(i have x''[t]=function(t) and
> y''[t]=function(t)). System of differential equations can be solved for 4
> inital conditions, x[0],y[0],x'[0] and y'[0]. With using function NDSolvei
> got functions of angles and angular velocities for upper and lower pendulum
> with respect to time, x[t],x'[t],y[t] and y'[t].
>
> To get a poincare section of double pendulum, i have to record position of
> y[t] and y'[t] whenever x[t] is equal to zero and the velocity of x'[t] is a
> positive number. In the end I must get some sort of phase diagram y[t] and
> y'[t].
> Because this is a Hamilton non-dissipative system, inital energy of the
> system is a constant of time and initial energy is a function of initial
> conditions. To get a real poincare diagram i must repeat the procedure
> described above for different initial conditions, but for the same energy
> level. I need mathematica to use some random numbers for initial conditions
> in a way that the initial energy of the system stays the same. So i must
> repeat procedure for poincare section(surface of section) for let's say 50
> different initial conditions and then display all results in one y[t],y'[t]
> diagram.
> Hope that someone can help me.
>
> Thank you,
> Gal Bevc

Hi,
one way of doing this is by using StepMonitor during the numerical
solution to look for zero-crossings of the variable x[t]:

(a)
Define an empty list, say
zeros = {};
This will be used to collect the zeros in your time interval.

(b)
Define an auxiliary variable "lastX" and set it equal to the initial
value of x (say xInitial),
lastX = xInitial;

(c)
In your NDSolve block, add the following:
...,StepMonitor:>(If[x[t]*lastX<0,AppendTo[zeros,t]];lastX=x[t])
This tests for zero crossings of x[t] between the current time step
and the previous one.

(d)
When NDSolve is done, the list "zeros" contains a set of approximate t
values with x[t]=0 that you can then use to refine using FindRoot.
Let's say your solution (in the form of rules {x->..., y->...}) is
stored in "sol", then say
Map[FindRoot[Evaluate[x[t]/.sol],{t,#}]&,zeros]

That should give you the values of t at which you would then evaluate
y[t] and y'[t] to make the points for the Poincare section.

Jens






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