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For a synchronized ODE system,I wanted to know if there is a program code in MATLAB available for plotting the conditional Lyapunov exponent. For example, synchronization of identical Rossler system with a coupling term $F$ present in both the response and the driver.The $F$ is not a coefficient of any variable. This is where I have a problem.

  • Do we have to include the coupling in the jacobian matrix and if yes how?
  • How to plot the Lyapunov exponent as a function of different values of $F$?

Thank you for any help.

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    $\begingroup$ Can you explain what a "synchronized ODE system" is? $\endgroup$ – David Ketcheson Jan 19 '12 at 6:04
  • $\begingroup$ Well,in the case of Rossler system if the response asymptotically follows the dynamics of the driver in the presense of an external coupling term F,then the system (response+driver) is said to be identically synchronized. $\endgroup$ – SKM Jan 22 '12 at 8:02
  • $\begingroup$ Can you provide the equations or a link? $\endgroup$ – David Ketcheson Jan 22 '12 at 13:32
  • $\begingroup$ The synchronization has to be done,i mean the system has to be constructed with the appropriate controller.I found this link for calculating lyapunov exponent for the unsynchronized case :mathworks.com/matlabcentral/fileexchange/… . I am trying to modify this code for the synchronized case. Upon synchronization,we take into consideration only the response system. So,in the program we shall have the sameset of equations but one of the equation will have an external forcing term say 5*F where F is a sine wave(for my case).How to calculate J? $\endgroup$ – SKM Jan 22 '12 at 20:40
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I don't know of any MATLAB code, but I believe the LyapOde software linked at the bottom of Paul Bryant's page may implement what you want. It is written in C. From its documentation:

The software also supports the calculation of Conditional Lyapunov Exponents or CLEs (see Pecora et al, Chaos Vol. 7, No. 4, (1997), 520). The CLEs are the Lyapunov exponents of a response system that receives unidirectional synchronizing stimulus from an identical drive system. Three types of stimulus are considered: continuous coupling, periodic impulse coupling, and substitution (full or fractional). More details are given in the discussion section below. The synchronizing stimulus can be applied to one or more of the variables. When any of the CLEs are positive or zero, they can be calculated in both synchronized and unsynchronized mode which typically give differing results. In the latter case the drive and response systems are allowed to freely deviate away from synchronization. Currently the software includes the equations for the standard Lorenz model, the Rossler system, the Rossler hyperchaos system, the 5-variable version of the Lorenz cyclic 1996 model, the Colpitts oscillator, and the driven Van der Pol oscillator.

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  • $\begingroup$ Thank you so much for taking the time to search.Great resource and useful. $\endgroup$ – SKM Jan 23 '12 at 6:30
  • $\begingroup$ both links are dead $\endgroup$ – wdg May 26 '15 at 14:34
  • $\begingroup$ Just confirmed that both links are now, in fact, alive! $\endgroup$ – m4r35n357 Oct 28 '20 at 10:50

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