Computation, continuation, and bifurcation of torus solutions for dissipative maps and ordinary differential equations

Abstract A periodic solution can be considered as a fixed point of a Poincare map. We shall demonstrate, how we can generalize this fixed-point-equivalence to quasi-periodic solutions with two periods. Thus the quasi-periodic solution or bi-periodic solution will be a fixed point of a more general Poincare map. The stability of the bi-periodic solution is then equivalent to the stability of the fixed point. We demonstrate, how we can follow the fixed point in dependence of any control parameter in the system. Furthermore we demonstrate both how the bi-periodic solution may bifurcate to more complicated solutions and how it may arise as a bifurcation from a periodic solution. Our work is algorithmic in spirit, and our statements are based on numerical evidence.

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