Exploring the dynamics of a third-order phase

June 12, 2018 8:51 text˙pll˙ijbc

International Journal of Bifurcation and Chaos c World Scientific Publishing Company

Exploring the dynamics of a third-order phase-locked loop model Cesar Manchein† , and Holokx A. Albuquerque∗ Departamento de F´ısica, Universidade Do Estado de Santa Catarina, Joinville, SC, 89219-710, Brazil † [email protected] ∗ [email protected] Luis Fernando Mello‡ Instituto de Matem´ atica e Computa¸ca ˜o, Universidade Federal de Itajub´ a, 37500-903 Itajub´ a (Brazil). ‡ [email protected] Received (to be inserted by publisher) We study the dynamics and characterize the bifurcation structure of a phase-locked loop (PLL) device modeled by a third-order autonomous differential equation with sinusoidal phase detector. The development of this work was performed using rigorous analysis and numerical experiments. Through theoretical analysis the bifurcation structures related to two fundamental equilibrium points of the system are described. By using extensive numerical experiments we investigate the intricate organization between periodic and chaotic domains in parameter space (named here parameter plane as the PLL model has only two control parameters) and obtain two following remarkable findings: (i) there are self-organized generic stable periodic structures along specific directions in parameter plane, whose periods are defined by a mathematical rule and, (ii) the existence of transient chaos phenomenon responsible for long chaotic temporal evolution preceding the asymptotic (periodic) dynamics for some particular control parameter pairs is characterized. Our theoretical and numerical results present an astonishing concordance. We believe that the present study, specially the parameter plane analysis, may have a great importance to experimental studies and general applications involving PLL devices when, for example, one would like to avoid the chaotic regimes.

1. Introduction The dynamics of phase-locked loop (PLL) devices have attracted considerable attention in the last four decades from both theoretical and experimental points of view. Although the PLL was originally proposed by Bellescize almost a century ago (in 1932 to be precise) [de Bellescize, 1932] still there are several aspects about its underlying dynamics which deserves to be investigated. A PLL device is defined as a control system that generates an output signal whose phase is related to the phase of an input signal [Harb & Harb, 2004b,a]. As far as we know, first rigorous mathematical definitions for PLL devices were given in [Leonov et al., 2015; Kuznetsov et al., 2015]. Usually, the configuration of PLL devices are composed by three fundamental building blocks: a phase detector, a time-invariant loop filter and a voltage-controlled oscillator [Gardner, 1979] (see also Refs. [Piqueira, 2009; Bueno et al., 2010] for discussion about a more sophisticated version of PLL device). One of the simpler PLL devices is composed by an electronic circuit consisting of a variable frequency oscillator and a phase detector in a feedback loop. The oscillator generates a periodic signal, and the phase detector compares the phase of that signal with the phase of the input 1

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Manchein, Albuquerque, and Mello

periodic signal, adjusting the oscillator to keep the phases matched. When the loop’s phase error has a constant value and the loop reached a stable equilibrium state the PLL is phase-locked [Stensby, 1993; Harb & Stensby, 1996]. In this case, the loop’s phase error represents an asymptotically stable solution of the nonlinear autonomous differential equation which describes the closed loop phase error. The spectrum of applications of a PLL is remarkable, specially in engineering problems involving synchronization processes between input and output signals [de Bellescize, 1932], modern electronic communications [Piqueira, 2009], chaos control [Harb & Harb, 2004a] and, optical communication devices [Best, 2007], to mention few. In addition, a PLL device is also applied to estimate the phase of a received periodic signal contaminated by random noise [Thacker et al., 2011]. Despite the obvious importance of a PLL device from the point of view of applications, there is a lack in literature about a detailed characterization of its dynamics in phase and parameter spaces composed by the set of dynamical variables and control parameters, respectively. The relevance of studying the parameter space of nonlinear systems is due to it allows one to understand how periodic behavior, chaos and bifurcations are related to each other in generic nonlinear dynamical systems. Indeed, parameters leading to different behaviors are completely correlated. Actually, periodic sub-domains embedded in chaotic ones appear in all scales and sometimes they assume universal shapes, as detailed below. For specific nonlinear systems such periodic structures appear aligned along preferred directions in parameter space and they are generated according to some mathematical rule as recently shown, by instance, for Watt governor models in [Marcondes et al., 2017]. Here we revisit the investigation of nonlinear dynamics and bifurcation scenario of a typical PLL device modeled by a nonlinear third-order autonomous differential equation, derived in the Ref. [Piqueira, 2009] and later explored in [Bueno et al., 2011]. Recently it was also studied in [Piqueira, 2017], where the author investigate nonlinear dynamical behaviors of a PLL device only for few pairs of control parameters located at tiny portion of the parameter plane. According to proper literature related to problems in Physics the PLL device modeled by a third-order autonomous differential equation and studied in this work, can be classified as a jerk system [Sprott, 2010]. In general, the third-order explicit autonomous differential equations applied to model jerk systems, represent an interesting subclass of dynamical systems that can exhibit many features of the regular and chaotic dynamics [Eichhorn et al., 1998]. We start our investigation characterizing the bifurcation structure of a PLL model by using an analytical technique to determine the stability of two fundamental equilibrium points through Lyapunov coefficients and, in this way, obtain all the possible bifurcations which occur at the equilibrium points of the nonlinear system. Besides the theoretical analysis, our aim is also apply numerical methods to unveil the bifurcation structures in the parameter space here named as parameter plane, because the continuous-time nonlinear model studied has only two control parameters. To this end, extensive numerical experiments are performed to characterize regular and chaotic behaviors. We start investigating a large portion of parameter plane of the nonlinear model (as two control parameters are varied simultaneously) by using the following numerical approaches: (i) the calculation of Lyapunov spectra (to extract the Largest Lyapunov Exponents (LLE)) based on Benettin’s algorithm [Benettin et al., 1980; Wolf et al., 1985], which includes the Gram–Schmidt re-orthonormalization procedure and, (ii) the estimation of local-number of spikes in a period of time-series of trajectories for different pairs of parameters used to describe dynamical behaviors in two-dimensional diagrams (so called Lyapunov and Isospikes diagrams, respectively) of the parameter plane of the continuous-time model. Magnifying tiny domains of the parameter plane of the nonlinear model we show the presence of generic self-organized periodic structures embedded in a single chaotic domain, namely Stable Periodic Structures (SPSs) [Celestino et al., 2014]. Specifically, one of them is wellknown to be typical or generic in dissipative nonlinear dynamical systems: Such kind of SPSs, so-named shrimp-shaped domains, are usually found in a wide range of both, discrete and continuous-time systems [Marcondes et al., 2017; Gallas, 1993; Lorenz, 2008; Celestino et al., 2011; Medeiros et al., 2011; Oliveira & Leonel, 2011; Francke et al., 2013; Manchein et al., 2013; Gallas, 2015, 2016; Wiggers & Rech, 2017; Horstmann et al., 2017] (and references therein), including ratchet systems, chemical reactions, population dynamics, electronic circuits, laser models and, neural networks, among others. To illustrate how complex the dynamics of the PLL model device can be, few pairs of control parameters are chosen to plot emblematic

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periodic (for periods q = 1, 2, 3, 4, 9) and chaotic attractor’s projections. Another remarkable observation obtained through direct numerical simulations is the existence of a phenomenon known as transient chaos [Lai & T´el, 2011] in the parameter plane of the model. It is well known in the proper literature that in problems involving observation, modeling, prediction and control of the system the transient dynamics can be more relevant than the asymptotic states. Transient chaos phenomenon has a wide range of applications as, for instance, the dynamics of decision making, dispersion and sedimentation of volcanic ash, doubly transient chaos of un-driven autonomous mechanical systems, and dynamical systems approach to energy absorption or explosion, to mention few [Lai & T´el, 2011]. As the system studied here is dissipative, i.e., the phase space volume contracts everywhere, the asymptotic states are attractors that may be regular, but the transient dynamics before the final state of the system is chaotic. This work is organized as follows: In Sec. 2 the nonlinear dynamical system used to model a PLL device is introduced, the properties of its two fundamental equilibrium points are detailed described and theoretical analyzes are performed to characterize the stability properties and all possible bifurcations which occur at the two fundamental equilibrium points. Section 3 discusses in details the numerical results where we show the abundant presence of SPSs embedded in chaotic domains and sometimes organizing themselves along preferred directions. In addition, we also explore the existence of transient chaos for a particular region of parameter plane. Finally, Sec. 4 is devoted to summarize our main conclusions indicating some of their possible implications.

2. The mathematical continuous-time model and the bifurcation structure analysis In order to study the dynamics of phase-locked loop devices, we consider the third-order autonomous differential equation model obtained by Piqueira [Piqueira, 2009, 2017] and written as ... φ + (3 − K)φ¨ + φ˙ + KG sin (φ) = 0, (1) where the variable φ is the phase difference between the phase-detector and a voltage controlled oscillator. The parameters K and G represent the low frequency gain of the filter and the range of the whole loop, respectively and, as originally proposed they only assume positive values, i.e, K > 0 and G > 0. ˙ φ) ¨ and performing few algebraic manipulations, the By defining the set of variables as (x, y, z) = (φ, φ, third-order differential equation (1) can be rewritten as a nonlinear system composed by three first-order differential equations, as follows x˙ =

dx = y, dt

y˙ =

dy = z, dt

z˙ =

dz = −(3 − K) sin(x) − y − (KG) z, dt

(2)

where (x, y, z) ∈ R3 are state variables and K and G are parameters which have exactly the same meaning as defined above. Note that (1) and (2) are periodic of period 2π in the variables φ and x, respectively. So one can restrict the study of the phase portraits of (2) to the set (−π, π] × R2 . The equilibrium points of (2) have the form (mπ, 0, 0), with m ∈ Z. But, due to the above restriction, it is enough to study the fundamental equilibrium points P1 = (0, 0, 0) and P2 = (π, 0, 0). We study all the possible bifurcations which occur at the equilibrium points of system (2). In this way the analyzes presented in [Piqueira, 2017] are extended and completed here. Hopf bifurcations give the simplest way in which the solutions of (2) have a periodic oscillatory regime with the birth (or death) of limit cycles. More precisely, we prove the following statements: (a) For the equilibrium point P1 = (0, 0, 0) the Hopf curve H1 is obtained in the parameter plane (G, H) ∈ R2 and the first Lyapunov coefficient l1 is calculated. It is established that this coefficient is always negative on H1 . See Fig. 1(a). (b) For the equilibrium point P2 = (π, 0, 0) the Hopf curve H2 is obtained in the parameter plane (G, H) ∈ R2 and the first Lyapunov coefficient l1 is calculated. It is established that this coefficient is always positive on H2 . See Fig. 1(b).

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3

K

H2 : K =

K

3 1−G

15

(G4 , K4 )

(G2 , K2 )

2

10

(G3 , K3 )

(G5 , K5 )

(G1 , K1 ) H1 : K =

3 1+G

5

(G6 , K6 )

3

1

0 0

G

1

0

2

0.2

0.6

0.4

(a)

0.8

G

1

(b)

Figure 1. Bifurcation diagrams of system (2) at the equilibrium points P1 (a) and P2 (b). (a) For a typical point (G1 , K1 ), with K1 < 3/(1 + G1 ), P1 is asymptotically stable; for a typical point (G2 , K2 ), with K2 = 3/(1 + G2 ), P1 is a weak attracting focus for the flow of (2) restricted to the center manifold; for a typical point (G3 , K3 ), with K3 > 3/(1 + G3 ), but close to the curve K = 3/(1 + G), P1 is unstable and a stable limit cycle appears from a Hopf bifurcation. (b) For a typical point (G4 , K4 ), with K4 > 3/(1 − G4 ), P2 is unstable; for a typical point (G5 , K5 ), with K5 = 3/(1 − G5 ), P2 is a weak repelling focus for the flow of (2) restricted to the center manifold; for a typical point (G6 , K6 ), with K6 < 3/(1 − G6 ), but close to the curve K = 3/(1 − G), P2 is stable for the flow of (2) restricted to a continuation of the center manifold and an unstable limit cycle appears from a Hopf bifurcation.

Now, we proceed the calculations of the first Lyapunov coefficient, l1 , for the equilibrium points P1 and P2 , which is defined by (see details in [Kuznetsov, 1998]) l1 =

1 Re hp, C(q, q, q¯) + B(¯ q , h20 ) + 2B(q, h11 )i, 2

(3)

where h11 = −A−1 B(q, q¯), h20 = (2iω0 I3 )−1 B(q, q), I3 is the identity 3 × 3 matrix, B and C are multilinear symmetric functions, p, q ∈ C3 are vectors such that Aq = iω0 q,

A⊤ p = −iω0 p,

hp, qi =

3 X

p¯i qi = 1,

(4)

i=1

A⊤

is the transpose of the Jacobian matrix A, which has a pair of purely imaginary eigenvalues λ2,3 = ±iω0 , ω0 > 0, and the other eigenvalue λ1 6= 0. Case 1. It is very easy to show that the characteristic polynomial of the Jacobian matrix of (2) at P1 is given by P (λ) = −λ3 + (K − 3)λ2 − λ − GK. Applying the Routh–Hurwitz stability criterion it follows that when (K − 3)(−1) = (−1)(−KG) ⇐⇒ K =

3 G+1

the Jacobian matrix of (2) at P1 has one real negative and a pair of purely imaginary eigenvalues. More precisely, for parameters on the set H1 = {(G, K) ∈ R2 : K = 3/(1 + G), G > 0} the eigenvalues of the Jacobian matrix of (2) at P1 are given by λ1 = −

3G < 0, 1+G

λ2,3 = ±iω0 ,

ω0 = 1.

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For parameters K > 0, G > 0 such that (G, K) ∈ / H1 the equilibrium point P1 is hyperbolic and its stability is easily determined (see Fig. 1(a)). The eigenvectors q and p defined in (4) have the form   i G+1 3iG ,− , q = (−1, −i, 1) , p = − 2 + (2 + 6i)G 2 2 + (2 + 6i)G and the multilinear symmetric functions B and C write as B(X, Y ) = (0, 0, 0) ,

C(X, Y, Z) =



3Gx1 y1 z1 0, 0, G+1



.

By elementary calculations, we obtain the complex vectors h11 = (0, 0, 0) and h20 = (0, 0, 0), which implies that 3iG . G2,1 = − (6 + 2i)G + 2i So, from (3) the first Lyapunov coefficient can be written as l1 = −

3G(G + 1) < 0. 8G(5G + 1) + 4

Case 2. Here it is also very easy to show that the characteristic polynomial of the Jacobian matrix of (2) at P2 is given by P (λ) = −λ3 + (K − 3)λ2 − λ + GK. Applying the Routh–Hurwitz stability criterion it follows that when 3 1−G the Jacobian matrix of (2) at P2 has one real positive and a pair of purely imaginary eigenvalues. More precisely, for parameters on the set H2 = {(G, K) ∈ R2 : K = 3/(1 − G), 0 < G < 1} the eigenvalues of the Jacobian matrix of (2) at P2 are given by (K − 3)(−1) = (−1)(KG) ⇐⇒ K =

3G > 0, λ2,3 = ±iω0 , ω0 = 1. 1−G For parameters K > 0, G > 0 such that (G, K) ∈ / H2 the equilibrium point P2 is hyperbolic and its stability is also easily determined (see Fig. 1(b)). The eigenvectors q and p defined in (4) have the form   3iG i G−1 ,− , q = (−1, −i, 1) , p = 2 − (2 + 6i)G 2 −2 + (2 + 6i)G λ1 =

and the multilinear symmetric functions B and C write as B(X, Y ) = (0, 0, 0) ,

C(X, Y, Z) =



3Gx1 y1 z1 0, 0, G−1



.

By elementary calculations, we obtain the complex vectors h11 = (0, 0, 0) and h20 = (0, 0, 0), which implies that 3iG . G2,1 = 2i − (6 + 2i)G So, from (3) the first Lyapunov coefficient can be written as l1 =

3(1 − G)G > 0. 8G(5G − 1) + 4

It is worth to mention that Case 1 was studied in [Piqueira, 2017]. The transversality conditions of the Hopf bifurcations for both cases can be obtained by easy calculations and they are equivalent to cross transversally the sets H1 or H2 in the parameter plane.

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From the Case 1, if ζ0 = (G0 , K0 ) ∈ H1 then l1 (ζ0 ) < 0. So system (2) has a transversal Hopf point at P1 for the parameter vector ζ0 . More specifically, the Hopf point at P1 is asymptotically stable (weak attracting focus for the flow of system (2) restricted to the attracting center manifold) and for a suitable ζ = (G, H) close to ζ0 there exists a stable limit cycle near the unstable equilibrium point P1 (see Fig. 1(a)). From the Case 2, if ζ0 = (G0 , K0 ) ∈ H2 then l1 (ζ0 ) > 0. Thus system (2) has a transversal Hopf point at P2 for the parameter vector ζ0 . More specifically, the Hopf point at P2 is unstable (weak repelling focus for the flow of system (2) restricted to the repelling center manifold) and for a suitable ζ = (G, H) close to ζ0 there exists an unstable limit cycle near the equilibrium point P2 (see Fig. 1(b)).

3. Numerical experiments 3.1. Lyapunov and isospikes diagrams This subsection is devoted to discuss the existence of different dynamical regimes as both parameters are varied. We present our results applying the fourth-order Runge-Kutta numerical method with fixed time-step equal to 10−2 to obtain the solutions of system (2) for a set of parameters and to compute the Lyapunov exponent spectrum and the number of spikes in one period of the time series. Through the Lyapunov spectra (which has been extensively applied to distinguish regular and chaotic motion), we obtain the largest Lyapunov exponents to plot the Lyapunov diagrams [Marcondes et al., 2017; Gallas, 2015; da Costa et al., 2016; de Sousa et al., 2016] (see also references therein) and using the number of spikes, i.e., the number of maxima in a period of the variable y(t) in the system (2), computed after a transient-time equal to 107 , we plot the isospikes diagram [Gallas, 2016; Hoff et al., 2013]. Both diagrams are obtained by discretizing the parameter-pair (G, K) in a grid of 103 × 103 values (the same grid is also used in Figs. 2, 3 and 4(b)), where a typical random initial condition (IC) given by (x0 , y0 , z0 ) = (π/2, 0, 1/2) was used to all parameter pairs. Some tests using other IC sets were performed and the Lyapunov and isospikes diagrams remain practically unchanged. Although, values of (K, G) only must be positive, as discussed in the previous Section, the choice for the (K, G) ranges used in this work is motivated by: (i) the existence of typical SPSs immersed in a chaotic domain and (ii) the comparison of theoretical Hopf bifurcation curve H1 with numerical simulations. The amplifications performed next (Figs. 3 and 4) are motivated by reason (i) and by the characterization of an organizing rule for a specific sequence of SPSs. In Figs. 2(a) and (b) we show the Lyapunov and isospikes diagrams, respectively. In (a) the color bar at right-side codifies the value of the largest Lyapunov exponent in a color gradient: white represents negative exponents (for equilibrium points), black one null exponent (related to periodic attractors) and continuously variation from yellow up to red for positive exponents (representing chaotic attractors). In (b) the color palette also at right-side codifies the number of spikes in one period of the time-series of trajectory. For example, blue color is related to a domain where in each period of time-series there exist 1 spike, while for green color domains there exist 2 spikes and, so on. The same logic rule can be extended to higher number of spikes until 11. Moreover, black color codifies chaotic behavior or the number of spikes larger than 11 and white color codifies equilibrium points as in (a). The theoretical Hopf bifurcation curve H1 , where system (2) presents a local bifurcation at the equilibrium point P1 = (0, 0, 0) with the birth or death of periodic attractors (see the magenta continuous-curve in Figs. 2(a) and (b)) obtained in Sec. 2, is also overlapped in these diagrams showing a remarkable agreement between the theoretical analysis and numerical experiments. For example, considering the typical point (G1 , K1 ) in Fig. 1(a), in Fig. 2(a) this point is a numerical attractor equilibrium point with negative largest Lyapunov exponent (in the white color domain). For the typical point (G2 , K2 ) in Fig. 1(a), in Fig. 2(a) this point is also a numerical attractor equilibrium point but now with slower temporal and numerical convergence in the largest Lyapunov exponent than the previous point. For the typical point (G3 , K3 ) in Fig. 1(a), this point in Fig. 2 is a numerical periodic attractor born by a Hopf bifurcation with null largest Lyapunov exponent (in the black, or blue, domain in Figs. 2(a), or (b), respectively). Those stability diagrams clearly show rich and complex dynamical behaviors that the PLL model (2) presents and, in addition, they endorse the goal of this work which is the characterization of dynamics of a PLL model device. Next, we explore the main features of those behaviors by using few magnifications of Lyapunov diagram, attractor’s projection and, time series to characterize transient chaotic behaviors.

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Figure 2. In (a) Lyapunov diagram for G × K for the largest exponent σ1 of the Lyapunov spectrum codified by colors, as indicated by color bar at the right-side of this panel. Black color (σ1 ∼ 0) indicates the existence of periodic attractors while the gradient from yellow to red (σ > 0) color is associated to chaotic attractors. In (b) the isospikes diagram is plotted for the same range of parameters used in (a) and the right-side discrete color palette indicates the number of spikes in one period of the variable y(t) of system (2). Black color codifies chaotic behavior and the number of spikes larger than 11 and, in both panels white color represents equilibrium points. H1 in both panels indicate the theoretical Hopf bifurcation curve plotted in magenta color and obtained for the equilibrium point P1 in Sec. 2 (see also Fig. 1(a)).

Clearly, in the stability diagrams of Fig. 2, we observe a profuse presence of the SPSs embedded in the chaotic domain concentrated near to a large periodic domain for 1.4 . K . 2.7 (see black domain in (a) or the multicolored domain in (b)). Away from this periodic domain, for K & 2.7, the chaotic domain predominate and the SPSs disappear, i.e., only chaotic attractors are found for such parameter set. Comparing both diagrams, we also observe different bifurcation routes to chaos, between periodic attractors in that large periodic region. For example, in the Lyapunov diagram, it is not possible to identify the bifurcation points which occur inside the large black region, however, in (b), the isospikes diagram, one is able to identify bifurcations between periodic attractors. Indeed, along different directions inside that region, we see period-doubling bifurcations. For example, if one follow a straight vertical-line starting from K = 0 for G = 17.5, it is possible to see the following bifurcation sequence to chaos: equilibrium point (white) → period-1 (blue) → period-2 (green) → period-4 (red) → period-8 (cian) → · · · → chaos. Different directions give other bifurcation sequences. With the aim to explore in more details the fine structure of the parameter plane in Fig. 2, we enlarge the region delimited by the green-dashed box in (a), and present it in Fig. 3(a). Again, a plethora of self-organized SPSs in specific directions of the parameter plane is observed. In Fig. 3(b), we also present the Lyapunov diagram for a tiny region of the diagram in Fig. 2(a), marked by a small greencontinuous box highlighted by a green-arrow. These typical diagrams show how curious and intricate is the organization of SPSs in the parameter plane of system (2). Those types of SPSs and their self-organization rules are commonly found in a large variety of dynamical systems. To be more specific, two exemplary self-organizations are presented in Figs. 3(b) and 4. In the first (Fig. 3(b)) a typical configuration of three SPSs connected by their antennas as previously observed in Refs. [Celestino et al., 2011; Francke et al., 2013] and [Manchein et al., 2013]. In the second, Fig. 4(a) is a bifurcation diagram used to identify the mathematical rule governing the organization of SPSs (highlighted by white stars in Fig. 5(b) which is a magnification of green-dashed box plotted in Fig. 3(a)) along a direction toward a periodic (accumulation) boundary [Bonatto & Gallas, 2008]. As far as we know this is the first work that explores the existence

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Figure 3. Magnification of a fraction of parameter plane highlighted by a green-dashed box in Fig. 2(a). In panel (a) several typical stable periodic structures embedded in chaotic domains are observed and, in panel (b) a peculiar formation known as a trio composed by three shrimp-shaped domains connected by antennas and found, for example, in parameter planes of ratchet systems [Celestino et al., 2011; Manchein et al., 2013].

and organization of such SPSs in a PLL model.

Figure 4. (a) Bifurcation diagram y(t) × K obtained for a vertical-line connecting white-stars for G = 6.88 in (b) which is a magnification of a tiny fraction of parameter plane plotted in Fig. 2(a) highlighted by green-dashed box in Fig. 3(a). Red numbers in the right-side of panel (a) indicate the period of respective periodic-window localized at left-side of each number, which is represented by a white-star over the respective stable periodic structure plotted in (b).

In Fig. 4(a) we present a bifurcation diagram obtained along a vertical line that connects the whitestars over main SPSs toward a periodic boundary in the Lyapunov diagram of Fig 4(b). Red numbers in (a) refer to the number of spikes of the respective SPSs with stars in (b). We can associate this sequence of spikes to a branch of a Stern-Brocot tree, observed in some models of dynamical systems [Freire & Gallas, 2011]. For example, the sum of spikes’ number of the two consecutive larger SPSs gives rise the spikes of the smaller intermediate ones. It is worth to mention that we can observe others SPSs self-organizing toward the same periodic boundary in Fig. 4(b).

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In Fig. 5, panels (a)-(e), we present exemplary periodic attractors’ projections in z × y phase space slice showing the asymptotic behavior after a long transient time and a chaotic attractor’s projection in panel (f). Each panel is obtained for a fixed pair of parameters (see the title of each panel of Fig. 5) in the Lyapunov diagram of Fig. 4(a). By analyzing this figure it becomes evident that larger periods represent more elaborated attractor’s topology and dynamics.

3.0

K = 0.50

4.5

K = 0.90 (b)

K = 1.30 (c)

y

(a)

6.0

−3.0 −3.0

z

1.0

K = 1.50

1 3.0

0.0 −5.0 10

z

2 4.0

K = 1.85 (e)

z

25

K = 2.60

−25 −30

z

(f)

y

(d)

3 6.0

−6.0 −6.0

−7.0 −8.0

z

4 −10 8.0 −15

z

9 15

chaos 30

Figure 5. Each panel in this figure is an attractor’s projection in the plane z × y for G = 12.5 and the respective K value indicated in the title. Panels (a)-(e) represent periodic orbits where the period of each attractor is indicated by red number in the right-bottom side of the respective panel. Panel (f) presents a chaotic attractor’s projection. The period in each attractor’s projection is equal to the number of spikes estimated in Fig. 2(b) for the same parameter’s combination.

3.2. Transient chaotic dynamics The existence of chaotic dynamics with arbitrarily long but finite lifetime is known as transient chaos and provides a kind of “pre-asymptotic state” which is different from the asymptotic state [Lai & T´el, 2011]. Consequently, the transient chaos is an example of phenomenon where long-time single realization and short-time ensemble averages are different. A dynamics that presents these “pre-asymptotic states” is characterized by topological changes of the attractor in the phase space for a given parameter combination for long-time evolution, and its signature in the Lyapunov diagram is presented by periodic domains with grainy borders in the interface between periodicity and chaos [Hoff et al., 2013]. There are many situations where the transient chaos phenomenon is studied as, for example, problems ranging from Classical to Quantum aspects as listed in Ref. [T´el, 2015] (see also references therein). In the problem studied here, the dynamics of PLL device model (2), presents transient chaos for a set of parameter pairs in the range 2.7 . K . 2.9 and G & 15, better visualized in Fig. 2(a) and (b) at top-right corner. For an experimental point of view it is essential to identify parameter space domains where the transient chaos takes place: specially, for example, if one would like to avoid nonlinear phenomena like crises and basin boundaries [T´el, 2015]. In Fig. 6(a), we present the typical time-series behavior for the parameter pair in the position of the white-star in Fig. 2(a), showing the feature of transient chaos. After a transient time, the chaotic motion (gray line) evolves into a periodic one (black line), corroborated by the temporal

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Manchein, Albuquerque, and Mello

Figure 6. In (a) transient chaos (gray line) and periodic motion (black line) characterized by a time series for the parameter pair written in the title of this figure and also represented by a white-star in the right-top side of Fig. 2(a). The Lyapunov spectrum σi (for i = 1, 2, 3) is superposed to the time series and the decay of value of σ1 is a strong evidence of transient chaos and the slow convergence to periodic motion. In (b) chaotic (gray line) and periodic attractor’s projection (black line) are superposed to compare the amplitude of each attractor.

change of the Lyapunov spectrum, also plotted in the same panel (red, green and blue lines representing the Lyapunov spectrum σi , for i = 1, 2, 3 from the largest to smallest one, respectively). In Fig. 6(b) we plot the attractor’s projection in the plane z × y: As observed in (a) gray lines represent transient chaotic regime while black lines is associated to the asymptotic periodic behavior. Following Ref. [Lai & T´el, 2011] the example presented follows the case in which the non-attracting chaotic set coexists with an asymptotic periodic attractor. This implies that, over certain domains in parameter plane of PLL device model, the system is virtually unpredictable over a long period of time. Obviously, for a practical system which uses PLL devices, this kind of behavior is undesirable. Actually, the present finding is essential to identify the parameter combinations or domains in parameter plane, related to long transient chaotic regimes to avoid the contamination of time-series of some observable sampled in real experiments involving the application of PLL devices.

4. Summary and conclusions We found that the phase-locked loop (PLL) device, modeled by the third-order autonomous differential equation (1) produces unexpected and rich dynamical behaviors and long transient chaotic motions for specific parameter pairs. Due the technological importance of PLL devices, our main objective has been to extend the recent interesting investigations in [Piqueira, 2017], where the author presents some theoretical and numerical results about one of two existent fundamental equilibrium points. We start by introducing a theoretical analysis of all the possible bifurcations which occur for both equilibrium points P1 = (0, 0, 0) and P2 = (π, 0, 0) of system (2). In this way, the previous analyzes presented in [Piqueira, 2017] are extended and completed here. In addition, we have characterized the existent Hopf bifurcations responsible to give the simplest way in which the solutions of (2) present a periodic oscillatory regime with the birth (or death) of limit cycles. For a matter of completeness, we also perform an extensive numerical investigation of PLL dynamics as both parameters in (2) are varied. The presence of typical stable periodic structures (SPSs) embedded in chaotic domains of parameter plane (G × K) is shown in Lyapunov and isospikes diagrams as illustrated by Figs. 2(a) and (b), respectively. Our numerical results corroborate with the generic nature of those SPSs, independently of the systems’ nonlinearity. Comparing the theoretical Hopf curve H1 obtained for P1 in

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Sec. 2 (see Fig. 1(a)), with the numerical results plotted in the Lyapunov and isospikes diagrams in Fig. 2, it is possible to realize the remarkable concordance between such results, even if the invariant measure used in each case is different. In addition, we discuss how the SPSs are self-organized along preferred directions (see Figs. 3(a) and 4) of the parameter plane of PLL model (1) and, show the presence of an intriguing structure composed by a trio of shrimp-shaped domains self-connected by antennas as recently reported in [Celestino et al., 2011; Francke et al., 2013; Manchein et al., 2013]. Another interesting finding is the appearance of chaos with finite lifetime usually known as transient chaos which provides an example of a “nonequilibrium state” that is different from the asymptotic state, and cannot be understood from the asymptotic behavior alone. In fact, by studying only the asymptotic regime of such dynamics would mean loosing the interesting, chaotic behavior contained in the transients specially, for the subject (PLL device model) of the present work, which has a strong technological appeal. For this reason, in order to avoid (or enhance) transient chaotic behaviors it is interesting to identify chaotic and regular dynamics or the mixture of them in the parameter plane used to choose the right parameter pairs to perform real experiments. After the present explanation, we hope this work may trigger research about application of PLL devices both experimentally and theoretically. Possible theoretical directions involve developing tools to characterize and predict where specific stable periodic structures might be located in stability diagrams and, more importantly, which type of nonlinearities are able to produce them. On the other hand, experimental setups using circuitry implementations may be produced to investigate how control or stabilize chaotic trajectories and as consequence reduce the effect of intrinsic experimental noise.

Acknowledgments The authors thank CNPq and CAPES (Brazilian agencies) for financial support. C.M. and H.A.A. also thank FAPESC (Brazil) for financial support. L.F.M. was partially supported by Capes/Est´agio Sˆenior no Exterior grant number 88881.119020/2016–01. In the course of this work L.F.M. was a visitor at Texas Christian University and gratefully acknowledges its warm hospitality.

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