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<- [[:chaosbook]]
====== Local stability ======

(ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#stability|Local stability]])  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-11 12:55//

Enter latest posts at the bottom - flows better. 

:?: Is that Schwarzenegger quote at the beginning of section 4.2 something he actually said?  When did he say that?  //[[jshyatt@gatech.edu|John Hyatt]] 2009-02-10// 

:!: Actually, it was a skit on [[http://prairiehome.publicradio.org/programs/20031129/scripts/guy_noir.shtml|Prairie Home Companion]]  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-10//

:?:  In the tutorial we went through on 1/20/2008, we only generated one eigenfunction for time considerations.  When I ran it on my own, I let the calculation run longer and it generated about 30 eigenfunctions before I wanted to move on.  How many of these eigenfunctions are worth generating for a particular solution and why? //Dustin 2009-01-25//

:!: 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. 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:

//variational equations//

//initial deviation vector//

propagate in time the //tangent map// by the evolution of the //Jacobian//

the mean exponential rate of divergence of deviation vectors is the //(maximal) Lyapunov exponent//

for spectrum, you need to do Gram-Schmidt or similar

//Smaler alignement vector (SALI)//: start with two different vectors, evolve, keep renormalizing the distance; take the minimum of their sum or their difference - defines a kite-shaped area; for chaotic motion goes to zero as the difference of the two leading exponents.

For regular motion the eigenvectors go to torus tangent space, but they do not align and SALI remain practically constant (for 2D maps, ie torus section in the plane, goes to zero by a power law). Can diagnose integrable regions this way; color code chaotic vs integrable regions.

//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.

**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.

**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~~
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