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 | +---------------------------------+