Type-II Intermittency in a Class of Two Coupled One-Dimensional Maps

The paper shows how intermittency behavior of type-II can arise from the coupling of two one-dimensional maps, each exhibiting type-III intermittency. This change in dynamics occurs through the replacement of a subcritical period-doubling bifurcation in the individual map by a subcritical Hopf-bifurcation in the coupled system. A variety of different parameter combinations are considered, and the statistics for the distribution of laminar phases is worked out. The results comply well with theoretical predictions. Provided that the reinjection process is reasonably uniform in two dimensions, the transition to type-II intermittency leads directly to higher order chaos. Hence, this transition represents a universal route to hyperchaos.

INTRODUCTION Many interesting phenomena have been discov- ered by studying the dynamics of weakly coupled, identical one-dimensional (l-D) maps.In particu- lar, coupled logistic [1][2][3] and coupled circle maps [4] have been extensively investigated from the point of view of the persistence of their bifurcation structures under the stabilizing or destabilizing influences of the mutual interaction.
It was observed by Froyland [1], for instance, that the period-doubling route to chaos may be *Corresponding author.233 replaced by a quasiperiodic transition when two identical logistic maps are coupled symmetrically.The reason for this is that at an early stage of the period-doubling cascade, a Hopf-bifurcation on the antisymmetric solution appears instead of period doubling.A similar replacement was found experimentally by Van Buskirk and Jeffries [5] who studied driven electronic resonance circuits consisting of an inductance in series with a pnjunction acting as a nonlinear capacitance.When varying the drive voltage the single resonator showed a period-doubling transition to chaos, whereas resistively coupled resonance circuits exhibited a quasiperiodic transition.A qualitative explanation of this phenomenon was provided by Erastova and Kuznetsov [6].Based on detailed analytic work, Reick and Mosekilde  [7] were subsequently able to demonstrate that the behav- ior is generic to symmetrically coupled period- doubling systems, and that it is robust towards slight deviations from complete symmetry.
In parallel studies, Fujisaka and Yamada [8] have shown how two identical chaotic oscillators under variation of the coupling strength can attain a state of chaotic synchronization, and a variety of applications of this phenomenon for chaos sup- pression, for monitoring of dynamical systems, and for different communication purposes have been suggested [9, 10].Most recently, the forma- tion of riddled basins of attraction [11][12][13] and the associated phenomenon of on-off intermittency in systems of two coupled 1-D maps [14,15] have attracted significant interest.
By contrast to the various bifurcation scenarios described above, the dynamics of coupled 1-D maps does not appear to have been as thoroughly analyzed in the cases where the individual map follows the intermittency route to chaos.In their classic paper on intermittent transition to tur- bulence in dissipative systems, Pomeau and Manneville [16] introduced three different types of intermittency, using a classification that reflects the types of local bifurcation by which a periodic orbit can lose its stability.Type-I and type-III intermittencies are associated with the passage of a real Floquet multiplier through +1 and 1, respectively.These types of intermittency have been theoretically explained in terms of simple 1-D maps [17], and they have also been widely demonstrated in the laboratory [18,19].
Type-II intermittency is essentially a two-dimen- sional phenomenon.It arises when a periodic orbit loses its stability in a subcritical Hopf-bifurcation, provided that a global reinjection mechanism exists that can send the trajectory back into the neighborhood of the destabilized periodic orbit.This type of intermittency has only been observed in a limited number of experiments [20,21], and it appears that simple, theoretical examples are lack- ing.Most of the existing models are high dimen- sional, and the strongly nonuniform reinjection processes that they produce cause the experimental (or numerical) results for the scaling behavior to deviate from simple theoretical calculations [22].
The purpose of the present paper is to show how type-II intermittency can arise in a class of weakly coupled 1-D maps which individually exhibits type-III intermittency.In much the same way as the coupling of two logistic maps can lead to the appearance of a Hopf-bifurcation instead of a period doubling, the coupling of the two maps in the present case leads to the replacement of a subcritical period doubling by a subcritical Hopfbifurcation.We demonstrate that within this class of maps, type-II intermittency is a generic phenom- enon, in that it occurs for various types of coupling, for large ranges of parameter values, and for arbitrarily small coupling constants.The statistics for the distribution of laminar phases is worked out, and our numerical results compare favorably with theoretical predictions.Provided that reinjection near the unstable two-dimensional manifold is reasonably uniform, there is a tendency for the transition to type-II intermittency to be associated with the abrupt generation of hyperchaos.with the bifurcation parameters # and b > 0. For b=0, the map represents the normal form of a subcritical period-doubling bifurcation with a stable fixed point x*= 0 surrounded by an unstable period-2 orbit Xl,2-nt-V/ for # < 0, and with an unstable fixed point for positive values of #.
Combined with the reinjection mechanism pro- vided by the exponential factor with b > 0, the subcritical period doubling produces the typical behavior of type-III intermittency.
We have previously studied this phenomenon in detail and found that the strongly nonuniform reinjection process produced by the map (2.1) causes an anomaly in the statistics of the laminar phases [23].This nonuniformity arises because the singularity associated with the maximum of the map controls the behavior around the minimum reinjection distance.To allow a better regulation of the reinjection process, the map may be modi- fied to read 3)e-eX: Around the fixed point, this map has precisely the same behavior as (2.1).However, by varying h' one can shift the position of the most probable reinjection point relative to the unstable fixed point and in this way achieve a more homogeneous reinjection process.
Figure shows the form of the map (2.2) for b =0.14, # =0.005, and 7 =0.02.Indicated on the figure is the critical point x. 3.167 together with its first and second iterates, f(Xc) and f2(Xc) respectively.With the assumed parameter values, f(xc)---8.534falls to the right of the point f(Xl) 7. 546 (=-Xo) where the map crosses the horizontal axis.Hence, f2(xc)> 0, and reinjection of the second iterate of the critical point directly into the unstable fixed point does not occur.
Together with a point x2 (f(x2)=f(Xl)) to the left of x, xl denotes the point whose second iterate falls in the unstable fixed point.For the reinjection process to be reasonably uniform, the average (numerical) slope of the map in Xl and x2 (weighted by the respective visiting frequencies) times the slope in f(xl) should be close to 1.This condition is approximately satisfied with the above parameters.
where the functions m (xn, yn) and n (x, y) specify the form of the coupling, and c determines its strength.In order for the coupled system to show type-II intermittency, the linear terms of the coupl- ing functions in the neighborhood of the fixed point must have a form such that the subcritical period doubling in the individual map is replaced by a subcritical Hopf-bifurcation in the coupled system.
To find the condition for this to occur we determine the Jacobian matrix at the fixed point (x, y) (0,0) and calculate its eigenvalues Al,2 =--(1 +/z) + + + Ox On) 20y + 4-0--y-y x x 0 m 0 n ] . (2.5)For symmetric coupling ((Om/Ox)=(On/Oy) and Under these conditions, the eigenvalues are always real, and a Hopf-bifurcation cannot occur.The criterium for the eigenvalues to become complex is OmOn (OmOn) 2 4 --y -x > Ox Oy (2.7) Hence, (Om/Oy) and (On/Ox) must have opposite signs.For an antisymmetric coupling with (On/ Ox) (Om/Oy) a and (Om/Ox) and the eigenvalues are complex conjugate as long as a 2 d2.
The bifurcation point, i.e., the value of # for which IA,21--1 is given by 2 (2.9) As a specific example in the following numerical study we shall take m (xn, y) y and n (x, y)- -xn corresponding to a=-I and d=0.This gives the 2-D map e-bx2  13)   To show that the Hopf-bifurcation is subcritical we introduce the complex notation zn x, / iyn Izle in, (2.14) multiply (2.11) by i, and add it to (2.10).When retaining terms up to third order in x and y this gives
Except for the slight shift in the bifurcation point caused by the coupling, this is precisely the same condition that determines whether the period-doubling bifurcation in the individual map is subcritical or not [23].Hence, for a subcriti- cal Hopf-bifurcation to occur we must have b< (2.17) (1/2)e 2 + -y In the numerical simulations to be discussed below typical parameter values are b=0.14, , 0.02, and c 0.001, and the condition (2.17) will always be satisfied.

NUMERICAL RESULTS
Figure 2 shows an example of the temporal behavior of the coupled map system over a period corresponding to 100,000 iterations.To illustrate the long laminar phases that occur immediately above the subcritical Hopf-bifurcation we have assumed #=0.0001.The vertical axis shows the variation of sign(xn)[X/l.The nonlinear scale serves to accentuate the variations during the laminar phases relative to the much larger excur- sions that take place during the turbulent bursts.The spiralling nature of the iteration process characteristic for type-II intermittency is revealed through the relatively slow oscillations that grow in size until the system bursts into a turbulent phase.
Figure 3 shows a phase plot in which we have followed the behavior of the system through 240 reinjection processes.Only each second iterate is displayed.The figure again illustrates the spiralling character of the iteration process in the neighborhood of the unstable fixed point.At the same time, the figure gives an impression of how the reinjec- tion points are distributed.
Figure 4 provides a much more detailed picture of the distribution of reinjection points in phase space.For the assumed parameter values, this distribution is characterized by a relatively homo- geneous region around the unstable fixed point.Iterates n FIGURE 2 Temporal behavior of the coupled map system immediately above the subcritical Hopf-bifurcation.One of the laminar phases is seen to last for nearly 30,000 iterations.Note that the scale on the vertical axis is nonlinear.Only each second iteration is shown.
-0.1 0 0.1 0.2 Xn FIGURE 3 Phase plot illustrating the spiralling character of the iteration process for type-II intermittency.The figure was obtained for #=0.0001, b-0.186, ,=0.13, and c-0.001.This region arises through the crossing bands extending along the vertical and horizontal axes and bounded by the critical curves of the coupled map system [24,25].In the limit of low coupling, the width of each of these bands is determined by the magnitude of./2(x).
Figures 5a and b display the radial and angular distributions of the reinjection points.The radial distribution P(ri)-N(ri)/ri is approximately con- stant up to reinjection distances ri-x/xZi + y2., _ 0.Sf2(x,).For rifZ(xc) a singularity occurs.This phenomenon is clearly associated with the focussing effect of the map near the critical point.
For ri >fZ(xc), the reinjection probability rapidly decreases with a second, although much weaker singularity for ri v/fZ(xc).The angular distribution of the reinjection points (Fig. 5b) was determined for ri 0.8 f2(xc).The distribu- tion appears at first sight to be approximately uni- form, although there is a 10% modulation with maxima occurring" around --pr/2, p 1,2, 3 and 4.However, a closer examination of the angular distribution reveals an extremely compli- cated structure with singularities associated with the crossing of higher-order critical curves through the reinjection area [24,25].This is Reinjection angle qbi/2g FIGURE 5 Radial (a) and angular (b) distributions of reinjection points.The gross features of the angular distribution is nearly homogeneous (for ri < 0.8 f (xc)).The radial distribution displays a couple of singularities for ri'fZ(xc) and ri x/f2 (xc) combined with a rapid decline for larger values of r..However, for reinjection distances ri < 0.8fZ(x:), the distribution is nearly homogeneous.
Parameters are the same as for Figure 4. x FIGURE 6 Network of critical curves of rank 30-50 crossing the reinjection zone.
illustrated in Figure 6 where we have drawn the network of critical curves of rank 30-50 in a region around the unstable fixed point.Critical curves of lower rank do not pass this region.Because of variations in the invariant density for the intermittent attractor, the reinjection probability is particularly high in the vicinity of the critical curves.One can also determine the angular variation in the density of critical curves over a distance r-O.8f2(xc) from the unstable fixed point.This variation clearly reveals the compli- cated mechanisms that control the finer aspects of the reinjection process.
Here, the last approximation applies for #l>> 1.
Figure 7 shows numerical results for P(/) obtained with two different sets of parameters.Curve (a) was obtained for # 0.005, b 0.14, 7 0.02 and c 0.001.Curve (b) applies to # 0.005, b 0.186, 7 0.13 and c 0.001.Both parameter sets have been adjusted so as to secure a uniform reinjection process close to the fixed point.
For distances r;>f2(x.), the assumption of a uniform reinjection into the unstable manifold no longer applies.Here, P(ri)c/ri, and c# P(l) e-2u l-(4.7) for #/<< 1.This behavior is clearly observed in Figure 8 where the two curves (a) and (b) refer to the same parameter values as above (Fig. 7).

DISCUSSION
We have developed a simple example to demon- strate the emergence of type-II intermittency in the subcritical Hopf-bifurcation of an antisymmetri- cally coupled pair of identical one-dimensional maps.By adjusting the parameters a uniform distribution of reinjection points can be obtained over a two-dimensional region of phase space around the unstable fixed point.Hence, the distribution of long laminar phases follows the theoretically predicted exponential form P(/) exp(-2#/) [17].This is in contrast to previous investigations [20][21][22]26] where the reinjection process has been very nonuniform.At the same time, these investigations have dealt with systems of higher specificity, and their value as illustra- tive examples is correspondingly smaller.Reminis- cent of the singularity produced by the critical point, the distribution of short laminar phases shows nongeneric variations.However, these variations also agree well with our theoretical predictions.
According to theory [17], the scaling behavior of the average length of the laminar phases im- mediately above the subcritical Hopf-bifurcation should follow Figure 9 shows a plot of (1) as a function of # for three different parameter combinations: 7=0.02 and b 0.140 for curve (a), 7 0.13 and b 0.186 for the intermediate curve, and 7=0.2 and b 0.193 for curve (c).In all cases e 0.001.The slopes of the curves in the logarithmic plot are all reasonably close to -1, confirming the prediction (5.1).By varying 7 and b one can change the range over which the reinjection process is approximately uniform.
It is interesting to note that due to its two- dimensional character, the behavior resulting from the subcritical Hopf-bifurcation (type-II intermittency) is hyperchaos with two positive Lyapunov exponents in the limit of low coupling.type-II intermittency appears.For larger values of #, intervals exist in which the two maps syn- chronize to produce a periodic or a quasiperiodic attractor..,:;:;:.:.: . .; , .s , : ,??[:.'!.:.i@ Again, we observe the intervals of synchroniza- existing for #=0.001, 3,=0.02,b=0.12, and tion between the maps.Most of these intervals e 0.10.The so-called folded-towel structure end in a supercritical Hopf-bifurcation followed displayed by this attractor arises through the by a crisis that reestablishes the hyperchaotic be- continuous stretching and folding in two inde- havior.This is typical for two-dimensional sys-pendent directions of phase space.As de- tems when the symmetry prevents them from scribed by Maistrenko et al. [25] the hyperchaotic collapsing into a nearly one-dimensional behavior, attractor is bounded to the so-called absorbing To conclude our discussion, finally Figure 12 area by the critical curves of the coupled map shows a phase plot of the hyperchaotic attractor system.FIGURE 12 Hyperchaotic attractor existing in the coupled map system for #=0.001, =0.02, b=0.12, and e =0.10.The type-II intermittency transition clearly represents a route to hyperchaos.

FIGURE 4
FIGURE 4 Distribution of reinjection points in phase space.Note the region of approximately homogeneous reinjection sur- rounding the unstable fixed point.Here, # =0.005, c 0.001, b 0.14, and 3' 0.02.

FIGURE 7 FIGURE 8
FIGURE7 For reinjection close to the unstable fixed point, the distribution of laminar lengths follows the theoretically predicted exponential function P(/) exp (-2//).The tails of the distributions fluctuate a little because of the relatively few instances of very long laminar phases.

Figure 10 1 FIGURE 9 5 FIGURE 10
Figure10shows the variation of the two Lyapunov exponents as functions of # for e=0.05.The other parameters are 7=0.02and b=0.14.These exponents display an abrupt jump at the bifurcation point #0 -1/282 where

Figure a shows a
Figure a shows a brute force bifurcation dia- gram for the system with the coupling strength c as a parameter and Figure 11 b shows the corresponding variation of the two Lyapunov exponents.

5 FIGURE 11
FIGURE 11 Bifurcation diagram (a) and Lyapunov exponents (b) with the coupling strength e as a parameter.Note the Hopf- bifurcations that occur in several of the periodic windows.Parameters are 3,=0.02,b=0.12, and #=0.001.