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chaosbook:stability [2010/07/01 05:50]
predrag Listening to Haris Skokos
chaosbook:stability [2010/07/07 01:34]
predrag on Lyapunov exponents
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 :!: For an equilibrium,​ you need to get all eigenvalues with positive real parts, and all negative eigenvalues whose magnitude is less or comparable to the largest expanding eigenvalue. More precise statement is the Kaplan-Yorke criterion (keep adding negative eigenvalues as long as the sum of leading eigenvalues is positive; for much more detail, see the [[:​gtspring2009:​predrag:​blog#​blog|2009-10-29 discussion of Lyapunov vectors]]). You do not need to worry about the remaining (60 thousand!) eigendirections for which the negative eigenvalues are of larger magnitude, as they always contract: nonlinear terms cannot mix them up in such a way that expansion in some directions overwhelms the strongly contracting ones. In plane Couette 20 or so eigenvalues have so far sufficed. ​  --- //​[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-10//​ :!: For an equilibrium,​ you need to get all eigenvalues with positive real parts, and all negative eigenvalues whose magnitude is less or comparable to the largest expanding eigenvalue. More precise statement is the Kaplan-Yorke criterion (keep adding negative eigenvalues as long as the sum of leading eigenvalues is positive; for much more detail, see the [[:​gtspring2009:​predrag:​blog#​blog|2009-10-29 discussion of Lyapunov vectors]]). You do not need to worry about the remaining (60 thousand!) eigendirections for which the negative eigenvalues are of larger magnitude, as they always contract: nonlinear terms cannot mix them up in such a way that expansion in some directions overwhelms the strongly contracting ones. In plane Couette 20 or so eigenvalues have so far sufficed. ​  --- //​[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-10//​
  
-**Haris Skokos to Predrag 2010-07-01** read I. Goldhirsch, P. L. SulemS. A. Orszag, "​Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method,"​ Physica D 27,  (1987) on distinction between Floquet multipliers computed from //J// as opposed to the eigenvalues of //J^T J//.+**Haris Skokos to Predrag 2010-07-01** read I. Goldhirsch, P. L. Sulem and S. A. Orszag, "​Stability and Lyapunov stability of dynamical systems: A differential approach and a numerical method,"​ Physica D 27,  (1987) on distinction between Floquet multipliers computed from //J// as opposed to the eigenvalues of //J^T J//.
  
 **Predrag 2010-07-01** Listening to Haris Skokos, various terms used: **Predrag 2010-07-01** Listening to Haris Skokos, various terms used:
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 //​Generalized alignement vector (GALI)//: take a wedge product of //k// initial vectors, measure the volume defined by them. Norm is the //​SQRT(det(A^TA))//,​ //A// composed of //k// vector rows. For hyperbolic motions all GALIs decay exponentially,​ as Lyapunov exponents. For regular, toroidal motions they fall off as power laws. //​Generalized alignement vector (GALI)//: take a wedge product of //k// initial vectors, measure the volume defined by them. Norm is the //​SQRT(det(A^TA))//,​ //A// composed of //k// vector rows. For hyperbolic motions all GALIs decay exponentially,​ as Lyapunov exponents. For regular, toroidal motions they fall off as power laws.
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 +**2010-07-06 Blaž to Predrag**, on Lyapunov exponents: I am studying your book Chaos: Classical and Quantum and I have a question about the explanation of the eq. (17.32). Isn't //​Lambda1(X0,​t)//​ the leading singular value (not eigenvalue) of the //​J^t(x_0)//,​ according to eq. (4.18)? I'm sorry if this is a false alarm.
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 +**2010-07-06 Predrag to Blaž** My frontal cortex is decoupled at the moment (I'm traversing a series of offices in Kafkastan in order to inherit a grave, etc etc) but you are probably right, and I am planning to rewrite some of the text on the linear stability. Due to my aversion to Lyapunov exponents (who needs them? they are unmeasurable,​ there is no "​gauge"​ you can stick into a chaotic flow and read of any of them), I had at least one wrong statement in ChaosBook on them, and most likely more. I recommend reading Goldhirsch, P. L. Sulem and S. A. Orszag (above) and Trevisan and Pancotti, "​Periodic orbits, Lyapunov vectors, and singular vectors in the Lorenz system",​ //J. Atmos. Sci.// **55** 390 (1998) for now, and if you have any suggestions as what I should rewrite (and how), I'm grateful for the input. Apologies for causing confusion
  
 ~~DISCUSSION~~ ~~DISCUSSION~~
  
chaosbook/stability.txt · Last modified: 2014/12/03 13:36 by predrag