Classification of eight types of critical curve for coupled periodic oscillators. The model we used in (a)–(d) is Lorenz oscillator [ , , ] with the parameter set ( , , ), and the model we used in (e)–(h) is Landau–Stuart oscillator [ , ] with the parameter . Under these parameters only one periodic attractor can be found.
Stability analysis of complete periodic synchronization with the eight critical curves in Fig. 1 . We use S, P, QP, LC, HC1, and HC2 to denote different desynchronous behaviors (see text for details). In simulation, usually initial states very close to the synchronous state are chosen, .
The same as Fig. 2 for .
[(a) and (b)] The spatial orders after the shortest wavelength bifurcation for the steady state and the periodic state, respectively. [(c)–(h)] The spatial orders after Hopf bifurcation from synchronous period for the quasiperiodic state, the low-dimensional chaotic state, the high-dimensional chaotic state with one positive and two zero Lyapunov exponents, the high-dimensional chaotic state with two positive and one zero Lyapunov exponents, the periodic state, and the chaotic state with the instability of the mode, respectively. (a) , [Fig. 3(f) ], (b) , [Fig. 3(c) ], (c) , [Fig. 3(b) ], (d) , [Fig. 3(b) ], (e) , [Fig. 3(b) ], (f) , [Fig. 3(d) ], (g) , [Fig. 3(b) ], and (h) , [Fig. 2(d) ].
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