1. Isochronal synchronization in mutually delay-coupled R¨ossler oscillators
Neal Woo
Reed College, 2014
Introduction
The synchronization of coupled chaotic systems has been a highly attractive problem in
nonlinear sciences, with applications to engineering, biological and chemical systems,
cryptocommunication, and more. In its two decades of study, the problem has moved
from the specific to the general, where even the collective behavior of huge networks
of nonidentical systems can be studied.
Despite a solid set of analytical tools, only recently have there been attempts to un-
derstand synchronization with time delay. These systems are common in nature and
inevitable if we are to model real, physical processes where coupling signals travel at
finite speed. Analysis of time-delay systems is challenging as they follow delay dif-
ferential equations (DDEs), which depend on an infinite-dimensional phase space of
history functions. While some methods such as Lyapunov theory have been extended
to DDEs, these systems are usually studied either by numerical integration or lineariza-
tion.
In this project, we study the dynamical behavior and synchronization of two mutu-
ally delay-coupled R¨ossler oscillators. Following the analysis by J¨ungling et al.1
, we
characterize the fixed points, periodic orbits, and synchronized states by linearization.
Specifically, we focus on the ‘stability dilemma’ for identically coupled systems, and
explore the effect of large delays on synchronous states.
Approach
We start with two identical R¨ossler systems x1, x2, with mutual dissipa-
tive coupling weighted by the parameter and with time-delay τ,
˙x1 = −y1 − z1 + (x2,τ − x1)
˙y1 = x1 + .15 y1
˙z1 = .2 + z1 (x1 − 10)
˙x2 = −y2 − z2 + (x1,τ − x2)
˙y2 = x2 + .15 y2
˙z2 = .2 + z2 (x2 − 10)
↔
˙x1 =f(x1) + K(x2,τ − x1)
˙x2 =f(x2) + K(x1,τ − x2)
(1)
where xi,τ ≡ xi[t − τ]. It is clear that the isochronal synchronization
state x1(t) = x2(t) remains a solution of (1); the problem is to determine
its stability given that the coupling term is invasive (nonvanishing in the
synchronous state) for τ = 0. Transforming to longitudinal and transver-
sal coordinates u = (x1+x2)/2 and v = (x1−x2)/2, the synchronization
manifold is defined by u = x1 = x2 and v = 0. For small synchro-
nization errors v ≈ 0 about some general solution ξ(t), we linearize by
f(ξ(t) ± v) f(ξ(t)) ± J(ξ(t))v, where J is the Jacobian matrix of f,
evaluated at ξ(t).
Fixed Points
The free R¨ossler system has an unstable fixed point at the origin x = 0.
Linearizing the normal coordinates, we get decoupled equations for the
small deviations and evaluate the Jacobian at the fixed point 0:
˙δu = J(0)δu + K(δuτ − δu)
˙δv = J(0)δv + K(−δvτ − δv) ,
J(0) =


0 −1 −1
1 .15 0
0 0 −10

 . (2)
We see that both δu3 and δv3 rapidly tend to zero by e−10 t. Treating the
first two terms as a complex-valued normal form, where z = δu1 + i δu2
and w = δv1 + i δv2, we then get
˙z = λ0 z + /2 (zτ − z)
˙w = λ0 w + /2 (−wτ − w) , (3)
where λ0 = .075 + .997 i is the complex eigenvalue of the 2×2 Jacobian
(ignoring δu3 and δv3). The eigenvalues of (3) are determined by
λ = λ0 − /2(1 − e−λ τ)
λ⊥ = λ0 − /2(1 + e−λ⊥τ) , (4)
and the fixed point 0 is stable if both Re(λ ) < 0 and Re(λ⊥) < 0. We
ran simulations in the ( , τ) domain and calculated real components of
the maximal Lyapunov exponents, as shown in Figure 1.
Re(λ⟂) < 0
Re(λ||) < 0
All Re(λ) < 0
All Re(λ) > 0
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
τ
ϵ
λ⟂� λ|| ��������� �������
Figure 1: Lyapunov-stable regions of longitudinal and transverse subspaces.
As a consequence of the form of (4), we see that transverse-stable
domains are centered at integer multiples of the uncoupled oscillator’s
mean period, T0 ≈ 6; the longitudinal domains occur at half-integer
multiples. We also see that, as τ increases, the stable domains decrease
in size. Oscillation death or quenching is found only in the small region
of overlap near τ ≈ T0/4, as shown in Figure 2.
(x1, x2)
(y1, y2)
(z1, z2)
-10 -5 5 10
-10
-5
5
x1(t) x2(t)
20 40 60 80 100
t
-20
-10
0
10
20
xi(t)
-10
0
10
x2(t)
-10
0
10
y2(t)
0
10
20
30
z2(t)
Figure 2: (τ = 1.5, = .6): Oscillation death in the overlap of longitudinal and
transverse-stable domains.
Periodic Solutions
The chaotic R¨ossler attractor contains an infinite set of unstable periodic
orbits (of period Tn ≈ n T0 for n ∈ Z+) that arise from the period-
doubling route to chaos. These ‘hidden orbits’ can be stabilized and
controlled in a number of ways, including by time-delayed self-feedback
with τ = Tn in uncoupled systems. This method of Pyragas control uti-
lizes the finite torsion of an orbit in order to draw trajectories toward the
periodic solution, where the coupling term vanishes.
Mutually coupled systems are fundamentally unable to be stabilized by
Pyragas control due to the opposite signs of the delay terms in (2), which
ensure that, whenever the torsion of an orbit and τ coincide to make the
coupling term approach 0 (either in δu or δv), the other system’s cou-
pling term remains and destabilizes the periodic orbit. This is in direct
analogy to the ‘either-or’ stability dilemma for fixed points. However,
stable periodic orbits can in principle be observed by detuning τ from
Tn. Unlike the fixed-point problem, it is difficult to solve for the lo-
cations of these periodic solutions, because the invasive coupling term
deforms their shape.
Searching for Synchronization
To map out the synchronous states, we sweep the parameter space up to
= .8 and τ = 42 ≈ 7 T0. For each (τ, ), we calculate σ = (x1 − x2)2
over several dozen cycles to estimate synchronicity.
Figure 3: Numerical simulations of σ(τ, ), the root-mean-squared value of the differ-
ence between identical systems x1 and x2 with randomized initial conditions
Qualitatively, we see isochronal synchronization of chaos for small τ
and ( > .1) in the purple strip in the left of Figure 3. This is sensible
because a small τ can be treated as a perturbation if the signal doesn’t
change rapidly on that timescale. The behavior is plotted in Figure 4:
(x1, x2)
(y1, y2)
(z1, z2)
-10 10 20 30
-10
10
20
30
x1(t) x2(t)
420 440 460 480 500
t
-20
-10
0
10
20
xi(t)
-10
0
10
x2(t)
-10
0
10
y2(t)
0
10
20
30
z2(t)
Figure 4: (τ = .25, = .6): Isochronal synchronization of chaos. The flattened, diag-
onal correlation plot indicates that the trajectories evolve in unison. On the right, we
recover the chaotic R¨ossler attractor.
The red plateaus in Figure 3 exhibit chaotic, anti-phase synchronization
as plotted in Figure 5:
(x1, x2)
(y1, y2)
(z1, z2)
-10 10 20 30
-10
10
20
30
x1(t) x2(t)
220 240 260 280 300
t
-20
-10
0
10
20
xi(t)
-10
0
10
x2(t)
-10
0
10
y2(t)
0
10
20
30
z2(t)
Figure 5: (τ = 3, = .15): Chaotic anti-phase synchronization. The negative-going
trend of the correlation plot indicates that the systems are always out of phase, while
its elliptical shape is symptomatic of leader-laggard switching between the oscillators.
Stable Periodic Orbits
We now focus on the regions (τ < 1.5 T0) and (2.5 T0 < τ < 3.5 T0) to
find detuned, stable periodic orbits, and observe the effects of large delay
on their dynamics.
Figure 6: Simulations of σ(τ, ) are represented in the colored region, with purple
indicating low σ (synchronous behavior). The colored dots indicate regions where a
significantly periodic signal is detected. The white dotted line corresponds to T0 on the
left, and 3 T0 on the right.
Synchronous periodic states appear to be found in the transitions between
isochronal and anti-phase synchronization, or the ‘peaks’ and ‘valleys’
of Figure 3. Figure 7 is an orbit from the robustly periodic transition
regions in the small-τ regime.
(x1, x2)
(y1, y2)
(z1, z2)
-10 10 20
-10
10
20
x1(t) x2(t)
400 420 440 460 480 500
t
-20
-10
0
10
20
xi(t)
-10
0
10
x2(t)
-10
0
10
y2(t)
0
10
20
30
z2(t)
Figure 7: (τ = .8599, = .1): Stable 3-cycle orbit.
We note that the large-τ phase space permits visibly fewer periodic solu-
tions; increasing τ tends to smear neat distinctions and generate unpre-
dictable phenomena. Figure 8 is from the large-τ regime which, though
deformed, maintains two pockets of periodic solutions.
(x1, x2)
(y1, y2)
(z1, z2)
-15 -10 -5 5 10 15
-15
-10
-5
5
10
15
x1(t) x2(t)
420 440 460 480 500
t
-20
-10
0
10
20
xi(t)
-10
0
10
x2(t)
-10
0
10
y2(t)
0
10
20
30
z2(t)
Figure 8: (τ = 14.75, = .09): Stable, bimodal 2-cycle orbit.
As expected, all of these stable solutions are found at τ = n T0. Fig-
ure 9 is just one example of the strange behavior for large-τ. The two
oscillators interchange leader-laggard roles in discrete cycles. The fact
that τ = 39.5 so closely matches the observed frequency illustrates the
complexity of isochronal synchronization in time-delay systems, espe-
cially when τ is larger than the Lyapunov time. Often, it is not the two
oscillators that are ‘synchronized’ together, but it is the system at time t
being synchronized to the ‘image’ of itself in the past.
(x1, x2)
(y1, y2)
(z1, z2)
-15 -10 -5 5 10 15
-10
10
20
x1(t) x2(t)
350 400 450 500
t
-20
-10
0
10
20
xi(t)
-10
0
10
x2(t)
-10
0
10
y2(t)
0
10
20
30
z2(t)
Figure 9: (τ = 39.5, = .775): Anomalous behavior
1
J¨ungling, T., Benner, H., Shirahama, H., & Fukushima, K.
Complete chaotic synchronization and exclusion of mutual Pyragas control in two delay-coupled R¨ossler-type oscillators, Phys. Rev. E. 84 (2011)