Abstract
We compare two approaches for determining the normal forms of Hopf bifurcations in retarded nonlinear dynamical systems; namely, the method of multiple scales and a combination of the method of normal forms and the center-manifold theorem. To describe and compare the methods without getting involved in the algebra, we consider three examples: a scalar equation, a single-degree-of-freedom system, and a three-neuron model. The method of multiple scales is directly applied to the retarded differential equations. In contrast, in the second approach, one needs to represent the retarded equations as operator differential equations, decompose the solution space of their linearized form into stable and center subspaces, determine the adjoint of the operator equations, calculate the center manifold, carry out details of the projection using the adjoint of the center subspace, and finally calculate the normal form on the center manifold. We refer to the second approach as center-manifold reduction. Finally, we consider a problem in which the retarded term appears as an acceleration and treat it using the method of multiple scales only.
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Nayfeh, A.H. Order reduction of retarded nonlinear systems – the method of multiple scales versus center-manifold reduction. Nonlinear Dyn 51, 483–500 (2008). https://doi.org/10.1007/s11071-007-9237-y
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DOI: https://doi.org/10.1007/s11071-007-9237-y