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Critical behavior of the Lyapunov exponent in type-III intermittency. (English) Zbl 1152.37311
Summary: The critical behavior of the Lyapunov exponent near the transition to robust chaos via type-III intermittency is determined for a family of one-dimensional singular maps. Critical boundaries separating the region of robust chaos from the region where stable fixed points exist are calculated on the parameter space of the system. A critical exponent $$\beta$$ expressing the scaling of the Lyapunov exponent is calculated along the critical curve corresponding to the type-III intermittent transition to chaos. It is found that $$\beta$$ varies on the interval $$0 \leqslant\beta < 1/2$$ as a function of the order of the singularity of the map. This contrasts with earlier predictions for the scaling behavior of the Lyapunov exponent in type-III intermittency. The variation of the critical exponent $$\beta$$ implies a continuous change in the nature of the transition to chaos via type-III intermittency, from a second-order, continuous transition to a first-order, discontinuous transition.

##### MSC:
 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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##### References:
 [1] Pomeau, Y.; Manneville, P., Intermittent transition to turbulence in dissipative dynamical systems, Commun math phys, 74, 189-197, (1980) [2] Pellegrini, L.; Tablino, C.; Albertoni, S.; Biardi, G., Different scenarios in a controlled tubular reactor with a countercurrent coolant, Chaos, solitons & fractals, 3, 537-549, (1993) · Zbl 0801.35140 [3] Mureithi, N.W.; Paidoussis, M.P.; Price, S.J., Intermittency transition to chaos in the response of a loosely supported cylinder in an array in cross-flow, Chaos, solitons & fractals, 5, 847-867, (1995) · Zbl 0928.76047 [4] Russo, L.; Altimari, P.; Mancusi, E.; Maffettone, P.L.; Crescitelli, S., Complex dynamics and spatio-temporal patterns in a network of three distributed chemical reactors with periodical feed switching, Chaos, solitons & fractals, 28, 682-706, (2006) · Zbl 1095.92074 [5] Mayer-Kress, G.; Haken, H., Attractors of convex maps with positive Schwarzian derivative in the presence of noise, Physica D, 10, 329-339, (1984) · Zbl 0602.58026 [6] Kawabe, T.; Kondo, Y., Scaling law of the Mean laminar length in intermittent chaos, J phys soc jpn, 65, 879-882, (1996) [7] Kodama, H.; Sato, S.; Honda, K., Reconsideration of the renormalization-group theory on intermittent chaos, Phys lett A, 157, 354-356, (1991) [8] Kodama, H.; Sato, S.; Honda, K., Renormalization-group theory on intermittent chaos in relation to its universality, Prog theor phys, 86, 309-314, (1991) [9] Khan, A.M.; Mar, D.J.; Westervelt, R.M., Spatial measurements near the instability threshold in ultrapure ge, Phys rev B, 45, 8342-8347, (1992) [10] Hye, W.; Rim, S.; Kim, C.; Lee, J.; Ryu, J.W.; Yeom, B., Experimental observation of characteristic relations of type-III intermittency in the presence of noise in a simple electronic circuit, Phys rev E, 68, 036203, (2003) [11] Calcavante, H.L.; Rios Leite, J.R., Averages and critical exponents in type-III intermittent chaos, Phys rev E, 66, 026210, (2002) [12] Kim, C.M.; Yim, G.S.; Ryu, J.W.; Park, Y., Characteristic relations of type-III intermittency in an electronic circuit, Phys rev lett, 80, 5317-5320, (1998) [13] Ono, Y.; Fukushima, K.; Yazaki, T., Critical behavior for the onset of type-III intermittency observed in an electronic circuit, Phys rev E, 32, 4520-4522, (1995) [14] Banerjee, S.; Yorke, J.A.; Grebogi, C., Robust chaos, Phys rev lett, 80, 3049-3052, (1998) · Zbl 1122.37308 [15] Kawabe, T.; Kondo, Y., Intermittent chaos generated by logarithmic map, Prog theor phys, 86, 581-586, (1991) [16] Cosenza, M.G.; González, J., Synchronization and collective behavior in globally coupled logarithmic maps, Prog theor phys, 100, 21-38, (1998) [17] Andrecut, M.; Ali, M.K., Robust chaos in smooth unimodal maps, Phys rev E, 64, 025203(R), (2001)
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