LIMIT CYCLES IN 3D LOTKA–VOLTERRA SYSTEMS APPEARING AFTER PERTURBATION OF HOPF CENTER

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We study the system ẋ = x(y+2z+(15/2η2)u), ẏ = y(x-2z-(7/2η2)u), ż = -z(x+y+(4/η2)u), u = x+y+z-1, and its two-parameter perturbations. We show that before perturbation there exists a one-parameter family of periodic solutions obtained via a nondegenarate Hopf bifurcation and after perturbation there remains at most one limit cycle of small amplitude and bounded period. Moreover, we found that a secondary Hopf bifurcation to an invariant torus occurs after the perturbation.