Horizontal Visibility graphs generated by type-II intermittency
aa r X i v : . [ n li n . C D ] O c t Horizontal Visibility graphs generated by type-II intermittency ´Angel M. N´u˜nez and Jose Patricio G´omez
Dept. Matem´atica Aplicada y Estad´ıstica. ETSI Aeron´auticos, Universidad Polit´ecnica de Madrid, Spain.
Lucas Lacasa
School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK (Dated: September 24, 2018)In this contribution we study the onset of chaos via type-II intermittency within the frameworkof Horizontal Visibility graph theory. We construct graphs associated to time series generated byan iterated map close to a Neimark-Sacker bifurcation and study, both numerically and analytically,their main topological properties. We find well defined equivalences between the main statisticalproperties of intermittent series (scaling of laminar trends and Lyapunov exponent) and those of theresulting graphs, and accordingly construct a graph theoretical description of type-II intermittency.We finally recast this theory into a graph-theoretical renormalization group framework, and showthat the fixed point structure of RG flow diagram separates regular, critical and chaotic dynamics.
PACS numbers: 05.45.Ac,05.45.Tp,89.75.Hc
I. INTRODUCTION
Some relatively recent strategies in nonlinear analysis techniques are based on the mapping of time series intographs according to different algorithms and criteria (see for instance [1–6]) and on the subsequent study of theassociated graphs. Topological features of the associated graphs are then related to dynamical aspects of the systemsthat generated the series and these methods are used for feature classification based analysis of complex signals.Amongst them, the so called Horizontal Visibility Algorithm (HVA) [1, 7] can be distinguished from others by beingcapable of distilling time series, condensing classes of them into a single graph whose structural properties representthe basic common dynamical properties of the series. The method uncovers structural features and forms sets of timeseries with the same feature by their representative HV graph ensemble, excluding from the ensemble those that lackthat feature. The kernel dynamics in each case is well captured by the associated graphs, such that when the HVmethod is applied to a time series of unknown source, inspection of the resulting graph provides basic informationabout its underlying dynamics. Some relevant applications of this approach include the discrimination of reversiblefrom irreversible dynamics [8], the characterization of chaotic and stochastic signals [7, 9] or, in general, applicationsto series classification problems where the HV is used as the feature extraction method (see [10] for a recent review).However, from a theoretic point of view, the method is still in its infancy and graph-theoretical descriptions ofnontrivial dynamics are in general open problems. This is specially important within this methodology, which hasbeen proved to be simple enough to be addressed analytically instead of being yet another (black box) classificationmethod, while at the same time being accurate and powerful for series classification.In the context of low-dimensional chaos, two of the canonical routes to chaos (Feigenbaum scenario and quasiperiodicroute) have been studied from this perspective and complete sets of graphs that encode the dynamics of theircorresponding classes of iterated maps have been introduced and characterized [11, 12]. The third canonical routeto chaos is the so called Pomeau-Manneville or intermittency route [13]. Under the generic term intermittencyseveral dynamical behaviours with a common feature can be considered. The common feature is the alternation of(pseudoperiodic) laminar episodes with sporadic break-ups or bursts between them called intermissions. Intermittentbehaviour can indeed be observed experimentally in many situations such as Belousov-Zhabotinski chemical reactions,Rayleigh-Benard instabilities, or turbulence [13–16] and has been deeply studied in the context of nonlinear sciences.As a result of this, a characterization of the onset mechanisms and main statistical properties of intermittency havebeen described and typified: from the classification of types I, II and III intermittency by Pomeau and Manneville[17] to other more recent types such as on-off intermittency [18] or ring intermittency [19].The theoretical description of type-I intermittency from a Horizontal Visibility perspective has been advancedrecently [20]. In the present contribution we extend the HV description to type-II intermittency and present thestructural, scaling and entropic properties of the graphs obtained when the HV formalism is applied to the type-IIintermittency case, further advancing the HV theory. We recall here that type-II as described by Pomeau andManneville in their seminar paper [17] is not only of theoretical interest, but indeed constitutes a physical mechanismwhich has been experimentally identified, for example, in coupled nonlinear oscillators [21] or hydrodinamic systems[22].In the following we first recall in section II the key aspects of type-II intermittency. In section III we outline the basicmethodology defined as the Horizontal Visibility algorithm and apply it to the study of trajectories generated byiterated maps close to a Neimark-Sacker bifurcation, where type-II intermittency takes place. An heuristic derivationof an analytical expression for the degree distribution P ( k ; ǫ ) of this kind of graphs is performed. This graph measureencodes the key scaling property of type-II intermittency: the mean length h ℓ i of the laminar episodes with ǫ manifestsin network realm as a comparable scaling with the same variable of the second moment h k i of the degree distribution P ( k, ǫ ). In turn, the scaling of Lyapunov exponent λ ( ǫ ) is recovered in network space, via Pesin-like identity, fromShannon block entropies h n over P ( k , k , ..., k n ; ǫ ), whose block-1 entropy h is only a first order approximation. Insection IV we recast the family of HV graphs generated by intermittent series into a graph-theoretical RenormalizationGroup (RG) framework and determine the RG flows close to and at the bifurcation point. As in other transition-to-chaos scenarios [11, 12, 20], we find two trivial fixed points of the RG flow (akin to the high and low temperature fixedpoints in thermal phase transitions) which are the attractors of regular and chaotic dynamics respectively, togetherwith a nontrivial fixed point associated to critical (null Lyapunov exponent) dynamics. II. TYPE-II INTERMITTENCY: DEFINITION AND BASIC STATISTICAL PROPERTIES
For definiteness we chose the case of type-II intermittency [13] as it develops for nonlinear iterated maps in thevicinity of a Neimark-Sacker bifurcation. As a canonical example of a discrete system exhibiting this type of dynamicslet us consider the iterated complex map: z t +1 = αz t + µ | z t | z t (1)with α = (1 + ǫ ) e ϕi and µ ∈ R . If we rewrite the variable z in its polar form z = xe θi , we can decompose the dynamicsof the system in a rotation of its argument θ and a nonlinear dynamics in its modulus x . We focus on the dynamicsof the modulus x , which is where intermittency appears: x t +1 = (1 + ǫ ) x t + x t = F ( x t ) (2)with a choice of µ = 1 for simplicity.This one dimensional iterated map has an unstable fixed point in x = 0 for 1 ≫ ǫ >
0. If no additional constraintswere imposed the map would diverge for initial conditions larger than zero, however, we can bound the phase spaceby introducing a modular congruence in the definition of this map x t +1 = (1 + ǫ ) x t + x t mod 1 . (3)The trajectories of this map are monotonically increasing functions up to a certain value 0 < x r < ǫ ) x r + x r . Then, the modular congruency reinjects it somewhere in the vicinity of the unstable fixed point x ∗ = 0, where they remain for a certain time (a certain number of iterations) t until they escape, go beyond x r andare reinjected once again (see figure 1 for a graphical illustration). Reinjections close to the unstable fixed pointtake long journeys to depart from its neighborhood, and are experimentally seen as pseudoperiodic (laminar) phases.Chaotic bursts are actually concatenation of short trends generated out from reinjections far from the unstable fixedpoint (see in figure 1 the alternation between long laminar phases and chaotic bursts, governed by the location of thereinjection value).Note that there is also a second value x r < x r < ǫ ) x r + x r from whichthe trajectories are also reinjected in the vicinity of x = 0 + . As a numeric guide, for ǫ = 10 − x r = 0 . . . . and x r = 0 . . . . . This map densely fill the phase space [0,1] and evidence sensitivity to initial conditions for ǫ >
0, regular dynamics for ǫ < ǫ = 0. ǫ actually determines the distance of the system to thebifurcation.A paradigmatic feature of type-II intermittency is the scaling of the mean length of the laminar trends h ℓ i ∼ ǫ − .This scaling is suggested by assuming the laminar trends to start at a x ≪ x ∗ = 0), such that x = (1 + ǫ ) x + x ≈ (1 + ǫ ) x , x ≪ . (4)This lead us by recurrence to x ℓ ≈ (1 + ǫ ) ℓ x = [1 + ǫℓ + O ( ǫ )] x ≈ (1 + ǫℓ ) x (5) t x(t) FIG. 1: A sample trajectory of the map defined in equation 3, with ǫ = 10 − . In type-II intermittency, trajectories monotonouslyincrease until reinjection takes place. Reinjections close to the unstable fixed point take long journeys to depart from itsneighborhood, and are experimentally seen as pseudoperiodic (laminar) phases. Chaotic bursts are actually concatenation ofshort laminar trends generated out from reinjections far from the unstable fixed point. with x ℓ the value at which we can consider the laminar trend is over. From the former expression we get: ℓ ≈ ǫ (cid:18) x ℓ x − (cid:19) (6)and averaging ℓ for different initial values x i of the trend, we get h ℓ i ≈ lim n →∞ n n X i =1 ǫ (cid:18) x ℓ x i − (cid:19) = (cid:0)(cid:10) x − (cid:11) x ℓ − (cid:1) ǫ − , (cid:10) x − (cid:11) = lim n →∞ n n X i =1 x i . (7)However, note that the definition of a laminar trend is somewhat ad-hoc, as we shall define what we consider to besuch a phase, that is, when do we consider that the trajectory has escaped laminarity. Accordingly, the mean lengthof laminar trends h ℓ i is not a variable that directly contains information particularly relevant from the point of viewof the graph theoretical description as we shall later see. A more objective measure is the time t between reinjections(or its mean value h t i ). An expression for this variable t as a function of the initial value x can be found by takingthe continuous limit and traducing our map to a flow: x t +1 − x t ≈ dxdt = x ( ǫ + x ) ⇒ Z x r x dxx ( ǫ + x ) = Z tt dt. (8)Assuming a relatively small initial time ( t − t ≈ t ) this integration is straightforward and lead us to the followingexpression: t ( x , ǫ ) = 1 ǫ log x r p ǫ + x x p ǫ + x r (9)where x is the initial or so called reinjection value (see fig. 2). If, on this expression, we make the substitution x r = x ℓ ≪ g ( t ) x t(x ) FIG. 2: Reinjection time t ( x ) for the map defined in equation 3, with ǫ = 10 − (see fig. 1). Red curve stands for the theoreticalexpression deduced in eq. 9. Black circles are direct numerical measures in a trajectory of length T = 10 . can now be deduced for the reinjection time t from the corresponding pdf f ( x ) of the reinjection values x , since weknow that: g ( t ) dt = f ( x ) dx ⇒ g ( t ) = f [ x ( t )] (cid:12)(cid:12)(cid:12)(cid:12) dx ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) . (10)From eq. 9 we have that x ( t, ǫ ) = ǫ / √ ae ǫt − , a = ǫ + x r x r . (11)In absence of a more detailed description, we may assume in a first approximation that f ( x ) is reasonably uniformlydistributed in the interval [0,1] which takes the role of the phase space of our system ( x → U [0 , ⇒ f ( x ) = 1).This yields the following density: g ( t ) = f [ x ( t )] (cid:12)(cid:12)(cid:12)(cid:12) dx ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) = aǫ / e ǫt ( ae ǫt − / ≈ ( a √ ǫ t − , t m ≫ t ǫ / √ a e − ǫt , t m ≪ t, (12)where t m is a characteristic time scale for which we can consider that the function behaves as a power law for shorttimes t ( t m ≫ t ) and exponentially for long times t ( t m ≪ t ). If we calculate the mean value h t i with this approachfor short and long times, we obtain: h t i = Z ∞ t tg ( t ) dt ≈ a √ ǫ Z t m t dt + Z ∞ t m ǫ / √ a te − ǫt dt == at m ǫ / + t m e − ǫt m √ a ǫ / + e − ǫt m − √ a ǫ − / ∼ ǫ − / , (13)which exhibits a scaling with ǫ that however quantitatively differs from the one found for ℓ (again, t m − t ≈ t m ). Wecan recover the scaling for ℓ with this approach just by assuming that, in order a trend to be considered as laminar, t g(t, ε ) FIG. 3: Log-log plot of the probability density function of times to reinjection g ( t ; ǫ = 10 − ) for the map defined in equation3. Red curve stands for the theoretical approximation. Black dots are extracted from the simulation of a trajectory of leght L = 10 . the reinjected value has to be in the vicinity of the fixed point: x ∈ (0 , b ] , b ∼ √ ǫ or otherwise, f ( x ) must be areasonably uniform pdf: x → U [0 , √ ǫ ] ⇒ f ( x ) = 1 / √ ǫ , The pdf, g ( ℓ ), that we obtain in this case is slightly different: g ( ℓ ) = f [ x ( ℓ )] (cid:12)(cid:12)(cid:12)(cid:12) dx ( ℓ ) dℓ (cid:12)(cid:12)(cid:12)(cid:12) = 2 ǫ e ǫℓ (2 e ǫℓ − / ≈ (cid:26) ℓ − / , ℓ m ≫ ℓ ǫ √ e − ǫℓ , ℓ m ≪ ℓ (14)This new pdf predicts a scaling for the mean length of the laminar trend of the form: h ℓ i = Z ∞ ℓ ℓg ( ℓ ) dℓ ≈ Z ℓ m ℓ ℓ − / dℓ + Z ∞ ℓ m ǫ √ ℓe − ǫℓ dℓ == 2 p ℓ m + ℓ m e − ǫℓ m √ ǫ + e − ǫℓ m − √ ǫ − ∼ ǫ − . (15)where ℓ m − ℓ ≈ ℓ m . A comparison of these expressions with the results of numerical simulations is shown in figure 3.Note that, if we expand our interval of initial conditions in the reinjection to the whole phase space [0 , t (lim b → ℓ = t ). III. TRANSFORMATION OF INTERMITTENT TIME SERIES INTO HORIZONTAL VISIBILITYGRAPHS
The Horizontal Visibility (HV) algorithm [7] assigns each datum x t of a time series { x t } t =1 , ,... to a node n t in itsassociated HV graph (HVg), where n t and n t ′ are two connected nodes if x t , x t ′ > x τ for all τ such that t < τ < t ′ .The resulting are outerplanar graphs connected through a Hamiltonian path [24] whose structural properties capturethe statistics enclosed in the associated series [10]. A relevant measure is the degree distribution P ( k ), that accountsfor the probability of a randomly chosen node to have degree k , which has been showed to encode key dynamicalproperties such as fractality, chaoticity or reversibility to cite some [10].For illustrative purposes, in figure 4 we represent a sketch of a type-II intermittent series along with its associatedHV graph. At odds with the phenomenology of type-I intermittency, whose mapping consisted of several repetitions FIG. 4: Sketch of an intermittent trajectory along with its associated HV graph. Nodes ’before reinjection’ correspond tovalues in the trajectory which have ’visibility’ over the values until the next reinjection. Nodes just after reinjection correspondto values in the trajectory which are bounded by two higher values. of a T -node motif (periodic backbone of period T associated to the ghost of the period 3 series) linked to thefirst node of the following laminar trend and interwoven with the chaotically connected nodes associated to thechaotic bursts between laminar trends [20], the case of type-II lacks any periodic backbone or chaotic burst, andconsists of nodes associated to reinjections and nodes linked to the last node of the previous reinjection. That is,the method naturally distinguishes reinjections from each other and do not include the (nonetheless ambiguous)distinction between laminarity and burstiness.Accordingly, we can classify nodes in three different categories (see fig. 4):a) nodes located just before a reinjection n r − , whose degree distribution will be later discussed in detail,b) nodes located just after a reinjection n r + , with a trivial degree distribution ( P r + ( k = 2) = 1) andc) the rest of the nodes n t , whose degree distribution is also trivial ( P t ( k = 3) = 1).Based on these observations, in what follows we derive some topological properties of these graphs and will show thatthey indeed incorporate the main statistical properties of type-II intermittency. A. Degree distribution P ( k ) Consider the degree distribution P ( k ), that describes the probability that a randomly chosen node of a graph has k links (degree k ). The previous features allow us to decompose the degree distribution of type-II intermittency graphsas a weighted sum of the aforementioned contributions: P ( k ) = f r P r − ( k ) + f r P r + ( k ) + (1 − f r ) P t ( k ) , (16)where f r is the reinjection fraction f r = lim τ →∞ n r τ , (17) kP(k) FIG. 5: (Circles) Log-log plot of the degree distribution P ( k ) of the Horizontal Visibility graph (HVg) mapped from a trajectorygenerated by equation 3 for ǫ = 10 − . (Red curve) theoretical expression for the same distribution as presented in eq. 16. and n r is the number of reinjections that have occurred up to time τ .On the other hand, the degree of the nodes before a reinjection can be decomposed in two different contributions k = t + k ′ . The first term t consists on the visibility the node has over the following laminar trend up to the nextreinjection and it is distributed like P tr − ( t ) ≈ g ( t ). The second term k ′ = k − t comes from the visibility the node hasover the rest of the reinjections and its distribution can be supposed to be the characteristic exponential distributionfound in HVg’s that come from stochastic/chaotic processes like a reinjection: P rr − ( k ) = δ ( k − k ′ ) e − λk , k ′ ∈ Z + − { } [9]. We have one further consideration: by construction, the degree of these nodes has an additional restriction: k r − ≥
4. As an approach, we can assume k ≫ k ′ or, equivalently, k ≈ t . Therefore, the degree distribution for thesenodes turns out to be: P r − ( k ) ≈ Z k P tr − ( t ) P rr − ( k − t ) dt = g ( t ) , k = t (18)In figure 5 we plot in log-log the approximate theoretical expression for the degree distribution P ( k, ǫ ), for a concretevalue of ǫ = 10 − , along with the numerics. B. Variance σ k = h k i − h k i In this section we are going to verify our previous hypothesis P r − ( k ) ≈ g ( t ) by deducing an expression for the meanvalue of the degree distribution h k i , and we are going to analyze how the second order moment of the distribution σ k inherits the scaling of h ℓ i . Let us start with the mean degree h k i . Recall that HV graphs associated to genericaperiodic series tend, as size increases, to a constant mean degree h k i = 4 [23] and that HVgs are, by construction,connected graphs where node i has degree k ≥ ∀ i (they have a Hamiltonian path). Then, h k i = ∞ X k =2 kP ( k ) = (1 − f r ) · f r · f r ∞ X k =4 kP r − ( k ) = 4 ⇒ ∞ X k =4 kP r − ( k ) = 4 + f − r = h k r − i . (19)This expression tell us that the mean value of the connectivity of the nodes before reinjetion h k r − i behaves as theinverse of the reinjection fraction f − r . Moreover, we can express the reinjection fraction as: f r = 1 − lim τ →∞ n r · h t i τ = 1 − f r · h t i ⇒ f − r = h t i + 1 ⇒ h k r − i = h t i + 5 (20) 〈 t 〉〈 l 〉σ (k) ε FIG. 6: Log-log plot of h t i (circles) and h ℓ i (squares) as a function of ǫ , numerically calculated from time series of 10 data.Diamonds correspond to the variance of the degree distribution of the associated HV graph (numerical results). In each case,solid lines are the predicted analytical scaling h t i ∼ ǫ − / (eq. 13), h ℓ i ∼ ǫ − (eq. 15) and h k i − h k i ∼ ǫ − (eq. 22), in goodagreement with the numerics. for which we have assumed that the number of nodes between reinjections is quite the same as the number ofreinjections multiplied by the mean value of the reinjection time ( N t = n r · h t i ), which gives a compact expression h t i = h k r − i − ∼ ǫ − / that indicates the behaviour of P r − ( k ) and g ( t ) coincides up to first order, and that f r goesto zero as ǫ / .If we now move to the second order moment, we get: σ k = h k i − h k i = ∞ X k =2 k P ( k ) −
16 = f r ∞ X k =4 k P r − ( k ) + 7(1 − f r ) , (21)and following our hypothesis we can approximate ∞ X k =4 k P r − ( k ) ≈ Z ∞ t t g ( t ) dt ≈ a √ ǫ Z t m t tdt + Z ∞ t m ǫ / √ a t e − ǫt dt == at m ǫ / + 2 √ a ǫ − / ∼ ǫ − / , where the last formula requires small values of ǫ and t m − t ≈ t m . As f r ∼ ǫ / for small values of ǫ , in this low limitwe have a leading order of σ k ∼ ǫ − (22)which reproduces the expected scaling (see figure 6 for a comparison with numerics). We conclude that the scalingof the mean length of laminar phases with ǫ is inherited in network space by a similar scaling in the variance of thedegree distribution. C. Scaling of Lyapunov exponent: Block entropies h n The second paradigmatic feature of type-II intermittency is the scaling of the Lyapunov exponent λ (which for a map x ( t + 1) = F ( t ) reads λ = lim t →∞ t P t − i =0 ln | F ′ ( x i ) | ) with respect to the distance to criticality λ ( ǫ ) ∼ ǫ / [13]. Note λ h ε FIG. 7: (Circles) Log-log plot of the Lyapunov exponent λ calculated numerically from a trajectory of length L = 10 with theexpression λ = lim t →∞ t P t − i =0 ln | F ′ ( x i ) | . Solid line is a regression with the predicted scaling: λ ( ǫ ) ∼ ǫ / . (Squares) Log-logplot of the block-1 graph entropy h ( ǫ ) (see the text) of the Horizontal Visibility graph associated to the same trajectory alongwith a regression, whose best power law fit reads h ∼ ǫ . . that Lyapunov exponents characterize a purely dynamical feature and, although some graph-theoretical extensionsof these exponents have been recently advanced [25], it is not evident at all how to cast this dynamical behaviourinto a graph-theoretical realm. However, note that Pesin identity relates positive Lyapunov exponents of chaotictrajectories with Kolmogorov-Sinai rate entropy in dynamical systems. Based on this identity, a relation betweenLyapunov exponents of maps and Shannon-like entropies over the degree distribution of the associated visibilitygraphs has been proposed [11, 20, 23, 25]. The block-1 graph theoretical entropy h [20] is defined as h = − ∞ X k =2 P ( k ) log P ( k ) , and block- n entropies take into account of higher order statistics of blocks P ( k , k , ..., k n ).An approximate leading order calculation for h can be performed assuming P r − ( k ) ≈ g ( t ) and recalling that therest of the terms in P ( k ) only contribute for k = 2 , h reduces to a linear combination − h = f r log f r + (1 − f r ) log (1 − f r ) + ∞ X k =4 f r P r − ( k ) log [ f r P r − ( k )] ≈ f r log f r + (1 − f r ) log (1 − f r ) + Z ∞ t f r g ( t ) log [ f r g ( t )] dt = 2 f r log f r + (1 − f r ) log (1 − f r ) + f r Z ∞ t g ( t ) log [ g ( t )] dt (23)In order to derive the leading order of h in ǫ , we recall that f r ∼ ǫ / for small values of ǫ . Let us pay attention tothe last integral, which can be evaluated by a time scale separation for short ( t m ≫ t ) and long times ( t m ≪ t ) as in(12): Z ∞ t g ( t ) log g ( t ) dt ≈ Z t m t a √ ǫ t − log a √ ǫ t − dt + Z ∞ t m ǫ / √ a e − ǫt log ǫ / √ a e − ǫt dt (24)0 n α (h n ) FIG. 8: (Circles) Semilog plot of the function (1 / − α ( n )) as a function of n , where 1 / α ( n ) are the respective scalingexponents of the Lyapunov exponent and the block entropy h n with respect to ǫ . We observe that these differences decreaseslogarithmically with n , suggesting the presence of a graph theoretical Pesin identity (see the text). = a √ ǫ a √ ǫ Z t m t dt − a √ ǫ Z t m log tt dt + ǫ / √ a log ǫ / √ a Z ∞ t m e − ǫt dt + ǫ / √ a Z ∞ t m t ( − ǫe − ǫt ) dt = aǫ / aǫ / t m − aǫ / t m ) + ǫ / √ a log ǫ / √ a e − ǫt m + ǫ / √ a e − ǫt m + t m ǫ / √ a e − ǫt m . (25)After a brief inspection, we conclude that for small values of ǫ , the leading term of h is not exactly ∼ ǫ / but ∼ − ǫ / log ǫ , what should yield an approximate scaling albeit with a smaller exponent. In figure 7 we plot in log-logthe numerical results of h as a function of ǫ , together with the numerics of λ ( ǫ ) in the same range ( ǫ ≪ h shows a reasonably similar scaling, although with a slightly smaller exponent (the numerical fit is α ≈ . n of the block in the graph-theoretical entropy h n : h n = − n X k ,...,k n P ( k , ..., k n ) log P ( k , ..., k n ) . (26)we numerically observe a tendency in the scaling behaviour h n ( ǫ ) ∼ ǫ α ( n ) that suggests lim n →∞ α ( n ) −→ / n →∞ h n = λ. Numerical evidence for these latter results are shown in figure 8.
IV. GRAPH-THEORETICAL RENORMALIZATION GROUP ANALYSIS
As in previous works [11, 12, 20, 23], we can define a Renormalization Group (RG) transformation R on the HVgraphs G as the coarse-graining of every couple of adjacent nodes where at least one of them has degree k = 2 intoa block node that inherits the links of the previous two nodes. In other words, the operation removes form thegraph every node of degree k = 2 along with its two links. Assuming infinitely long series (in order to avoid therescaling procedure in standard RG), in what follows we argue that the flow induced by iteratively performing thisRG operation classifies dynamics coming from above and below transition, although the phase portrait turns to bevery different from the one found in type-I intermittency [20].1 FIG. 9: Cartoon of an HV graph at criticality G ( ǫ = 0) (mapped from a trajectory at the onset of chaos in a Neimark-Sackerbifurcation, ǫ = 0 in equation 3). By construction the graph is invariant under renormalization (see the text), such that R{ G ( ǫ = 0) } = G ( ǫ = 0). As any perturbation in ǫ generates a RG flow that takes the HV graph away from this equilibriumtowards the stable attractors (parsimoniously towards the random graph G rand for positive perturbations and instantaneouslyto the chain graph G for negative perturbations), G ( ǫ = 0) constitutes an unstable fixed point of the Renormalization Groupflow associated to critical dynamics on the map and acts as a boundary between regular ( ǫ <
0) and chaotic ( ǫ >
0) dynamics. • When ǫ <
0, trajectories are damped and rapidly converge to a constant series x ( t ) = 0 ∀ t , so after a transientthe associated HVG G will be a chain graph with k = 2 for all nodes [11]. A graph which is indeed the RGattractor of regular dynamics, invariant under renormalization R{ G } = G [23] and represents a minimum forthe entropy functional h = 0. • When ǫ > n r + , k = 2) are eliminated progressively and, step by step, the laminar phases disappear, leaving essentiallya series of ’nodes before reinjection’ which form mainly an uncorrelated random series. The analogous flow ingraph space slowly converges towards G rand , the universal HV graph associated to an uncorrelated series, whichconstitutes a stable fixed point of the RG flow lim p →∞ R ( p ) { G ( ǫ > } = G rand = R ( G rand ) [23]. As it has beenproved that G rand is a maximally entropic HV graph [23], we conclude that the RG flow ∀ ǫ > • At the bifurcation ( ǫ = 0) the trajectories are monotonically increasing series bounded at −∞ by a largevalue (the ghost of the intermittent regime), which maps into a HV graph which is itself indeed invariantunder renormalization G c = G ( ǫ = 0) = R{ G ( ǫ = 0) } (see figure 9 for a graphical illustration). Its degreedistribution can be described for a graph of N nodes (where the limit N → ∞ will be taken afterwards) as P ( k = 2) = P ( k = N −
1) = 1 /N ; P ( k = 3) = ( N − /N , yielding also a null entropy h = 0 in the limit ofdiverging sizes. This constitutes the third fixed point of the RG flow. Note that this fixed point is differentfrom G , nonetheless, they both share the same entropy. The reason is that there is no actual parsimonioustrajectory connecting G ( ǫ = 0) to G ( ǫ <
0) = G , once you apply an infinitisemal perturbation making ǫ < x ( t ) = 0 ∀ t , whose associated HVg is directly G . There is hence no RG flow or entropy production associated with the path. On the other hand, any positiveperturbation in ǫ is amplified in the RG flow, taking the graph to G rand . We conclude that G ( ǫ = 0) is anunstable fixed point of the RG flow.2 V. CONCLUSIONS
To conclude, in this work we have further advanced the Horizontal Visibility graph theory by providing ananalytical and numerical graph theoretical description of type-II intermittency route to chaos, extending previousresults for type-I intermittency [20]. We have shown that the key ingredients of type-II intermittency (concretescalings of the mean length of laminar trends and Lyapunov exponent with respect to ǫ ) are recovered in the networkrealm by comparable scalings of the variance of the degree distribution and Shannon block entropy respectively.Finally, we have recasted the problem into a graph-theoretical renormalization group framework and have shownthat the graph-theoretical fixed points of the RG flow distinguish regular, chaotic and critical dynamics. Whereasthe trivial fixed points of the RG flow are universal attractors of regular and chaotic dynamics respectively, theunstable graph-theoretical fixed point, the RG flows and the associated entropy production paths are unique fortype-II intermittency, being for instance a different scenario than what we found in type-I intermittency [20] or inother routes to chaos [12, 23]. Acknowledgements.
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