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MathGroup Archive 1997

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Q: complex Log[z] and linear dynamics

  • To: mathgroup at smc.vnet.net
  • Subject: [mg5789] Q: complex Log[z] and linear dynamics
  • From: calvitti at zmo.ces.cwru.edu
  • Date: Sat, 18 Jan 1997 14:58:40 -0500
  • Organization: Case Western Reserve University
  • Sender: owner-wri-mathgroup at wolfram.com

anyone know why the following compuation fails:

the linear, planar dynamical system dx/dt = A.x where 
A = {{a,-b},{b,a}} with complex eigenvalues a+ib, a-ib is isomorphic to
the complex dynamical system dz/dt = (a+ib).z where z is a complex
number.

the solution to this system is z(t) = Exp[(a+ib)*t]*z(0) where z(0) is
the initial condition....so if z(t) and z(0) are connected by a
solution and the real part of a+ib is nonzero (the origin is a spiral
attractor/repellor depending on the sign of 'a'), then there is a
unique time 't' that should be obtainable by 

                 t = Log[(z(t)/z(0))]/(a+ib)

yet (at least in Mathematica 3.0) with the choice a+ib = -0.5 + i 2.0
and the initial conditions z(0) = 1 i get the correct (real) time only
when z(t) is about t = +/- 1.5 'away from' z(0) and i get complex valued
't' when z(t) is 'farther away' than that....for example, z(4) =
-0.0197 + i 0.1339 and yet i obtain a complex 't' when using the
formula in the previous paragraph. i've checked the modulus of the
complex time but that's also wrong.

is there a singularity in Log[z], is this a numerical problem, or did
i make an error somewhere?...thanks for the info.

 +---------------------------------+
 |          Alan Calvitti          |
 |       Control Engineering       |
 | Case Western Reserve University |
 +---------------------------------+





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