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Volume 8, Number 5, 2018, Pages 1307-1325                                                                DOI:10.11948/2018.1307
Hopf Bifurcation and new singular orbits coined in a Lorenz-like system
Haijun Wang,Xianyi Li
Keywords:Lorenz-like system, singularly degenerate heteroclinic cycle, heteroclinic orbit, Lyapunov-like function.
Abstract:
      We seize some new dynamics of a Lorenz-like system: $\dot{x} = a(y - x)$, \quad $\dot{y} = dy - xz$, \quad $\dot{z} = - bz + fx^{2} + gxy$, such as for the Hopf bifurcation, the behavior of non-isolated equilibria, the existence of singularly degenerate heteroclinic cycles and homoclinic and heteroclinic orbits. In particular, our new discovery is that the system has also two heteroclinic orbits for $bg = 2a(f + g)$ and $a > d > 0$ other than known $bg > 2a(f + g)$ and $a > d > 0$, whose proof is completely different from known case. All the theoretical results obtained are also verified by numerical simulations.
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