Fig. 4

Behavior of Eq. (4) as \(\tau=\tau _{\theta}/\tau_{w}\) changes. \(\mathbf{x}^{(1)}=(1,0)\), \(\mathbf{x}^{(2)}=A(\cos(1),\sin(1))\), \(\rho=1/2\). (A) \(A=1\), so that both fixed points have the same stability properties. Curves show maximum and minimum value of \(v_{1}\) or \(v_{2}\). Red line shows stable equilibrium, black, unstable equilibrium, green circle show stable limit cycles and blue unstable. Two points are marked by black filled circles and the Hopf bifurcation is depicted as HB. Apparent homoclinic is labeled HC. (B) Symmetric pairs of limit cycles for two different values of τ on the curves in (A) projected on the \((v_{1},v_{2})\) plane. (C) \(A=1.5\) so that the stability of the two equilibria is different. The maximum value of \(V_{2}\) is shown as τ varies. Upper curves (2) bifurcate from \((v_{1},v_{2},\theta )=(0,2,2)\) and lower curves (1) from \((2,0,2)\). Colors as in panel A. LP denotes a limit point and Hs denotes a Hopf bifurcation for the symmetric equilibrium \((1,1,1)\). (D) Orbits taken from the two bifurcation curves in (C) projected onto the \((v_{1},v_{2})\) plane