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Critical behavior of the Lyapunov exponent in type-III intermittency
Available online at www.sciencedirect.com
Chaos, Solitons and Fractals 36 (2008) 150–156 www.elsevier.com/locate/chaos
Critical behavior of the Lyapunov exponent in type-III intermittency O. Alvarez-Llamoza b
a,b,*
, M.G. Cosenza b, G.A. Ponce
c,d
a Departamento de Fı´sica, FACYT, Universidad de Carabobo, Valencia, Venezuela Centro de Fı´sica Fundamental, Grupo de Caos y Sistemas Complejos, Universidad de Los Andes, Me´rida 5251, Me´rida, Venezuela c Departamento de Fı´sica, Universidad Nacional Auto´noma de Honduras, Honduras d Departamento de Ciencias Naturales, Universidad Pedago´gica Nacional Francisco Moraza´n, Tegucigalpa, Honduras
Accepted 7 June 2006
Communicated by Prof. M.S. El Naschie
Abstract 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 b 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 b varies on the interval 0 6 b < 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 b 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. 2006 Elsevier Ltd. All rights reserved.
Intermittency is one of the main scenarios for the transition to chaos in nonlinear dynamical systems. Intermittent chaos is characterized by the irregular switchings between long periodic like signals, called laminar phase, and comparatively short chaotic bursts. The phenomenon has been extensively studied since the original work of Pomeau and Manneville [1] who classified it into three types according to the local Poincare´ map associated to the system. Type-I intermittency occurs by a tangent bifurcation when the Floquet’s multiplier for the Poincare´ map crosses the circle of unitary norm in the complex plane through +1; type-II intermittency is due to a Hopf’s bifurcation which appears as two complex eigenvalues of the Floquet’s matrix cross the unitary circle off the real axis; and type-III intermittency is associated to an inverse period doubling bifurcation whose Floquet’s multiplier is 1. Although other mechanisms may lead to intermittency, these three cases are the most simple and the most commonly found in low-dimensional systems. Many experimental evidences for these types of intermittency have appeared in the literature [2–4]. The statistical signature of intermittency is usually given by the scaling relations describing the dependence of the average length
*
Corresponding author. Address: Departamento de Fı´sica, FACYT, Universidad de Carabobo, Valencia, Venezuela. E-mail address: [email protected] (O. Alvarez-Llamoza).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.017
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151
(denoted by hli) of the laminar phases with a control parameter (denoted by ) that measures the distance from the bifurcation point. Chaotic dynamics is characterized by the positive sign of the largest Lyapunov exponent (denoted by k), although this quantity is in general more difficult to measure from experimental data than statistical variables such as the average laminar length. Pomeau and Manneville [1] assumed a uniform reinjection probability into the laminar phase and calculated the scaling behavior of both the average laminar length and the Lyapunov exponent for those three types of intermittent chaos. In particular, for type-III intermittency Pomeau and Manneville predicted the relations hli / 1, and k / 1/2, both when ! 0. However, theoretical and numerical works [5–8], as well as recent experiments [9–13], have shown deviations from the Pomeau–Manneville’s prediction for the scaling behavior of the average laminar length for type-III intermittency; they found hli / m, with 1/2 < m < 1. These deviations are attributed to the presence of mechanisms leading to nonuniform probability reinjections. On the other hand, the scaling behavior of the Lyapunov exponent in type-III intermittency has rarely been explored.
Fig. 1. Bifurcation diagrams of the iterates of the map Eq. (1) as a function of the parameter l for two values of the order of the singularity z, showing robust chaos. Type-I or type-III intermittencies appear at the boundaries of the robust chaos intervals. (a) z = 0.5; (b) z = 0.5.
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In this paper we calculate the scaling properties of the Lyapunov exponent at the transition to chaos via type-III intermittency in a family of one-dimensional singular maps. We show that the behavior of the Lyapunov exponent near this transition shows appreciable deviations from the prediction of Pomeau and Manneville. We find that k / b, where the critical exponent b varies continuously in the interval 0 6 b < 1/2 as a function of the map parameters. The Lyapunov exponent can be regarded as an order parameter characterizing the order-chaos transition via type-III intermittency along a critical boundary on the space of parameters of the system. As a simple model of a dynamical system displaying type-III intermittency, we consider the following family of singular maps: xtþ1 ¼ f ðxt Þ ¼ l jxt jz ;
ð1Þ
where t 2 Z, jzj < 1, and l is a real parameter. The exponent z describes the order of the singularity at the origin. These maps are unbounded, i.e., x 2 (1, 1), and exhibit robust chaos on a single interval of the parameter l. A chaotic attractor is said to be robust if, for its parameter values, there exist a neighborhood in the parameter space with absence of periodic windows and the chaotic attractor is unique [14]. Robustness is an important property in applications that require reliable operation under chaos, in the sense that the chaotic behavior cannot be destroyed by arbitrarily small perturbations of the system parameters. Robust chaos is also found in logarithmic maps [15,16]. It should be noted that robust chaos not associated to type-III intermittency has also been discovered in smooth, continuous one-dimensional maps [17]. The Schwarzian derivative of the family of maps Eq. (1) is always positive, i.e., 2 f 000 3 f 00 1 z2 Sf 0 ¼ > 0; ð2Þ 0 2 f f 2x2 for jzj < 1. Thus the family of maps Eq. (1) do not belong to the standard universality classes of unimodal maps and they do not exhibit a sequence of period doubling bifurcations. Instead, the condition Sf > 0 leads to the occurrence of an inverse period doubling bifurcation, where a stable fixed point losses its stability at some critical value of the parameter l to yield robust chaos. Fig. 1 shows the bifurcation diagrams of the iterates of map Eq. (1) as a function of the parameter l for two different values of the singularity exponent z. Two stable fixed points satisfying f(x*) = x* and jf 0 (x*)j < 1 exist for each value of z: x < 0 and xþ > 0, both are seen in Fig. 1. For z 2 (1, 0), the fixed point x becomes unstable at the parameter value z
1
l ðzÞ ¼ jzj1z jzj1z ;
ð3Þ
–
–
Fig. 2. Critical boundaries l(z) and l+(z) of the robust chaos region for the singular maps on the space of parameters (l, z). The thick, dark line indicates the boundary lIII(z) for the transition to chaos via type-III intermittency. The thin, light line corresponds to the boundary for the onset of type-I intermittency.
O. Alvarez-Llamoza et al. / Chaos, Solitons and Fractals 36 (2008) 150–156
153
through an inverse period doubling bifurcation that gives rise to chaos via type-III intermittency, while the fixed point xþ originates from a tangent bifurcation at the value z
1
lþ ðzÞ ¼ jzj1z þ jzj1z ;
ð4Þ
and the transition to chaos at this value of l takes place through type-I intermittency. On the other hand, for z 2 (0, 1) the behaviors of the fixed points are interchanged: x experiences a tangent bifurcation at the parameter value l(z) and a type-I intermittent transition to chaos occurs; while the fixed point xþ undergoes an inverse period doubling bifurcation at the value l+(z), setting the scenario for a type-III intermittent transition to chaos. Fig. 2 shows the critical boundaries l(z) and l+(z) for the transition to chaos. These boundaries separate the region on the parameter plane (l, z) where robust chaos takes place from the region where stable fixed points of the maps Eq. (1) exist. The transition to chaos via type-III intermittency occurs at the critical parameter values lIII(z) = l(z) for z 2 (1, 0), and lIII(z) = l+(z) for z 2 (0, 1). The critical boundary lIII(z) on the space of parameters (l, z) is indicated in Fig. 2. Near the critical boundary lIII(z) within the chaotic region, the map Eq. (1) shows type-III intermittency.
a
λ
b
λ
Fig. 3. Lyapunov exponent k as a function of the parameter l for two values of z, calculated over 5 · 104 iterations after neglecting 5000 iterates representing transient behavior for each value of l. (a) z = 0.5; (b) z = 0.5.
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The width of the interval for robust chaos on the parameter l for a given jzj < 1 is 1
DlðzÞ ¼ lþ l ¼ 2jzj1z :
ð5Þ
Fig. 3 shows the Lyapunov exponent k as a function of the parameter l for the family of maps Eq. (1), for two values of z. The boundaries l(z) and l+(z) correspond to the values k = 0. The Lyapunov exponent is positive on the robust chaos interval Dl(z). The transition to chaos through type-I intermittency is smooth, as seen in Fig. 3. In contrast, the transition to chaos via type-III intermittency is manifested by a discontinuity of the derivative of the Lyapunov exponent at the parameter values corresponding to the critical curve lIII(z). This discontinuity is due to the sudden loss of stability of the fixed point associated to the inverse period doubling bifurcation that occurs at the boundary lIII(z). The Lyapunov exponent can be regarded as an order parameter that characterizes the transition to chaos via type-III intermittency. This transition can be very abrupt as seen in Fig. 3. As in a phase transition, the behavior of the Lyapunov exponent near the boundary lIII(z) within the chaotic region can be characterized by a scaling relation k bðzÞ ;
ð6Þ
where = jl lIIIj 1 and b(z) is a critical exponent expressing the order of the transition. Fig. 4 shows a log–log plot of the Lyapunov exponent vs. for different values of the singularity z, verifying relation Eq. (6). The exponent b(z) is calculated from the slopes of each curve in Fig. 4. Fig. 5 shows the resulting graph of the critical exponent b as a function of z. The values of b range from 0 to values less than 1/2, in a clear deviation from the Pomeau–Manneville’s prediction b = 1/2 [1]. Fig. 5 reveals that the nature of the transition to robust chaos associated to type-III intermittency in this system may change from a first-order, discontinuous phase transition (b = 0) to a second-order, continuous phase transition (b 2 (0, 1)) as the singularity z varies in the interval (1, 1). As z ! 1 or z ! 1, the discontinuity of the singular map becomes more pronounced, and the transition to type-III intermittent chaos becomes more abrupt. Note that the exponent b is not defined at z = 0. The order of the singularity z of the map Eq. (1) determines the order of the tangency of the composed map f (2) corresponding to the local Poincare´ map at the inverse period doubling bifurcation. Varying the singularity z also affects the reinjection mechanism and produces nonuniform probability reinjections into the laminar phase of the iterates of map Eq. (1). This is reflected in the variation of the critical exponent b with z and determines the universality class of the order-chaos transition via type-III intermittency. In conclusion, we have characterized the scaling behavior of the Lyapunov exponent at the transition to chaos in a family of singular maps exhibiting type-III intermittency by means of a critical exponent, as in Eq. (1). The values of the critical exponent b has been shown to lie in the interval 0 6 b < 1/2 as the singularity z of the map varies in (1, 1), in contrast with earlier predictions of the value of this exponent. Consequently, the nature of the singularity of the map
Fig. 4. Log–log plot of the Lyapunov exponent as a function of = jl lcj for several values of z. From top to bottom: z = 0.65 (circles), z = 0.55 (crosses), z = 0.45 (squares), and z = 0.25 (diamonds).
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155
Fig. 5. Critical exponent b as a function of z for the transition to chaos via type-III intermittency. Error bars are indicated.
may change the character of the transition from order to chaos via type-III intermittency from being a discontinuous, first-order transition, to a continuous, second-order transition. It is to be expected that the scaling behavior shown here for the Lyapunov exponent could also be detected in experimental situations with type-III intermittent chaos, where deviations from the scaling properties of the average laminar length have already been found.
Acknowledgements This work was supported by Consejo de Desarrollo Cientı´fico, Humanı´stico y Tecnolo´gico of the Universidad de Los Andes, Me´rida, Venezuela, under Grant No. C-1396-06-05-B. G.A.P. acknowledges support from the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, through the Centro Latinoamericano de Fı´sica, Rio do Janeiro, Brazil.
References [1] Pomeau Y, Manneville P. Intermittent transition to turbulence in dissipative dynamical systems. Commun Math Phys 1980;74:189–97. [2] Pellegrini L, Tablino C, Albertoni S, Biardi G. Different scenarios in a controlled tubular reactor with a countercurrent coolant. Chaos, Solitons & Fractals 1993;3:537–49. [3] Mureithi NW, Paidoussis MP, Price SJ. Intermittency transition to chaos in the response of a loosely supported cylinder in an array in cross-flow. Chaos, Solitons & Fractals 1995;5:847–67. [4] Russo L, Altimari P, Mancusi E, Maffettone PL, Crescitelli S. Complex dynamics and spatio-temporal patterns in a network of three distributed chemical reactors with periodical feed switching. Chaos, Solitons & Fractals 2006;28:682–706. [5] Mayer-Kress G, Haken H. Attractors of convex maps with positive Schwarzian derivative in the presence of noise. Physica D 1984;10:329–39. [6] Kawabe T, Kondo Y. Scaling law of the mean laminar length in intermittent chaos. J Phys Soc Jpn 1996;65:879–82. [7] Kodama H, Sato S, Honda K. Reconsideration of the renormalization-group theory on intermittent chaos. Phys Lett A 1991;157:354–6. [8] Kodama H, Sato S, Honda K. Renormalization-group theory on intermittent chaos in relation to its universality. Prog Theor Phys 1991;86:309–14. [9] Khan AM, Mar DJ, Westervelt RM. Spatial measurements near the instability threshold in ultrapure Ge. Phys Rev B 1992;45:8342–7. [10] Hye W, Rim S, Kim C, Lee J, Ryu JW, Yeom B, et al. Experimental observation of characteristic relations of type-III intermittency in the presence of noise in a simple electronic circuit. Phys Rev E 2003;68:036203.
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[11] Calcavante HL, Rios Leite JR. Averages and critical exponents in type-III intermittent chaos. Phys Rev E 2002;66:026210. [12] Kim CM, Yim GS, Ryu JW, Park Y. Characteristic relations of type-III intermittency in an electronic circuit. Phys Rev Lett 1998;80:5317–20. [13] Ono Y, Fukushima K, Yazaki T. Critical behavior for the onset of type-III intermittency observed in an electronic circuit. Phys Rev E 1995;32:4520–2. [14] Banerjee S, Yorke JA, Grebogi C. Robust chaos. Phys Rev Lett 1998;80:3049–52. [15] Kawabe T, Kondo Y. Intermittent chaos generated by logarithmic map. Prog Theor Phys 1991;86:581–6. [16] Cosenza MG, Gonza´lez J. Synchronization and collective behavior in globally coupled logarithmic maps. Prog Theor Phys 1998;100:21–38. [17] Andrecut M, Ali MK. Robust chaos in smooth unimodal maps. Phys Rev E 2001;64:025203(R).
Chaos, Solitons and Fractals 36 (2008) 150–156 www.elsevier.com/locate/chaos
Critical behavior of the Lyapunov exponent in type-III intermittency O. Alvarez-Llamoza b
a,b,*
, M.G. Cosenza b, G.A. Ponce
c,d
a Departamento de Fı´sica, FACYT, Universidad de Carabobo, Valencia, Venezuela Centro de Fı´sica Fundamental, Grupo de Caos y Sistemas Complejos, Universidad de Los Andes, Me´rida 5251, Me´rida, Venezuela c Departamento de Fı´sica, Universidad Nacional Auto´noma de Honduras, Honduras d Departamento de Ciencias Naturales, Universidad Pedago´gica Nacional Francisco Moraza´n, Tegucigalpa, Honduras
Accepted 7 June 2006
Communicated by Prof. M.S. El Naschie
Abstract 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 b 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 b varies on the interval 0 6 b < 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 b 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. 2006 Elsevier Ltd. All rights reserved.
Intermittency is one of the main scenarios for the transition to chaos in nonlinear dynamical systems. Intermittent chaos is characterized by the irregular switchings between long periodic like signals, called laminar phase, and comparatively short chaotic bursts. The phenomenon has been extensively studied since the original work of Pomeau and Manneville [1] who classified it into three types according to the local Poincare´ map associated to the system. Type-I intermittency occurs by a tangent bifurcation when the Floquet’s multiplier for the Poincare´ map crosses the circle of unitary norm in the complex plane through +1; type-II intermittency is due to a Hopf’s bifurcation which appears as two complex eigenvalues of the Floquet’s matrix cross the unitary circle off the real axis; and type-III intermittency is associated to an inverse period doubling bifurcation whose Floquet’s multiplier is 1. Although other mechanisms may lead to intermittency, these three cases are the most simple and the most commonly found in low-dimensional systems. Many experimental evidences for these types of intermittency have appeared in the literature [2–4]. The statistical signature of intermittency is usually given by the scaling relations describing the dependence of the average length
*
Corresponding author. Address: Departamento de Fı´sica, FACYT, Universidad de Carabobo, Valencia, Venezuela. E-mail address: [email protected] (O. Alvarez-Llamoza).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.017
O. Alvarez-Llamoza et al. / Chaos, Solitons and Fractals 36 (2008) 150–156
151
(denoted by hli) of the laminar phases with a control parameter (denoted by ) that measures the distance from the bifurcation point. Chaotic dynamics is characterized by the positive sign of the largest Lyapunov exponent (denoted by k), although this quantity is in general more difficult to measure from experimental data than statistical variables such as the average laminar length. Pomeau and Manneville [1] assumed a uniform reinjection probability into the laminar phase and calculated the scaling behavior of both the average laminar length and the Lyapunov exponent for those three types of intermittent chaos. In particular, for type-III intermittency Pomeau and Manneville predicted the relations hli / 1, and k / 1/2, both when ! 0. However, theoretical and numerical works [5–8], as well as recent experiments [9–13], have shown deviations from the Pomeau–Manneville’s prediction for the scaling behavior of the average laminar length for type-III intermittency; they found hli / m, with 1/2 < m < 1. These deviations are attributed to the presence of mechanisms leading to nonuniform probability reinjections. On the other hand, the scaling behavior of the Lyapunov exponent in type-III intermittency has rarely been explored.
Fig. 1. Bifurcation diagrams of the iterates of the map Eq. (1) as a function of the parameter l for two values of the order of the singularity z, showing robust chaos. Type-I or type-III intermittencies appear at the boundaries of the robust chaos intervals. (a) z = 0.5; (b) z = 0.5.
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O. Alvarez-Llamoza et al. / Chaos, Solitons and Fractals 36 (2008) 150–156
In this paper we calculate the scaling properties of the Lyapunov exponent at the transition to chaos via type-III intermittency in a family of one-dimensional singular maps. We show that the behavior of the Lyapunov exponent near this transition shows appreciable deviations from the prediction of Pomeau and Manneville. We find that k / b, where the critical exponent b varies continuously in the interval 0 6 b < 1/2 as a function of the map parameters. The Lyapunov exponent can be regarded as an order parameter characterizing the order-chaos transition via type-III intermittency along a critical boundary on the space of parameters of the system. As a simple model of a dynamical system displaying type-III intermittency, we consider the following family of singular maps: xtþ1 ¼ f ðxt Þ ¼ l jxt jz ;
ð1Þ
where t 2 Z, jzj < 1, and l is a real parameter. The exponent z describes the order of the singularity at the origin. These maps are unbounded, i.e., x 2 (1, 1), and exhibit robust chaos on a single interval of the parameter l. A chaotic attractor is said to be robust if, for its parameter values, there exist a neighborhood in the parameter space with absence of periodic windows and the chaotic attractor is unique [14]. Robustness is an important property in applications that require reliable operation under chaos, in the sense that the chaotic behavior cannot be destroyed by arbitrarily small perturbations of the system parameters. Robust chaos is also found in logarithmic maps [15,16]. It should be noted that robust chaos not associated to type-III intermittency has also been discovered in smooth, continuous one-dimensional maps [17]. The Schwarzian derivative of the family of maps Eq. (1) is always positive, i.e., 2 f 000 3 f 00 1 z2 Sf 0 ¼ > 0; ð2Þ 0 2 f f 2x2 for jzj < 1. Thus the family of maps Eq. (1) do not belong to the standard universality classes of unimodal maps and they do not exhibit a sequence of period doubling bifurcations. Instead, the condition Sf > 0 leads to the occurrence of an inverse period doubling bifurcation, where a stable fixed point losses its stability at some critical value of the parameter l to yield robust chaos. Fig. 1 shows the bifurcation diagrams of the iterates of map Eq. (1) as a function of the parameter l for two different values of the singularity exponent z. Two stable fixed points satisfying f(x*) = x* and jf 0 (x*)j < 1 exist for each value of z: x < 0 and xþ > 0, both are seen in Fig. 1. For z 2 (1, 0), the fixed point x becomes unstable at the parameter value z
1
l ðzÞ ¼ jzj1z jzj1z ;
ð3Þ
–
–
Fig. 2. Critical boundaries l(z) and l+(z) of the robust chaos region for the singular maps on the space of parameters (l, z). The thick, dark line indicates the boundary lIII(z) for the transition to chaos via type-III intermittency. The thin, light line corresponds to the boundary for the onset of type-I intermittency.
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through an inverse period doubling bifurcation that gives rise to chaos via type-III intermittency, while the fixed point xþ originates from a tangent bifurcation at the value z
1
lþ ðzÞ ¼ jzj1z þ jzj1z ;
ð4Þ
and the transition to chaos at this value of l takes place through type-I intermittency. On the other hand, for z 2 (0, 1) the behaviors of the fixed points are interchanged: x experiences a tangent bifurcation at the parameter value l(z) and a type-I intermittent transition to chaos occurs; while the fixed point xþ undergoes an inverse period doubling bifurcation at the value l+(z), setting the scenario for a type-III intermittent transition to chaos. Fig. 2 shows the critical boundaries l(z) and l+(z) for the transition to chaos. These boundaries separate the region on the parameter plane (l, z) where robust chaos takes place from the region where stable fixed points of the maps Eq. (1) exist. The transition to chaos via type-III intermittency occurs at the critical parameter values lIII(z) = l(z) for z 2 (1, 0), and lIII(z) = l+(z) for z 2 (0, 1). The critical boundary lIII(z) on the space of parameters (l, z) is indicated in Fig. 2. Near the critical boundary lIII(z) within the chaotic region, the map Eq. (1) shows type-III intermittency.
a
λ
b
λ
Fig. 3. Lyapunov exponent k as a function of the parameter l for two values of z, calculated over 5 · 104 iterations after neglecting 5000 iterates representing transient behavior for each value of l. (a) z = 0.5; (b) z = 0.5.
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O. Alvarez-Llamoza et al. / Chaos, Solitons and Fractals 36 (2008) 150–156
The width of the interval for robust chaos on the parameter l for a given jzj < 1 is 1
DlðzÞ ¼ lþ l ¼ 2jzj1z :
ð5Þ
Fig. 3 shows the Lyapunov exponent k as a function of the parameter l for the family of maps Eq. (1), for two values of z. The boundaries l(z) and l+(z) correspond to the values k = 0. The Lyapunov exponent is positive on the robust chaos interval Dl(z). The transition to chaos through type-I intermittency is smooth, as seen in Fig. 3. In contrast, the transition to chaos via type-III intermittency is manifested by a discontinuity of the derivative of the Lyapunov exponent at the parameter values corresponding to the critical curve lIII(z). This discontinuity is due to the sudden loss of stability of the fixed point associated to the inverse period doubling bifurcation that occurs at the boundary lIII(z). The Lyapunov exponent can be regarded as an order parameter that characterizes the transition to chaos via type-III intermittency. This transition can be very abrupt as seen in Fig. 3. As in a phase transition, the behavior of the Lyapunov exponent near the boundary lIII(z) within the chaotic region can be characterized by a scaling relation k bðzÞ ;
ð6Þ
where = jl lIIIj 1 and b(z) is a critical exponent expressing the order of the transition. Fig. 4 shows a log–log plot of the Lyapunov exponent vs. for different values of the singularity z, verifying relation Eq. (6). The exponent b(z) is calculated from the slopes of each curve in Fig. 4. Fig. 5 shows the resulting graph of the critical exponent b as a function of z. The values of b range from 0 to values less than 1/2, in a clear deviation from the Pomeau–Manneville’s prediction b = 1/2 [1]. Fig. 5 reveals that the nature of the transition to robust chaos associated to type-III intermittency in this system may change from a first-order, discontinuous phase transition (b = 0) to a second-order, continuous phase transition (b 2 (0, 1)) as the singularity z varies in the interval (1, 1). As z ! 1 or z ! 1, the discontinuity of the singular map becomes more pronounced, and the transition to type-III intermittent chaos becomes more abrupt. Note that the exponent b is not defined at z = 0. The order of the singularity z of the map Eq. (1) determines the order of the tangency of the composed map f (2) corresponding to the local Poincare´ map at the inverse period doubling bifurcation. Varying the singularity z also affects the reinjection mechanism and produces nonuniform probability reinjections into the laminar phase of the iterates of map Eq. (1). This is reflected in the variation of the critical exponent b with z and determines the universality class of the order-chaos transition via type-III intermittency. In conclusion, we have characterized the scaling behavior of the Lyapunov exponent at the transition to chaos in a family of singular maps exhibiting type-III intermittency by means of a critical exponent, as in Eq. (1). The values of the critical exponent b has been shown to lie in the interval 0 6 b < 1/2 as the singularity z of the map varies in (1, 1), in contrast with earlier predictions of the value of this exponent. Consequently, the nature of the singularity of the map
Fig. 4. Log–log plot of the Lyapunov exponent as a function of = jl lcj for several values of z. From top to bottom: z = 0.65 (circles), z = 0.55 (crosses), z = 0.45 (squares), and z = 0.25 (diamonds).
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155
Fig. 5. Critical exponent b as a function of z for the transition to chaos via type-III intermittency. Error bars are indicated.
may change the character of the transition from order to chaos via type-III intermittency from being a discontinuous, first-order transition, to a continuous, second-order transition. It is to be expected that the scaling behavior shown here for the Lyapunov exponent could also be detected in experimental situations with type-III intermittent chaos, where deviations from the scaling properties of the average laminar length have already been found.
Acknowledgements This work was supported by Consejo de Desarrollo Cientı´fico, Humanı´stico y Tecnolo´gico of the Universidad de Los Andes, Me´rida, Venezuela, under Grant No. C-1396-06-05-B. G.A.P. acknowledges support from the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, through the Centro Latinoamericano de Fı´sica, Rio do Janeiro, Brazil.
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