12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

HOMOCLINIC BIFURCATION, EDGE STATES, AND BOUNDARY CRISIS IN PLANE
COUETTE FLOW
Julius Rhoan T. Lustro
Department of Mechanical Engineering
University of the Philippines Diliman
Magsaysay Avenue, Diliman, Quezon City,
Metro Manila 1101, Philippines
jtlustro@up.edu.phYudai Shimizu
Measurement Instrumentation and Digital
Development Innovation Division
Toyota Motor Corporation
1, Toyota-cho, Toyota, Aichi 471-8571, Japan
yudai19891023@gmail.com

Genta Kawahara
Graduate School of Engineering Science
Osaka University,
1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japankawahara@me.es.osaka-u.ac.jp

ABSTRACTWe describe a rich bifurcation scenario that is related to the
Nagata steady solution and time-periodic solutions in a plane
Couette flow with streamwise period slightly longer than theminimal unit. We identify the presence of three homoclinic
bifurcations that are linked with the creation (or destruction) of
the time-periodic solutions. These periodic orbits function as
homoclinic orbit to the lower-branch Nagata steady solution
during homoclinic bifurcation at critical values of Reynolds
number. In this computational domain, the lower branch of the
Nagata steady solution acts as edge state at lower Reynolds
number. At higher Reynolds number, the edge state switchesfrom the lower-branch Nagata steady solution to the vigoroustime-periodic solution during the creation of this limit cycle dueto the homoclinic bifurcation. As a consequence, the stablemanifold that forms the boundary which separates the basins of
attraction of the laminar attractor and the time-periodic/chaotic
attractor also switches respectively. Thus, the invariant solutionthat can trigger the transition to turbulence follows accordingly
this switching of the edge state.

2006; Schneider et al., 2008; Vollmer et al., 2009). The edge
state has a profound property that its stable manifold, which
includes the edge state itself, forms, at least locally, the laminarturbulent boundary. The initial conditions originating within thislaminar-turbulent boundary will have trajectories that neither
decay to the laminar nor become turbulent, but will be
dynamically attracted to the edge state. For a simple edge state
which has only one unstable direction, applying a strong enoughperturbation results to a trajectory outside this boundary. As
such, edge state can become a mechanism for controlling theflow (Kawahara, 2005).
In this paper we describe homoclinic bifurcation that relates
the Nagata steady solution and the time-periodic solutions in aplane Couette flow with slightly longer streamwise period thanthe minimal unit by Kawahara and Kida (2001). An earlier study
by Ehrenstein and Koch (1995) showed homoclinic bifurcation
occurring in Blasius boundary-layer flow. They mentioned thatthe homoclinic orbits (Guckenheimer and Holmes, 1983; Palis
and Takens, 1993; Ott, 2002) that are formed during homoclinic
bifurcation are often related to intermittent physical bursting,which is also what van Veen and Kawahara (2011) suggested
about the homoclinic orbits to a time-periodic edge state which
they discovered in minimal plane Couette flow. The Blasius
boundary-layer flow utilizes the Navier-Stokes equation which
is simplified within the boundary layer. In contrast, we utilize
the full Navier-Stokes equation in this study. In the next sections
we discuss the flow configuration and computational schemes,
then followed by the bifurcation diagram and their dynamical
relevance.

INTRODUCTION
Dynamical systems theory has been an invaluable tool in thelast two decades in providing a theoretical explanation on the
problem of transition to turbulence happening in wall-boundedshear flows (Eckhardt et al., 2007; Eckhardt et al., 2008;
Kawahara et al., 2012). Transition in shear flows belongs to thesubcritical category, where it has been found in numerical andexperimental studies that crossing a threshold of a finiteamplitude disturbance can promote the transition process at
Reynolds number below its critical value of linear instability.
Under dynamical systems theory, the flow under subcriticaltransition category is considered as a system being regulated byinvariant sets in phase space. In this context for shear flows, astable invariant set which corresponds to the laminar flow
coexists with a stable or marginally unstable invariant set whichcorresponds to the chaotic motion (turbulence) in the transitional
regime. In the middle of these two sets exists an invariant set
which forms the boundary that separates the basins of attraction
between the laminar set and the chaotic set. This particular set isa saddle-type, which can either be an equilibrium or a timeperiodic, and is often referred to as edge state (Skufca et al.,

FLOW CONFIGURATION & NUMERICAL METHODS
We study plane Couette flow which is considered as flow of
Newtonian fluid between two parallel plates moving at aconstant speed 𝑈. The plates are at a 2ℎ distance apart and havea no-slip and impermeable surface. The streamwise, wallnormal, and spanwise coordinates are given as the 𝑥 −, 𝑦 −, and𝑧 − directions, respectively, with their origin on the midplane.
The Reynolds number is defined as 𝑅𝑒 = 𝑈ℎ/𝜈, where 𝜈 is the
kinematic viscosity of the fluid. Spatial periodicity is imposedon the flow in both the 𝑥 − and 𝑧 − directions. The streamwise
period is given as 𝐿𝑥 = 1.93𝜋ℎ while the spanwise period isgiven as 𝐿𝑧 = 1.2𝜋ℎ.

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1. The bifurcation diagram of the Nagata steady solution in a plane Couette flow with slightly longer streamwise period than theminimal unit displayed as the value of the normalized input energy rate 𝐼 as a function of the 𝑅𝑒. Edge states are colored red.
where 𝑢 is the streamwise component of the velocity and 𝝎 isthe vorticity vector.

We solve the incompressible Navier-Stokes equation using
a spectral method. The numerical scheme is Crank-Nicolson
method for the viscous terms and Adams-Bashforth method forthe nonlinear terms, which is similar to the one used by Kim etal. (1987). Chebyshev-polynomial expansion is applied in the𝑦 − direction, while dealiased Fourier expansions are employed
in both the 𝑥 − and 𝑧 − directions. The numerically solved timeperiodic solutions are contained in a subspace that is invariant
under two spatial symmetries: (a) reflection in the streamwise
and spanwise directions followed by a spanwise shift of 𝐿𝑧 /2,and (b) reflection in the midplane followed by a streamwise shiftof 𝐿𝑥 /2. Numerical computations are carried on a resolution of
16 × 33 × 16 in the 𝑥 −, 𝑦 −, and 𝑧 − directions, respectively.
Newton-GMRES (generalized minimal residual) method is used
to compute for both the Nagata steady solution and time-periodic
solutions, and Arnoldi iteration is used to examine the stabilityof the solutions to infinitesimal disturbance. These numerical
procedures which use the time-stepping scheme mentioned
above are essentially the same as what Sánchez et al. (2004) and
Viswanath (2007) used. Edge-tracking method (Itano and Toh,
2001; Skufca et al., 2006; Schneider et al., 2008) is applied tosearch for a suitable initial condition to feed into the NewtonGMRES solver.
The trajectory of an initial point is determined by computing
the normalized input and dissipation energy rates per unit time
given respectively as

BIFURCATION SCENARIO TOWARDS TRANSITIONTO TURBULENCE
Figure 1 shows the bifurcation diagram of the Nagata steadysolution in the plane Couette flow in this study. The Nagata
steady solution originates from a saddle-node bifurcation at
𝑅𝑒 ≈ 161.70. The dashed lines denote unstable solution, whilesolid lines denote stable solution. The three time-periodic
solutions PO1, PO2, and PO3 are plotted using their maxima
and minima values of 𝐼 in one period and are represented bycrosses, dots, and inverted open triangles, respectively. The
location of the three homoclinic bifurcations HB1, HB2, and
HB3 are shown by the green vertical dashed lines. The lower
branch of the Nagata steady solution is unstable and has only one
real unstable eigenvalue, while the upper branch is initiallystable. Inspection of the eigenvalues reveals that at 𝑅𝑒 ≈ 163.36
the upper branch becomes unstable, i.e., a Hopf bifurcation
occurs. A time-periodic solution (black crosses in Figure 1)
arises from the upper branch of the Nagata steady solution due
to the Hopf bifurcation. This periodic orbit PO1 disappears dueto the occurrence of homoclinic bifurcation HB1 at 𝑅𝑒𝐻𝐵1 ≈163.58.
A new homoclinic bifurcation HB2 occurs at 𝑅𝑒𝐻𝐵2 ≈198.50 and another time-periodic solution (black dots in Figure1) appears. This periodic orbit PO2 is initially stable and it
encounters an instability at 𝑅𝑒 ≈ 219.00 and a period-doublingcascade begins. The period-doubling cascade results to a chaotic
attractor, which encounters a boundary crisis BC at 𝑅𝑒𝐵𝐶 ≈238.01. The boundary crisis BC at 𝑅𝑒𝐵𝐶 ≈ 238.01 is manifestedby the contact of the chaotic attractor CA with another timeperiodic solution (red inverted open triangles in Figure 1). Thisvigorous periodic orbit PO3, which comes in contact with CA

𝐿𝑥 𝐿𝑧

1
𝜕𝑢
𝜕𝑢
𝐼=
∫∫( |
+
|
) 𝑑𝑥𝑑𝑧2𝐿𝑥 𝐿𝑧 /ℎ𝜕𝑦 𝑦=−ℎ𝜕𝑦 𝑦=+ℎ0(1)0𝐿𝑥 +ℎ 𝐿𝑧

1𝐷=∫ ∫ ∫ |𝝎|2 𝑑𝑥𝑑𝑦𝑑𝑧
2𝐿𝑥 𝐿𝑧 𝑈 2 /ℎ

(2)0 −ℎ 02

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 2. Variation of period 𝑇 of the three time-periodic solutions with respect to 𝑅𝑒 for (a) PO1, (b) PO2, and (c) PO3.

Figure 3. Approach of the trajectory of (a) PO1, (b) PO2, and (c) PO3 to the trajectory of the lower-branch Nagata steady solution at 𝑅𝑒of the encircled points in Figure 2.

periodic orbits at the 𝑅𝑒 encircled in Figure 2 shows striking
resemblance with the flow structures of the lower-branch Nagata
steady solution at the same 𝑅𝑒. Figures on the left (a, c, and e)
are on the extended portions of the trajectory of the periodic
orbits PO1, PO2, and PO3, respectively, that are approaching
the lower-branch Nagata steady solution. Figures on the right (b,d, and f) are on the lower-branch Nagata steady solution at 𝑅𝑒
corresponding to those of (a, c, and e), respectively. Gray corrugated isosurfaces of the null of streamwise velocity represent
streaks. The red and blue isosurfaces of the second invariant ofa velocity gradient tensor represent clockwise and counterclockwise streamwise vortex tubes, respectively.
Figure 5 illustrates the occurrence of homoclinic bifurcation.
The black dot is the lower-branch Nagata steady solution, which
is a fixed point. The light blue and light green lines denote theunstable and stable manifolds to the fixed point (saddle-type),
respectively. For 𝑅𝑒 < 𝑅𝑒𝐻𝐵2 homoclinic bifurcation just notoccur yet and all trajectories from the fixed point goes to thelaminar attractor. For 𝑅𝑒 = 𝑅𝑒𝐻𝐵2 , a homoclinic bifurcation
occurs and PO2, a limit cycle, appears. This limit cycle grows insize as 𝑅𝑒𝐻𝐵2 is being approached, yielding an orbit of infinite
period. This infinite-period orbit actually serves as homoclinic
orbit to the fixed point. For 𝑅𝑒 > 𝑅𝑒𝐻𝐵2, all trajectories from thefixed point are eventually attracted to the limit cycle PO2 that isdenoted by the dark blue curve. Depending on the direction of
the appearance of the periodic orbit, either above or below the𝑅𝑒 where the homoclinic bifurcation occurs, the trajectories are
attracted either to the laminar attractor or to the other attractor

during BC, appears from a homoclinic bifurcation HB3 at
𝑅𝑒𝐻𝐵3 ≈ 228.32. The critical Reynolds number for BC is shownby the purple vertical dashed line in Figure 1. This perioddoubling cascade that leads to a chaotic attractor and subsequent
boundary crisis with a time-periodic solution (which is an edge
state) is reminiscent of similar bifurcation scenario in plane
Couette flow of different computational domains.

HOMOCLINIC BIFURCATION, EDGE STATES, &BOUNDARY CRISIS
A time-periodic solution, which is a limit cycle, appears or
disappears at the 𝑅𝑒 where the homoclinic bifurcation occurs.
The periodic orbit appearing during homoclinic bifurcation
constitutes the homoclinic orbit to the lower-branch Nagata
steady solution. Near the homoclinic bifurcation the trajectory of
the periodic orbit that approaches the lower-branch Nagata
steady solution is extended and goes to infinity as the 𝑅𝑒 forhomoclinic bifurcation is being approached asymptotically, as
seen in Figure 2. This monotonic increase in the period 𝑇 of theperiodic orbits as the 𝑅𝑒 for homoclinic bifurcation is being
approached is an example of a global bifurcation known as
infinite-period bifurcation (Strogatz, 1994; Wiggins, 1998). In
Figure 3, it is shown that this extension of the period 𝑇 near thesite of the homoclinic bifurcation is the part of the periodic orbit
trajectory that spends more time during the approach to thelower-branch Nagata steady solution. Indeed, as shown in Figure
4, inspection of the flow structures of these extended parts of the3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
which is the limit cycle that appears. As such as seen in thebifurcation diagram in Figure 1, the occurrences of homoclinic
bifurcations are associated with the existence of edge state. Theedge states are indicated with red color in Figure 1. At lower 𝑅𝑒,
the lower-branch Nagata steady solution serves as the edge state
between the laminar attractor and the other attractor, which is thelimit cycle. This means that at lower values of 𝑅𝑒 the steadyedge state and its stable manifold forms the boundary that
separates the basins of attraction between the laminar attractor
and the limit cycle attractor. For 163.36 < 𝑅𝑒 < 163.58, thesteady edge state and its stable manifold forms the basinboundary between the laminar attractor and PO1. For 198.50 <𝑅𝑒 < 228.32, the steady edge state and its stable manifold forms
the basin boundary between the laminar attractor and PO2 andits subsequent period-doubling branch. Unfortunately, we did
not find any limit cycle or bifurcation event for 163.58 < 𝑅𝑒 <198.50, and so no edge state exists in that range of 𝑅𝑒.
At higher 𝑅𝑒, we observe a switching of edge states. This
switching of edge states happens during homoclinic bifurcation
HB3 at 𝑅𝑒𝐻𝐵3 ≈ 228.32, where the edge state is changed fromthe lower-branch Nagata steady solution to the newly createdlimit cycle PO3 due to HB3. In addition to the creation of thevigorous time-periodic solution PO3 during HB3, HB3 also
serves as a mechanism that triggers the switching of edge states.
For 228.32 < 𝑅𝑒 < 238.01, this vigorous periodic orbit PO3,
which is unstable, becomes the edge state between the laminar
attractor and the period-doubling branch of PO2 that turns
subsequently into the chaotic attractor CA. As a consequence,
aside from the edge state switching, the formation of theboundary that separates the basins of attraction between thelaminar attractor and the period-doubling branch of PO2/chaotic
attractor CA also switches from the steady edge state to thisvigorous time-periodic edge state PO3 (together with theirrespective stable manifolds) during HB3.

Figure 5. Sketch of the appearance of homoclinic bifurcation
HB2 for increasing 𝑅𝑒 in the vicinity of 𝑅𝑒𝐻𝐵2 ≈ 198.50.

In Figure 1 we see that during boundary crisis BC at 𝑅𝑒𝐵𝐶 ≈238.01 all of the four red inverted open triangles which represent
the maxima and minima values of PO3 fall onto the black dots
which represent CA. PO3 is mentioned as a vigorous timeperiodic solution earlier because its 𝐼 and 𝐷 energy values
sometimes exhibit that of turbulent behavior. Figure 6 shows thatthe trajectory of the chaotic attractor CA comes nearer to thetime-periodic edge state PO3 in the vicinity of the 𝑅𝑒 of theboundary crisis BC, where CA comes in contact with PO3 at
𝑅𝑒𝐵𝐶 ≈ 238.01. The approach is more evident as the distance
between the minima of CA and PO3 shrinks for increasing 𝑅𝑒
towards its critical value (see the bottom part of Figure 6a andFigure 6b). This shrinking distance is measured quantitatively bycomputing 𝑑 between the minima of CA and PO3 as shown in
Figure 6 using the quantities below.

𝑇

𝑑𝐼 =

√∫0 [𝐼𝐶𝐴 (𝑡 + 𝜏) − 𝐼𝑃𝑂3 (𝑡)]2 𝑑𝑡
𝑇∫0 [𝐼𝐶𝐴 (𝑡)]2 𝑑𝑡

(3)

𝑇

𝑑𝐷 =

√∫0 [𝐷𝐶𝐴 (𝑡 + 𝜏) − 𝐷𝑃𝑂3 (𝑡)]2 𝑑𝑡
𝑇∫0 [𝐷𝐶𝐴 (𝑡)]2 𝑑𝑡(4)

The subscripts CA and PO3 correspond to the chaotic
attractor and time-periodic edge state, respectively, while 𝑡denotes the time period shift that is optimised so that 𝑑𝐼 and 𝑑𝐷
may be minimal at each 𝑅𝑒. Figure 7 shows that both 𝑑𝐼 (redcircles) and 𝑑𝐷 (blue circles) becomes smaller and smaller as thevalue of the critical 𝑅𝑒 of boundary crisis is being approached.
These values of 𝑑𝐼 and 𝑑𝐷 are at the minimum, i.e., closest tozero, when the boundary crisis happens at 𝑅𝑒𝐵𝐶 . We note that
this approach is only a partial and not a full one because of theinherent instability of the CA and the unstable characteristics ofPO3. We confirm that there is a time dependence of thisapproach to the local minima of PO3 and that the threshold timeis short in comparison with the period of PO3. Nevertheless, thenumerical approach achieved here is sufficient to give evidence
for the real approach of CA to PO3.

Figure 4. Visualization of the flow structures of the extended
portions of the trajectory of the periodic orbits (on the left) andthe lower-branch Nagata steady solution (on the right) very near
HB1, HB2, and HB3, from top to bottom respectively.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 7. A quantitative measure of the approach of the chaotic
attractor CA to the time-periodic edge state PO3 during theboundary crisis BC.

Couette flow (Skufca et al., 2006; Kreilos and Eckhardt, 2012;
Shimizu et al., 2014; Lustro et al., 2019).
Homoclinic bifurcation is a global bifurcation which occurs
when a periodic orbit touches the lower-branch Nagata steadysolution, which is a fixed point. At the homoclinic bifurcation
point, the periodic orbit takes the role of homoclinic orbit to this
fixed point. Therefore, this homoclinic orbit to the fixed point is
not associated to chaotic dynamics, but rather to a limit cycle,
i.e., the periodic orbit. This lack of chaotic dynamics is due tothe absence of the transversal intersection of the stable andunstable manifolds, i.e., homoclinic orbits, to the fixed point. A
Shilnikov criteria, i.e., finding a homoclinic orbit that istransversal to a plane containing the fixed point must be imposedin order to obtain this transversal intersection of the stable andunstable manifolds (Guckenheimer and Holmes. 1983; Wiggins,
1988; Shilnikov, 1970)
In our recent work in minimal plane Couette flow (Lustro etal., 2019), we showed that the onset of transient turbulence thateventually relaminarizes happens due to boundary crisis. Suchboundary crisis is generated by the tangency of homoclinic orbits
to a time-periodic edge state. Hence, the tangency of homoclinic
orbits to a time-periodic edge state provides the mechanism forthe onset of transient turbulence in minimal plane Couette flow.
This is because of the presence of Smale horseshoe in transversal
homoclinic orbits following the Smale-Birkhoff theorem inchaotic dynamical systems (Guckenheimer and Holmes, 1983;
Palis and Takens, 1993; Strogatz, 1994; Ott, 2002). This transient turbulent behavior that is observed in this study after theboundary crisis BC may perhaps also be associated withhomoclinic orbits to the time-periodic edge state PO3, if they
exist. The finding of such homoclinic obits to PO3 is left for
future work.

Figure 6. Approach of the chaotic attractor CA to the timeperiodic edge state PO3 during the boundary crisis BC.

Transient turbulence which eventually relaminarises is seen
for 𝑅𝑒 above 𝑅𝑒𝐵𝐶 ≈ 238.01. Such appearance of transientchaotic behavior is associated with the occurrence of boundary
crisis in chaotic dynamical systems (Ott, 2002; Lai and Tel,2011). For other plane Couette flow systems, such transient
turbulence that eventually relaminarizes was also observed after
the occurrence of boundary crisis, such as in Kreilos andEckhardt (2012), Shimizu et al. (2014), and Lustro et al. (2019).
In both of our works in the minimal plane Coutte flow previously(Lustro et al., 2019) and in a slightly longer plane Couette flowin this study, the boundary crisis is found to be associated withthe approach of the chaotic attractor with a time-periodic edge
state.
The switching of edge states as well as the invariant sets
which form the boundary that separates the basins of attraction
between the laminar attractor and the time-periodic/chaotic
attractor in this study are novel observations on transitional shear
flows. To our knowledge, these observations are also new
findings for other chaotic dynamical systems. Both of theseswitching events happen due to the presence of homoclinic
bifurcation at a critical value of 𝑅𝑒 ( 𝑅𝑒𝐻𝐵3 ≈ 228.32). For
transitional plane Couette flow, it was reported that the structureof the basin boundary between the laminar attractor and thechaotic set is fractal. Such fractal basin boundary, which is also
known as edge of chaos (Skufca et al., 2006), is due to thebifurcation from a simple, smooth basin boundary. The same
relationship between this fractal basin boundary and chaotic
saddle formation has also been reported in transitional plane

CONCLUSION
We reported homoclinic bifurcations in plane Couette flow
with slightly longer streamwise period than the minimal unit.
Three homoclinic bifurcations are shown to be related to theappearance or disappearance of their corresponding timeperiodic solutions. At 𝑅𝑒 where the homoclinic bifurcations
occur, the corresponding periodic orbits function as homoclinic
orbits to the lower-branch Nagata steady solution. The most5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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Lustro, J. R. T., Kawahara, G., van Veen, L., Shimizu, M.,
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Palis, J., and Takens, F., 1993, Hyperbolicity and Sensitive
Chaotic Dynamics at Homoclinic Bifurcations, Cambridge
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energetic of these periodic orbits, which we called the vigorous
periodic orbit PO3, serves as the edge state between the laminar
attractor and time-periodic/chaotic attractor at higher values of
𝑅𝑒 . At lower values of 𝑅𝑒 the lower-branch Nagata steadysolution serves as the edge state between the laminar attractor
and time-periodic attractor (limit cycle). The switching of edge
states from the lower-branch Nagata steady to the vigorous
periodic orbit PO3 occurs due to the creation of PO3 duringhomoclinic bifurcation HB3.
The periodic orbit PO2 is initially stable but encounters an
instability which leads to a period-doubling cascade that later
becomes a chaotic attractor CA. A boundary crisis happens
when the chaotic attractor CA touches the periodic edge state
PO3 at 𝑅𝑒𝐵𝐶 ≈ 238.01. For 𝑅𝑒 > 238.01, transient turbulence
which eventually relaminarizes is observed consistently with theprevious result for the minimal unit. This boundary crisis has
been confirmed qualitatively and quantitatively.
The rich bifurcation scenario presented here for a fluid flow
system with two kinds of edge states adds to the repertoire of
dynamical systems approach to understanding the subcriticaltransition to turbulence in shear flows. Finally, the discovery ofhomoclinic bifurcation here is meaningful for bifurcation theory
as it provides evidence of such event in a system governed by
the full Navier-Stokes equation.

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