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Onset of chaos in differential delay equations. (English) Zbl 0644.65050
The authors study the onset of chaos in the differential delay equation $\dot x(t)=-x(t)+bx(t-2)/(1+x^{10}(t-2)),$ where $$b>0$$ is a parameter. A numerical technique is developed which allows to study the global behaviour of the flow defined by the equation. Some typical results are: for $$b\approx 1.34$$ a Hopf bifurcation from the steady states $$x_ 1=(b-1)^{0.1}$$ and $$x_ 2=-x_ 1$$ to stable T-periodic (T$$\approx 5.5)$$ orbits takes place. Here the initial data that generates the T-periodic orbit is approximately found by the homotopy continuation method. For $$b\approx 1.56$$ a period doubling bifurcation to stable 2T- periodic orbits occurs. For $$b\approx 1.72$$ the 2T-periodic orbits become unstable and 4T-periodic orbits appear. For 1.725$$\leq b\leq 1.74$$ several period doubling bifurcations take place and orbits of periods 8T, 16T and 32T are observed. Further on, there is a $$b_ 0$$ between 1.74 and 1.77 such that for $$b=b_ 0$$ there is a homoclinic tangency between the stable and unstable manifolds of the T-periodic orbit. For $$b=1.77$$ there is a transverse homoclinic orbit. For $$b<1.8$$ the unstable manifold of the Poincaré map is 1-dimensional. For $$b\approx 1.8$$ a saddle-node bifurcation from the T-periodic orbit takes place and the unstable manifold becomes 2-dimensional. The creation of homoclinic orbits leads to infinitely many periodic orbits and thus to the onset of chaos.
Reviewer: M.M.Konstantinov

##### MSC:
 65L05 Numerical methods for initial value problems 34C25 Periodic solutions to ordinary differential equations 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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