Stability of Hopf Bifurcations in TimeDelayed FullyConnected Pll Networks
Abstract
Dynamics in delayed differential equations (DDEs) is a well studied problem mainly because DDEs arise in models in many areas of science including biology, physiology, population dynamics and engineering. The change of nature in the solutions in the parameter space for a network of PhaseLocked Loop oscillators was studied in Symmetric bifurcation analysis of synchronous states of timedelayed coupled PhaseLocked Loop oscillators. Communications in Nonlinear Science and Numerical Simulation, Elsevier BV, Volume 22, Issues 13, May 2015, Pages 793820, where the existence of Hopf bifurcations for both cases, symmetrypreserving and symmetrybreaking synchronization was well stablished. In this work we continue the analysis exploring the stability of smallamplitude periodic solutions emerging near Hopf bifurcations in the Fixedpoint subspace, based on the reduction of the infinitedimensional space onto a twodimensional center manifold. Numerical simulations are presented in order to confirm our analitycal results. Although we explore network dynamics of secondorder oscillators, results are extendable to higher order nodes.
Introduction
Networks of oscillators have been studied for decades because their models can represent dynamics in a very wide range of fields, as astronomy, biology, neurology, economics, and the stock market, [10, 17, 16, 2, 18]. Much of the research has focused to understand the influence that changes in the parameter space have over the global dynamics. In that sense, the stability of the synchronization in the network have been explored from many approaches [16, 21, 20, 6]. In this contibution, we study the stability of smallamplitude periodic solutions which emerge near Hopf bifurcations of the symmetrypreserving solutions for a Nnode PLL network, using the centre manifold theorem extended to functional differental equations, and the normal form in the center space. This work is mainly based on previous results obtained on the existence of symmetrypreserving and symmetrybreaking Hopf bifurcations in a Nnode secondorder PLL network, by Ferruzzo et al. [5].
It is important to note that when the lag in the communications between nodes is considered, the ordinary differential equation (ode) system which describe the network dynamics, becomes a delayed differential equation (dde), whose solution lies, in general, in the function space, and its characteristic equation has infinitely many roots. This particular kind of functional differential equations appear in many engineering problems [15, 19, 8].
We consider the Full Phase model introduced in [5] to analyse stability of smallamplitude periodic orbits near Hopf bifurcations emerging in the parameter space at the Fixedpoint subspace, for the non degenerative case (). It has been shown that these bifurcations can occur when a pair of complex conjugate eigenvalues crosses the imaginary axis in iether direction, from the left to the right and from the right to the left. The main approach used for the analysis is the decomposition of the infinitedimentional space into a 2dimensional center space spanned by the eigenvectors correponding to the simple imaginary eigenvalues , , and into an infinitedimentional space “orthogonal” to the first one (the orthogonality condition will be defined below). We will follow closely the theory and procedures presented in [15, 7, 22, 1, 9].
The Fullphase model
The general model for a node, fullyconnected, secondorder oscillator network, in terms of the th node output phase , is:
(1) 
where:
(2) 
, , and .
The equilibria , in equation (1), are:
(3) 
. For our analysis, we consider three main assumptions:

The critical eigenvalue of the linearization of (1) at equilibria crosses the imaginary axis with non vanishing velocity, i.e. .

The purely imaginary eigenvalue is simple.

The linearization of (1) at equilibria, has no eigenvalues of the form , .
The Taylor expansion of (1) at equilibria is:
(4) 
where . Truncate the series up to the thirdorder term:
(5) 
, here, for the sake of notation we changed .
The vector field form , , can be obtained by choosing and , then, the restriction , or, gives:
(6) 
Following [11, 7], we can represent the dynamics in (6) by the abstract differential equation:
(7) 
We define , the Banach space of continuous functions from into , equipped with the usual norm
, in (7), lies in and satisfies , where is a semigroup of family of operators, , and is a vector of parameters. The linear operator is:
(8) 
where , and,
(9) 
, , , and
In order to build the decomposition of the infinitedimensional space, we need the adjoint operator associated to the linear part of the linearization and a inner product, via a bilinear form. Associated to the linear part of (7), the formal adjoint equation is
(10) 
The strongly continuous semigroup , defines the infinitesimal generator:
(11) 
such that , . The natural inner product has the form [12]:
and ; thus, we have [7]:

is an eigenvalue of if and only if is and eigenvalue of .

If is a basis for the eigenspace of and is a basis for the eigenspace of , construct the matrices and . Define the bilinear form:
(12)
The Fixed Point space
Due to the symmetry of (1) the space where solutions lie can be decomposed into the Fixedpoint subspace where symmetrypreserving solutions emerge and a subspace with symmetrybreaking solutions, this was shown in [5]. We analyze stability of the smallamplitude periodic solutions near Hopf bifurcations in the Fixed point space, these bifurcations satisfy assumptions (a)(c) for . In this subspace, equation (6) has the form:
(13) 
then matrices and in (11) become:
(14) 
(15) 
and in (9) takes the form , with , and :
(16) 
We need the complex eigenfunctions , , associated to the critical eigenvalues , and with and . These eigenfunctions can be computed solving the boundary value problem , and , which, after substituting the operator , becomes:
(17) 
and
(18) 
with general solutions:
(19) 
The coefficients can be obtained by considering the boundary conditions,
(20) 
the “orthonormality” condition , and setting and , see [15, 13] for more details.
It is also possible to decompose the solution to equation (7) into , where and lie in the center subspace, such that , and in the infinitedimensional component subspace, thus, we have
(21) 
(22) 
where
(23) 
The center manifold
Following [14, 15, 22], we know that w can be approximated by the secondorder expansion:
(24) 
thus, by differentiating and substituting equation (22) keeping up to second order terms, we obtain:
(25) 
and from equation (22),
(26) 
From the definition of , equivalent to (11), we see that
(27) 
then, from equation (24), (25), (26), and (27), we can obtain the unknown coefficients , and , solving:
(28) 
and,
(29) 
where , , and .
Equation (28) is written as the inhomogeneus differential equation:
(30) 
where
with general solution:
(31) 
After substituting the general solution into (30) we solve for and , and then from the boundary value problem we solving for ,
(32) 
(33) 
where
(34) 
and .
The expressions for and , necessary in (23), are:
(35) 
note that we only need since the nonlinear function in (16) only depends on ; then by substituting (35) into (21), we obtain:
(36) 
or
(37) 
In [9] is computed the coefficient , which determines stability of the normal form (37),
(38) 
where . Periodic orbits near Hopf bifurcation at the critical eigenvalue , will be stable if and unstable if .
Numerical Results
We reproduced some of the computations for the Hopf bifurcations in the Fixed point space for the case presented in [5], because we will compute stability for these bifurcation curves using results obtained in the previous section. In figure 1 (part of figure 10, in [5]) are shown the symmetrypreserving bifurcations curves in the parameter space for , for both cases: bifurcations with in black color, and with in red color; we also choose three testing point for numerical simulation , , and .
In figure 2 is shown the coefficient computed using equation (38), in the parameter space , for , related to the Hopf bifurcations curves shown in figure 1. The black curve corresponds to stability of periodic orbits near Hopf bifurcations with (black curves in figure 1), as we can see, these periodic solutions are all stable (). The red curve corresponds to stability of periodic orbits near Hopf bifurcations with (red curves in figure 1), these periodic orbits are unstable for , and stable for . Are also shown point , anc . At points and , small amplitude periodic orbits are stable, whilst at point , they are unstable.
In order to confirm our results, we computed branches of periodic solutions near the Hopf bifurcations points , , and , using DDEBIFTOOL [4, 3], along with the Floquet multipliers for a specific periodic solution chosen in the branch.
A branch of periodic solutions with small amplitude, emerging from the Hopf bifurcation point is shown in figure 3(a). In 3(b) it is shown the periodic solution profile at . The Floquet multipliers related to are shown in figure 3(c). It is clear that this periodic solution is stable, since there is no Floquet multiplier outside the unity circle.
For the point , the branch of periodic solutions is shown in figure 4(a), in figure 4(b), it is shown the profile for the periodic solution chosen at , these solution is unstable, because there is a Floquet multiplier outside the unity circle, see figure 4(c).
Finally, the branch of periodic solutions near the Hopf bifurcation point is shown in figure 5(a). The periodic solution choosen in the branch is at , its profile is shown infigure 5(b). All the Floquet multipliers shown in figure 5(c) are within the unity circle, therefore the solution is stable.
Conclusions
The reduction of the inifinitedimensional space onto the center manifold in normal form, was applied to the Fixed point space for the Fullphase model in order to analyse the stability of smallamplitude periodic orbits near simple Hopf bifurcations, in both cases, for and , we found that in the first case periodic orbits which are stable () can emerge, and, in the other case, unstable () periodic orbits can emerge for , and stable periodic orbits for . The numerics show that the analytical results are correct.
Although, we computed the coefficient for a specific value of , the procedure shown is valid for all the parameter space where simple Hopf bifurcations appear.
Finally, it is important to spotlight some points for further research: First, what is the nature of the solutions at the special point , at which the coefficient changes sign. Second, analyze stability of the degenerate Hopf bifurcations at the Fixed point space for , which are codimension 2, pure imaginary eigenvalue and zero eigenvalue; and third, the stability of the symmetrybreaking degenerate Hopf bifurcations which have multiplicity .
acknowledgment
We would like to thank the Escola Politécnica da Universidade de São Paulo and FAPESP for their support.
References
 SueAnn Campbell. Calculating centre manifolds for delay differential equations using maple. In Delay Differential Equations, pages 1–24. Springer US, 2009.
 Matthew Earl and Steven Strogatz. Synchronization in oscillator networks with delayed coupling: A stability criterion. Phys. Rev. E, 67(3), Mar 2003.
 K. Engelborghs, T. Luzyanina, and D. Roose. Numerical bifurcation analysis of delay differential equations using DDEBIFTOOL. ACM Trans. Math. Softw., 28(1):1–21, March 2002.
 Koen Engelborghs, Tatyana Luzyanina, and Giovanni Samaey. DDEBIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations. In Numerical Analysis and Applied Mathematics Section. Department of Computer Science, K.U.Leuven, Leuven, Belgium, October 2001.
 Diego Paolo Ferruzzo Correa, Claudia Wulff, and José Roberto Castilho Piqueira. Symmetric bifurcation analysis of synchronous states of timedelayed coupled phaselocked loop oscillators. Communications in Nonlinear Science and Numerical Simulation, 22(1â3):793 – 820, Aug 2015.
 Fotios Giannakopoulos and Andreas Zapp. Bifurcations in a planar system of differential delay equations modeling neural activity. Physica D: Nonlinear Phenomena, 159(34):215 – 232, 2001.
 David E. Gilsinn. Bifurcations, center manifolds, and periodic solutions. In Delay differential equations, pages 155–202. Springer, New York, 2009.
 DE Gilsinn. Estimating critical Hopf bifurcation parameters for a secondorder delay differential equation with application to machine tool chatter. NONLINEAR DYNAMICS, 30(2):103–154, OCT 2002.
 John Guckenheimer and Philip Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, volume 42. New York Springer Verlag, 1983.
 I. Győri and G. Ladas. Oscillation theory of delay differential equations. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1991. With applications, Oxford Science Publications.
 J. K. Hale. Functional differential equations. SpringerVerlag, New York, 1971.
 J. K. Hale and S. M. Verduyn Lunel. Introduction to functional differential equations. SpringerVerlag, London, 1993.
 Jack K. Hale. Theory of Functional Differential Equations (Applied Mathematical Sciences). Springer, 1977.
 Brian D. Hassard, Nicholas D. Kazarinoff, and Yieh Hei Wan. Theory and applications of Hopf bifurcation, volume 41 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1981.
 Tamás KalmárNagy, Gábor Stépán, and Francis C. Moon. Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations. Nonlinear Dynamics, 26(2):121–142, 2001.
 Wenxue Li, Hongwei Yang, Liang Wen, and Ke Wang. Global exponential stability for coupled retarded systems on networks: A graphtheoretic approach. Communications in Nonlinear Science and Numerical Simulation, 19(6):1651–1660, Jun 2014.
 A. Martins and L.H.A. Monteiro. Frequency transitions in synchronized neural networks. Communications in Nonlinear Science and Numerical Simulation, 18(7):1786–1791, Jul 2013.
 A. Ponzi and Y. Aizawa. Selforganized criticality and partial synchronization in an evolving network. Chaos, Solitons & Fractals, 11(7):1077 – 1086, 2000.
 E Stone and SA Campbell. Stability and bifurcation analysis of a nonlinear DDE model for drilling. JOURNAL OF NONLINEAR SCIENCE, 14(1):27–57, JANFEB 2004.
 Andrea Vuellings, Eckehard Schoell, and Benjamin Lindner. Spectra of delaycoupled heterogeneous noisy nonlinear oscillators. EUROPEAN PHYSICAL JOURNAL B, 87(2), FEB 3 2014.
 Yuan Yuan and XiaoQiang Zhao. Global stability for nonmonotone delay equations (with application to a model of blood cell production). Journal of Differential Equations, 252(3):2189–2209, Feb 2012.
 Siming Zhao and Tamáss KalmárNagy. Center Manifold Analysis of the Delayed Lienard Equation. In Delay Differential Equations, pages 1–17. Springer US, 2009.