数理解析研究所講究録 1231 巻 2001 年 133-144

133
A Bifurcation of a Two-Dimensional Symmetric Flapping Model$j\mathfrak{B}$$\mathrm{P}_{\mathrm{d}}3$$\mathrm{A}\mathrm{S}$

Makoto IIMA and Tatsuo YANAGITA \dag er*

Nonlinear Studies and Computation, Research Institute for Electronic Science,
Hokkaido University, Sapporo 060-0812, Japan(Dated: September 7, 2001)

A twO-dimensional symmetric flapping model is studied in terms of thebifurcation. This model consists of two wings attached together at a hinge,
and the flapping motiom of the wings is symmetric with respect to the horizontal line. The center of mass of this model can move accroding to the lift
generated by an interaction of the wing and vortices separated from boundary layer. The bifurcation parameter is a time scale of the diffusion, which
is simplified to contrast the transition of the type of the motion. Bifurcation
diagram shows that there are unstable regions for a steady flapping motionof zero-mean velocity, and that there is a region where two types of a steadystable flapping motion coexisit. We illustrate these types of the motion.

I.

INTRODUCTION

The problem of amotion of abody interacting with fluid is familiar to us: acoin motion
dropped in water[l, 2], insect flight[3-7], and the flapping flag in wind[8] . In the case of
tw0-dimensional inviscid fluid without vorticity, the motion of the body in fluid is describedby ordinary differential equations $[9, 10]$ . In these cases listed above, however, their various
motion should be understood as avortex-body interaction: the motion of abody is dominanedby ainteraction between the vortices separated from boundary layer (separation vortex) andthe body.
In the case of the insect flight, they can not even sustain their weight without suchinteraction[3]. Experimental studies using areal insect and amechanical model [4, 5, 1113], and numerical simulations $[14, 15]$ show that the separation vortex from the leading-edge
produces an extra lift. The lift-enhance mechanism, called delayed stall, is proven to be exist
in areallife insects, but the detail of the mechanism has not been understood.
An analysis of asimple flapping motion in two dimensional fluid which is calculated bydirect numerical simulation (DNS) reveals some aspects of the vortex-body interaction in a
flapping flight such as dipole jet emission during one type of hovering[16], and afrequency
selection mechanism in auniform flow[17].
Coin motion falling in fluid is also determined by the vortex-body interaction. Threedimensional expriment[l] and pseudo two dimensional experiment[2] show that separationvortcies interacting with the body is acritical element for determining their motion. In thecase of flapping flag, separation vortex is also observed
Here we focus our attention on the motion of abody with flapping wings in two dimesional
fluid [18]. The motion of the center of mass (CM) due to vortex-wing interaction is considered.
Within the authors’ knowledge, there is no study of aflapping model considering the motionof CM. The condition of the free motion of CM permits us to consider stability, as well as
the effect of the body inertia[19], where the ratio of fluid density and the inertial mass of the
body in anon-dimensional form is the critical parameter to determine the type of the motion
as well as the coin motion 2].
Our question considered here is simple: what happens to the motion of CM if the flapping
motion is symmetric, there is no gravity, and any kind of airspeed is disregarded ?The answer
appears trivial, because symmetic flapping motion seems to give symmetric lift contribution.
Sunada et al. calculated avortex-wing interaction without the dynamics of separation vortex,

Electronic address: makotoQaurora. es.hokudai. ac.jp; URL: http:.es.hokudai. ac.
ronic address:
yanagitaOaurora. es.hokudai. ac.jpj URL: http:.es.hokudai. ac.
$//\mathrm{a}\mathrm{u}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{a}$$\mathrm{j}\mathrm{p}/" \mathrm{m}\mathrm{a}\mathrm{k}\mathrm{o}\mathrm{t}\mathrm{o}$$\uparrow \mathrm{E}1\mathrm{e}\mathrm{c}\mathrm{t}$$//\mathrm{a}\mathrm{u}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{a}$$\sim..--\wedge\wedge-\cdot\wedge$-$\mathrm{j}\mathrm{p}/$134
and state that there is no mean lift in asymmetric flapping[20]. Wang performed DNS of theflow following aflapping elliptic wing in uniform flow. Her result shows that the lift functionof time is symmetric if the angle of attack is zer0[17]. On the contrary, we found that thesymmetry of lift production can be broken due to the vortex-wing interaction [18].
In this paper, we study this symmetry-breaking phenomenon in terms of the bifurcation.
In section , the detail of the model is described. We briefly review atypical vortex-body, and we show theinteraction in section III. The bifurcation diagram is shown in section
dynamics of the two flapping phases in section IV C.
$\mathrm{I}\mathrm{I}$$\mathrm{I}\mathrm{V}$II.

THE TWO-DIMENSIONAL FLAPPING MODEL

FIG. 1: The geometry of our model. The wings are represented by point vortices drawn as circles, andthe body is represented by the square located at the hinge . Slip boundary conditions are requiredat the point midway between each pair of adjacent boundary vortices (represented by crosses). The
boundary vortices at the outer edges of the wings become detached, thereby becoming separationvortices (represented by triangles). At each time step one vortex is shed from each wing in thismanner. Mirror symmetry with respect to the vertical center line represented by the broken line forboth the flow and the wings is assumed.
$O$

Our flapping-flight model “butterfly” has wings represented by two rigid lines swinging on ahinge in two dimensional space, as shown in fig.l . For simplicity, we assume mirror symmetry
for the motion of the wing and fluid with respect to the vertical center line and that the“butterfly” can move in the vertical direction only. The fluid around the wing is assumed to
be incompressible and inviscid, but it is possible for vortices to be shed from the outer edgeof each wing.
In order to model separation vortices, we used adiscrete vortex method $[21, 22]$ , in which
fluid motion and flow on the wings are approximated using vortex blobs. Asingle vortex blob
, where is theat $x=(x_{0},y_{0})$ induces the flow
circulation of the vortex, $r=\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}$ , and is asmoothness parameter. For
the details of the model, see ref. [19].
We note that this model implements acut-0ff viscosity. Due to viscosity, the size of eachvortex gradually grows, while the total circulation it possesses is preserved. This effect ofviscosity can be implemented in adiscrete vortex method in several ways[23]. Our purpose is
to isolate the salient feature of the bifurcation, not to simulate an accurate fluid dynamics byaflapping motion. Thus we simplify the core-spreading dynamics of aseparation vortex asfollows. In our model, each separated vortex is shed from awing at aparticular time. Then,. We consider that the radius
we define acharacteristic “lifetime” of separated vorticesof the vortex suddenly becomes infinite when $\tau=$ lift, where is the length of time after. Changing the effectwhich the vortex comes into existence;of the separated vortices by considering different values of the lifetime, we can show that theinteraction between the separated vortices plays an important role in the symmetry breaking.
,Here we have used parameters corresponding to abutterfly [24]: wing mass
, wing length, hinge wing distancebody mass, period of flapping cycle $T=0.1\mathrm{s}$ . To evaluate the two dimensional air density is
not straightforward because of the difference between this two dimensional model and real
. The
butterfly in three-dimensional space. Here, we used the parameter$(u(x,y),v(x, y))=(- \frac{\Gamma(y-y\mathrm{o})}{2\pi(r^{2}+\delta^{2})}, \frac{\Gamma(x-x_{\mathrm{O}})}{2\pi(r^{2}+\delta^{2})})$$\Gamma$$\delta$$T_{1\mathrm{i}\mathrm{f}\mathrm{e}}$$\tau$$\delta(\tau)=\delta_{0}(\tau<T_{1\mathrm{i}\mathrm{f}\mathrm{e}});\infty(\tau>T_{1\mathrm{i}\mathrm{f}\mathrm{e}})$$m=3.5\cross 10^{-6}\mathrm{k}\mathrm{g}$$M=6.0\cross 10^{-5}\mathrm{k}\mathrm{g}$$\mathit{1}=3.0\cross$$10^{-2}\mathrm{m}$$l_{\mathrm{d}}=5.0\cross$$10^{-3}\mathrm{m}$$\rho=7.0\cross 10^{-3}\mathrm{k}\mathrm{g}/\mathrm{m}^{2}$

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tw0-dimensional density itself, however, is not acritical parameter to determine the behaviorof “butterfly”, which is characterized by anondimensional number defined and discussed in[19].
For the time integration, the 4th order Runge-Kutta method and the Euler method withtime stepis used to calculation flow due to the wing flapping and the motionof the center of mass, respectively. We checked that the result does not depend significantly on, $N$ , and . The value of simulation parameters, $N$ , andused in the following sections
are $1.0\cross 10^{-4},50$ , and $1.0\cross 10^{-4}$ , respectively. The flapping is assumed to be characterized
by asinusoidal function, which represents the angle made by each wing with respect tothe horizontal line intersecting the hinge: $\theta(t)=\Delta\theta\cos(2\pi t/T)$ where $also=0.36\cross 2\pi$ .
In order to concentrate our attention on the bifurcation, mechanism, we consider the casewithout agravitational force.
$\chi$$\Delta t=1.0\cross 10^{-4}(\mathrm{s})$$\triangle t$$\Delta t$$\delta_{0}$$\delta_{0}$$\theta(t)$III.

ASYMMETRY-BREAKING CASE

In this section, we briefly illustrate two typical cases of the result of the simulation using a
nondimensional number. For the detail of the dynamics described in this section,
see ref. [18].
The dynamical interaction between separated vortices and the wing shows asymmetric lift
production in one case, $\eta=1.0$ , while we only see atrivial symmetric lift production in theother case, $\eta=0.25$ (fig.2(a) and fig.2(b)).
In the case of y7 $=0.25$ , the generation of the coherent vortices during flapping motion issymmetric, as shown in fig.3. On the other hand, in the case of $\eta=1.0$ , asymmetric generationof the lift force has been observed, as shown in fig.2(b). The most characteristic dynamics
is observed in the second downstroke. Acoherent vortex
generated in the first upstroke
remains under the wing during the production of the coherent vortex
created by the second
downstroke (fig.4(c)). As aresult,andtogether induce stronger outer flow around thewing. This flow generates alarger upward lift force.
Once the body attains asufficiently large velocity, the distance between awing and thecoherent vortex it sheds in each downstroke becomes much shorter than the distance between
this wing and the coherent vortex it sheds in the upstroke. As aresult, $L(t)$ becomes asymmetric, unlike in the case of y7 $=0.25$ (fig.2(a) and fig.2(b)). Through this mechanism, thecenter of mass of the “butterfly” moves asymmetrically despite the symmetric nature of the
flapping, and the direction of its overall motion is determined by the direction in which thewings initially flap.
The symmetric-breaking mechanism requires two coherent vortex interacting with fluid.
Since the number of the coherent vortex is controlled by , we can expect asharp transition
from the state of the symmetric lift production to the state of the asymmetric lift production
as is increased. Fig.5 shows the averaged lift in the period $[T, 2T]$ as afunction of $\eta\in[0,2.0]$ .
The averaged lift shows atransition at $\eta=\eta_{0}\simeq 0.62$ . When, the averaged lift is verysmall compared to the averaged lift in the case of. Two typical dynamics in the first
several periods, each of which is discussed above, corresponds to the case ofand thecase of, respectively.
Fig.5 also shows that this model gives qualitatively only two state. In other words, even
if two or more coherent vortex (of finite radius) are allowed to exist, the symmetry-breaking
mechanism is essentially the same.
The number of the coherent vortex is roughly estimated by, because ahalf
stroke (upstroke or downstroke)produces one coherent vortex. If $\eta<0.5$ , no coherent vortex
can include all the separation vortices produced during ahalf stroke, because the lifetimeof separation vortex is so short that some separation vortices included in acoherent vortex
disappears (precisely, the radius of them is infinite). Such imperfect coherent vortex do notwork effectively for inducing flow. The graph in fig.5 shows that in such case, the symmetriclift production is not broken in the second period.
On the other hand, if $\eta>1.0$ , there is roughly more than two coherent vortices which canwork for lift production. Although the effect of the extra vortex is shown as aconcave around$\eta=1.5$ , it is not significant compared with the change at. Therefore extra coherent
vortices do not play acritical role$\eta\equiv T_{1\mathrm{i}\mathrm{f}\mathrm{e}}/T$$\mathrm{V}_{2}$$\mathrm{V}_{3}$$\mathrm{V}_{3}$$\mathrm{V}_{2}$$\eta$$\eta$$\eta<\eta_{0}$$\eta>\eta_{0}$$\eta<\eta_{0}$$\eta>\eta_{0}$$[T_{1\mathrm{i}\mathrm{f}\mathrm{e}}/2T]--[\eta/2]$$\eta=\eta_{0}$

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(a)

(b)

$\overline{\underline{\mathrm{z}}}$$\overline{\underline{\mathrm{z}}}$$\hat{\frac{\tilde{\mathrm{o}}}{\vee \mathrm{x}}}$$\hat{\frac{\check{\mathrm{o}}}{\vee \mathrm{x}}}$$\mathrm{g}\lrcorner$$\underline{\not\in}\lrcorner$

Time

Time

FIG. 2: The vertical lift force as afunction of time, (a) In the case of $\eta=0.25$ , symmetrical lift
generation is observed, (b) In the case of $\eta=1.0$ , asymmetrical generation of the lift force is found.
This creates directed motion of the body despite the symmetrical nature of flapping.

FIG. 3: Snapshots of the configuration of separated vortices and the wings in the case of $\eta=0.25$ .
The separated vortices and streamlines are indicated by dots and contour lines, respectively. The two:the beginning of the first downstroke. Separated
straight lines represent the wings, (a)vortices form acoherent vortex , which generates an upward vertical force. (b):during
the first upstroke. Acoherent vortex V2 is formed, and generates adownward force. In this case,symmetric lift generation is observed (see fig.2(a)).
$\mathrm{t}=(5+3/8)\mathrm{T}$$\mathrm{t}=(5+7/8)\mathrm{T}$$\mathrm{V}_{1}$In the region $0.5<\eta<1.0$ , there are one perfect coherent vortex and one imperfect coherent
vortex. The perfect one works for lift production fully, while imperfect one does not. The resultthat the critical value is $\eta\simeq\eta_{0}=0.62$ means that the existence of such secondary vortex,
is imperfect, is essential for the symmetry-breaking dynamics.
evenLastly, we note that the case $\eta=2.0$ corresponds to the case of inviscid fluid. Thus thesymmetry-breaking is not the artifact due to our model viscosity.
$\mathrm{i}\mathrm{f}\cdot \mathrm{i}\mathrm{t}$$\mathrm{I}\mathrm{V}$. BIFURCATIONIn this section, we study the stability of this model. As shown in section III, the cut-0ff
viscosity is useful to characterize the vortex-wing interaction. Anondimensional parameter,, is relevant to control the number of coherent vortex. Using , we construct abifurcation, whilediagram. We found amore complicated structure of stable state in the case of
the dynamics in the second period appears to be asimple symmetric lift generation.
$\eta$$\eta$$\eta<\eta_{0}$A.

Existence of steady states
1Convergence

Fist of all, we prove the existence of the steady state of this model. Fig.6 shows mean
for $\eta=0.25$ ,0.5.0.6, 0.7 and 1.0. Their convergen ce
velocity in one period ($V \rangle_{t}\equiv\frac{1}{T}\int_{t}^{t+T}V\mathrm{d}t$

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0.15

0.15,

$-0.\dot{\dagger}5- 0.15- 0^{\cdot}..\prime \mathrm{s}^{-}$$- 0.15_{\sim}-\overline{0}..\overline{15}$—--$–\overline{0}.-15$(a)$0.\prime 5$

(b)

$—$

FIG. 4: Same as fig.3 but y7 $=1.0$ . (a):the beginning of the first downstroke. Separated
vortices form acoherent vortex Vi, which generates an upward vertical force when it is above thewing. The broken arrow indicates the path of, which turns around and moves under the wings.
(b):during the first upstroke. Acoherent vortex V2 is formed and remains under the wing
due to the flow induced by. (c):the second downstroke. Acoherent vortex
is formedby the second downstroke, and V2 induces stronger outward flow around the wing, which produces
larger upward lift, (d):the body is accelerated in the second downstroke, and moves to ahigher position. Acoherent vortex V4 is formed in the second upstroke.
$\mathrm{t}=3/8\mathrm{T}$$\mathrm{V}_{1}$$\mathrm{t}=7/8\mathrm{T}$$\mathrm{V}_{1}$$\mathrm{t}=11/8\mathrm{T}$$\mathrm{V}_{3}$$\mathrm{t}=17/8\mathrm{T}$is attained in apractical sense when $t>30T$ . It should be noted that the mean velocity inthe first periods does not correspond to the terminal velocity. The difference of the shape ofbetween the case of $\eta=0.5$ and the case of $\eta=0.6$ can not be discerned when $t<5T$ ,while the limit of
for $\eta=0.5$ , about 0.9, is qualitatively different from the case $\eta=0.6$ :its limit is 0.0.
This case shows that the behavior of this model is not so simple, in spite of the highsymmetry this model possesses. This model has stable steady state, but the relation between
the stable steady state and the typical dynamics is not straightforward: the stable steadystate characterized by non-zero velocity can not always be obtained by the symmetry-breaking
mechanism explained in sec.III, which is clearly shown by the case $\eta=.0.5$ . Therefore, afurther
study about stable steady state is necessary for understanding this model, besides the analysis
of typical dynamics gives prescribed forms of anontrivial vortex-wing interaction.
The stable steady state can be described by velocity and the circulation of awing F. For
this model, the stable steady state corresponds to astable steady flapping state (SSFS) withperiod .
$\langle V\rangle_{t}$$\langle V\rangle_{t}$

$V$

$T$

2Independency of initial phase

Next, we check the initial-phase dependency for SSFS. We performed simulations for the setof initial conditions:$S_{\mathrm{i}\mathrm{n}}$$\mathrm{S}_{\mathrm{i}\mathrm{n}}=\{\theta_{i}(t)|\theta_{i}(t)\equiv\Delta\theta\cos(2\pi t/T+\phi);\phi=\frac{2\pi i}{N}(i=0,1,$\ldots ,N);N $=10\}$ ,

(4.1)

where
is the flapping angle.
We show the trajectoryin the phase space defined by the circulation of the wing andthe velocity$6\mathrm{i}(\mathrm{t})$$T_{\eta,\phi}$$T_{\eta,\phi}\equiv\{(x, y)|x=\Gamma(t);$y $=V(t);$ t $\in[29T,$ 30T.,

(4.1)

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$\overline{\underline{\mathrm{z}}}$$1\hat{\mathrm{o}_{\mathrm{O}}}$$\overline{\vee \mathrm{x}}$$.\lrcorner\underline{\epsilon}$$[mathring]_{\underline{\Phi\infty \mathrm{t}\mathrm{d}}}$$\triangleleft\Phi>$$\eta$

FIG. 5: Averaged lift in the period $[T, 2T]: \frac{1}{T}\int_{T}^{2T}L(t)dt$ . As increase, more coherent vortices of. In case of y7 $>1.0$ , the average lift does not
finite radius coexist. Atransition is found ataround $\eta=1.5$ .
observedisareducealthoughdifference,show aqualitative$\eta$$\eta\simeq 0.62$

2.5
2$\geq$$\overline{8}$1.5$\overline{>\Phi}$

1$[mathring]_{\Xi}\varpi \mathrm{c}$

0.5

0
20T
Time

IOT

OT

40T

30T

for $\eta=0.25$ (open circle), 0.5 (cross), 0.6 (open square), 0.7
FIG. 6: Mean velocity functionis almost constant.
(plus), and 1.0 (star). When $t>30T$ ,$\langle V\rangle_{t}$$\langle V\rangle_{t}$and $V(t)$ depend on and . If SSFS is an attractor of the model, thewhere functions
trajectory converges to one or several closed curve(s).
In fig.7, two types of SSFS are shown. In case of $\eta=0.6$ , all the initial condition (4.1) gives
the same SSFS Si, which means that all the trajectory. converges to . Moreover,is symmetric to itself with respect to the origin: again, the symmetry is not broken in thisand , are observed. Because
case. On the other hand, in case of $77=1.0$ , two SSFS,
is equivalent to the mirror transform with respect to the horizontal line,
changing into
are symmetric to each other with respect to the origin.
andTherefore, it is shown that this model gives one or more SSFS depending on the nondimensional parameter . In the next subsection, we draw the bifurcation diagram.
$\Gamma(t)$$\eta$$\phi$$S_{1}$$T_{\eta,\phi\in \mathrm{S}_{\mathrm{i}\mathrm{n}}}$$S_{1}$$S_{2}$$\phi$$S_{2}$$S_{3}$$\phi+\pi$$S_{3}$$\eta$

B.

Bifurcation diagram

Because this model gives aSSFS, we can draw adiagram of the SSFS. This diagram shows
the structure of the SSFS.

139$\geq\backslash$

2.5
2.0
1.5
1.0
0.5

8o.o

$\overline{\Phi}$$>- 0.5$

-1.0
-1.5
-2.0
-2.5
-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5

Circulation
FIG. 7: Plot of $T_{\eta=0.6,\phi}(S_{1})$ and $T_{\eta=1.0,\phi}$ ( and ). Initial phase is changed according to (4.1).
In case of $\eta=0.6$ , all the initial phases result in aunique SSFS. In case of $\eta=1.0$ , initial phases
separates into two groups, both of which are symmetric with respect to the origin.
$S_{2}$$S_{3}$

1The method

To construct the diagram, we need SSFS for agiven . We used two kinds of simulation to$\leq 2.0$ . We regard(defined in (4.2)) for the range
obtain SSFS. One is,
SSFS as the trajectory in $[29T, 30T]$ , and characterize it by its mean value of the velocity. We
plot the absolute value of the mean velocity
as afunction of .
The other is the trajectory of
where
is aslow-changing function defined asfollowings:$\eta$$0.20\leq \mathrm{y}7$$T_{\eta=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$$\phi$$|\langle V\rangle_{t=29T}|\equiv\overline{V}$$T_{\eta=\eta(t),\phi}$$\eta_{1}$$\eta_{1}+\frac{\eta\sim\circ-\eta_{1}}{t_{2}-t_{1}}(t-t_{1})$$\eta(t)=\{$$\eta_{2}$$\eta_{2}+\frac{\eta_{3}-\eta_{2}}{t_{3}-t_{2}}(t-t_{2})$$\eta_{3}$$\eta$$\mathrm{r}\mathrm{j}(\mathrm{t})$$(0<t<t_{1})$ ,,,$(t_{3}\leq t<t_{4})$ ,$(t_{1}\leq t<t_{2})$(4.3)$(t_{2}\leq t<t_{3})$$(t_{4}\leq t<T_{\mathrm{e}\mathrm{n}\mathrm{d}})$,where Tend is the simulation time (see fig.8).

FIG. 8: Plot of the function$\eta(t)$.

We performed simulations for sets, listed in table 1. Using, we can check theexistence of another SSFS which can not be reached by initial condition Sin,. In this case, weplot the sequence of the absolute value of the mean velocity;.
In all the analysis in this section, we fixed $\phi=0$ .
$T_{\eta=\eta(t),\phi}$$|\langle V\rangle_{t}|$$(t=0, T, 2T, \ldots, T\mathrm{e}\mathrm{n}\mathrm{d}-T)$

140
TABLE
Run
7/1
10.57
0.60
2
3
0.57

Table of the parameters for the definition of$\mathrm{I}\ovalbox{\tt\small REJECT}$

y72

7/3

$t_{1}$$t_{2}$$t_{\mathit{3}}$

0.61
0.64
0.50

0.57
0.60
0.57

8T
8T
8T

30T
30T
30T

70T
70T
70T

$\mathrm{r}\mathrm{i}(\mathrm{i})$$t_{4}$(see (4.3)).

$T_{\mathrm{e}\mathrm{n}\mathrm{d}}$

92T
92T

symbol in fig.

10OT

$+$

$\mathrm{I}\mathrm{O}\mathrm{O}^{\ovalbox{\tt\small REJECT}}\mathrm{T}$$\mathrm{x}$10OT$92\ovalbox{\tt\small REJECT} \mathrm{T}$$*$2Result

Fig.9 shows we have one state of non-zero mean velocity in the region $\eta>1.0$ . In this region,
the mean velocity of the SSFS is almost independent of . This is contrast to the averaged liftof the second flapping period shown in fig.5, in which the averaged lift shows areduce around$\eta=1.5$ . Fig.9 shows
is almost constant if $\eta>1.0$ . Again, this means that the interactionof two coherent vortices and wing are essential to achieve the SSFS (see sec.III). In the region$0.70<\eta<1.00$ , the state is qualitatively the same as the state in the region $\eta>1.0$ , although
the value of is not the same due to the short lifetime ,,To analyze the SSFS in the region $0.35<\eta<0.70$ , we need the data obtained bybecause there is another SSFS which can not be obtained by the simulation started from initialas (quasi-) SSFS, and plot themconditions (4.1). As well as , we regard the function
in fig.10.
In the region $0.57<\eta<0.61$ , we see aSSFS characterized by zer0-mean velocity by the
result of RUN 1shown in fig.10 (except the first period due to the initial condition). However,
it is suggested that the branch observed in the region $0.7<\eta$ is also connected to another
SSFS characterized by non-zero mean velocity. The result of RUN 2 shows atransition fromthe SSFS characterized by zer0-mean velocity to another SSFS characterized by non-zeromean velocity. In the initial eight period of these runs $(\eta=0.60)$ , aSSFS characterized byzer0-mean velocity is achieved, which is consistent with the case of RUN 1. However, after atransient state occuring when $\eta=0.64$ , (quasi-)SSFS converges to another SSFS characterized
by non-zero mean velocity (about 0.12) in the last eight period in which $\eta=0.60$ again.
after the transient state described above $(70T<t<100T)$ can be regarded as SSFS,
If (their branch is close to abranch indicated by (where $\eta\geq 0.66$ ) shown in fig. 10. We can see along transient term for the run when is near the critical point: in the region $0.53<\eta<0.55$ ,does not converge at $t=30T$ .
In the region $\eta<0.35$ , asingle state of zer0-mean velocity is observed. In this region, theSSFS is trivial: the flapping motion gives symmetric lift function without mean motion, as
exampled in section III.
$\eta$$\overline{V}$$\overline{V}$$\eta$$\langle V\rangle_{t}$$(V)_{t}$$\overline{V}$$V\rangle_{t}$$\overline{V}$$\eta$$\overline{V}$3Diagram, we can construct the bifurcation diagram drawn
By knowledge of the functions andin fig.ll. The bifurcation diagram has mirror symmetry with respect to the line $(V)=0$ dueto the symmetry of the system. There is apitch-folk type bifurcation at A $(\eta\simeq 0.35)$ . In the, there are one SSFS, , characterized by $(V)=\pm V_{0}(\neq 0)$ .
region between Aand
These branches exist all the region where is larger than A. In addition to that, the analysis
andbetweenin the above subsection, there is another SSFS characterized bycoexist. Unstable mode is also shown, but(y7 $\simeq 0.60$ ). In this region, two SSFSs aandthey are one example deduced by the stable modes. SSFS aand correspond to essentiallydifferent types of flapping dynamics. In the next section, we discuss the dynamics.
$\overline{V}$$|\langle V\rangle_{t}|$$\mathrm{B}(\eta\simeq 0.57)$$\mathrm{a}$$\eta$$\langle V\rangle=0$$\mathrm{B}$$\mathrm{C}$$\mathrm{b}$$\mathrm{b}$C.

Dynamics of two flapping phases

In this subsection, we describe the dynamics of the two different SSFS. Parameter is fixed
0.6 for achieving two different SSFS, which is obtained by Run Z. asequence $[7T, 8T]$ for
zer0-mean velocity state(b in fig.11), and asequence [ $99T,$ 10OT] for non-zero mean velocity
state(a in fig. 11).
$\eta$$\cdot$

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$|>$

$\eta$

FIG. 9: Plotted graph of . In the region $\eta>0.62$ ,mode in the region $\eta<0.35$ . In the region $\eta>1.00$ ,our cut-0ff viscosity method.
$\overline{V}$has asingle
has two stable states, whiletakes almost the same value, which validates$\overline{V}$$\overline{V}$$\overline{V}$$-\wedge-$$|>\underline{>\vee}\wedge$$\eta$(open square is the run for $N=50$ and filled square is the run for
FIG. 10: Plotted graph ofis plotted as jointed-point graph.
$N=150)$ andin the region $0.20<\eta<0.80$ . Function
In the region $0.55<\eta<0.62$ , two stable modes is observed.
$\overline{V}$$(\mathrm{V})\mathrm{t}$$(\mathrm{V})\mathrm{t}$fig. 12(a) and fig.12(b) shows two typical snapshots of zer0-mean velocity state. The difference of their caputure time is $1/2T$ . The flow pattern and the distribution of point vortices
are almost symmetric with respect to the horizontal line denoting the height of the center ofmass. The symmetry of the relative position among point vortices and the wing results in thesymmetric oscillation pattern in the phase space, which is shown in fig .
Because the value of , 0.6, is sufficient for this model to form one coherent vortex, thecoherent vortex shown in fig.12 works fully to generate lift. This interaction between one
vortex and the wing is essential for lift production.
fig. 12(c) and fig. 12(d) shows two typical snapshots of non-zero velocity state. The differenceof their time is also $1/2T$ . In this case, the flow pattern is asymmetric with respect to thehorizontal line.
Acoherent vortex is shown in the both figures, but the relative position of the vortex andthe wing is different. In fig.12(c), the distance of the coherent vortex and the wing is small,which means strong lift generation. On the other hand, the distance is large in fig.12(d), and(negative) lift generation is small. This asymmetry causes an asymmetric shape of trajectoryin the phase space shown in fig.13. However, it should be noted that the total sum of theinstantaneous lift during one period is zero because of the definition of SSFS.
The magnitude of the parameter $\eta=0.60$ is not sufficient to make asymmetry-breakin$\eta$

142

FIG. 11: Schematic picture of the bifurcation diagram. Solid line is astable mode, and broken line isan example of unstable mode which is deduced by the structure of the stable mode.

(b)

(a)
0\prime 5

(d)

(c)

FIG. 12: Snapshots of the two SSFS in the case of $\eta=0.60$ . Horizontal broken line denotes the heightand (b) are symmetricandof the center of mass.
:a(non-zero mean); $t=3/4T$ andeach other with respect to the horizontal line.
$(\mathrm{a})\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{b}(\mathrm{z}\mathrm{e}\mathrm{r}\sim \mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n});t=3/4T$$(\mathrm{b})t=7/4T:(\mathrm{a})$$(\mathrm{c})\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}$$(\mathrm{d})t=3/4\mathrm{T}$1.5
1$\geqq$80.5$\overline{>\Phi}$0-0.5-0.2 -0.1

0 0.1 0.2 0.3 0.4
Circulation

FIG. 13: Phase space of two SSFS for $\eta=0.6$ . Solid line corresponds to the SSFS characterized byzer0-mean velocity. Broken line corresponds to the SSFS characterized by non-zero mean velocity.

143
dynamics discussed in sec.III. That is the reason why all the initial conditions of this caseresult in asymmetry state discussed in sec.IV A. On the other hand, SSFS characterized by
non-zero mean velocity is achieved if y7 $>\eta_{0}\simeq 0.62$ . In this region, two coherent vortices
interact with the wing, symmetry breaks, and all the initial conditions result in one non-zeromean velocity state.
The appearance of this dynamics is controlled by the nondimensional parameter , and its. However, when this model is once locked in the nonoccurance is clearly separated atzero mean velocity state. our result shows that the non-zero velocity state can be achieved
even if. One method to achieve the lock-in is to set as afunction of time, $i.e.$ , Run
2. We can show anon-zero mean velocity state by asimilar dynamics to that in. III even
if is constant and is smaller than 770. Amethod is to stop the flapping motion by changing
as$\eta$$\eta=\eta_{0}$$\eta<\eta_{0}$$\eta$$\mathrm{s}\mathrm{e}\mathrm{c}$$\eta$$\theta(t)$$\Delta\theta\cos(2\pi t/T)(t\leq 7T+\frac{1}{2}T, 8T+\frac{1}{2}T<t)$$\theta(t)=\{$$-\Delta\theta$$(7T+ \frac{1}{2}T<t\leq 8T+\frac{1}{2}T)$

.

,

(4.4)

Astop of flapping in atemporal rising of CM causes aflow turning around wing. This flow
helps to form asmall coherent vortex under the wing. This small coherent vortex remains
under the wing after the wing restarts flapping, and the dynamics of this vortex and anothercoherent vortex produced in the next downstroke becomes similar to the symmetry-breaking
case.
V.

CONCLUDING REMARKSOur model, two plates attached together at ahinge, and flapping symmetric with respect tothe horizontal line, shows interesting aspects of CM motion due to flapping. Given physicalparameters, this model converges one or more discrete type(s) of steady periodic motion.
Choosing aparameter 77, the characteristic time of the cut-0ff viscosity in non-dimensional
form, the stable state shows abifurcation. The bifurcation diagram shows that i) atrivial
state (zer0-mean velocity) is not always stable; ) two stable states can coexist in some regionof .
Adynamics which makes our model achieve the symmetry-breaking[18] is observed in aregion y7 $>\eta_{0}\simeq 0.62$ , but the state characterized by non-zero mean velocity can be realized. This means that the state characterized by non-zero mean velocity can beeven if y7achieved by not only the typical dynamics shown in ref.[18] but another one. Long time simulations shows that the flapping state characterized by zer0-mean velocity can be unstable, andthat the model converges to astable state characterized by non-zero mean velocity. Distinct
dynamics to achieve the transition is not observed in this case.
The phenomena observed in the region
might not be related to the ‘real’ situation,
because our model simplifies viscosity. However, our simplified viscosity gives arelevant bifurcation parameter controlling the number of separated vortex. By this parameter, we cansee asharp change of the stable state characterized by zer0-mean velocity and discuss thecontribution of the coherent vortex directly. Thus it is easily known that the stability of thestate is controlled by the amount of point vortices consisting of coherent vortex, contrary to
the symmetry-breaking dynamics where the necessarily and sufficient condition of it is thecoexistence of two coherent vortices.
In the standpoint of the the vortex-body interaction, this model presents aviewpoint of theanalysis of these phenomena. The phase of the motion of the body interacting with fluid isclassified as afunction of nondimensional parameters $[1, 2]$ . However, our model shows thatgiven set of parameter does not always map into one type of motion. This result is meaningfulwhen the dynamics of insect flight is considered, because they should choose abest type of
flapping-flight mechanism corresponding to their circumstances. The existence of bistablemode in vortex-body interaction seems rather universal phenomenon: asimilar behavior isobserved in the case of one-dimensional flapping string in tw0-dimensional soap film [8].
Studies in terms of stability are required to understand phenomena related to the vortexbody interaction. To do that, asimple model with high-symmetry such as ours is quite useful,because classification of typical phenomena observed in vortex-body interactions is easy in thistype of model and the classification is useful to analyse the vortex-body interaction.
$\mathrm{i}\mathrm{i}$$\eta$$\leq\eta_{0}$$\eta<\eta_{0}$144
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$\mathrm{E}\mathrm{x}\mathrm{p}$$\mathrm{E}\mathrm{x}\mathrm{p}$$\mathrm{B}352$$\mathrm{E}\mathrm{x}\mathrm{p}$$\mathrm{E}\mathrm{x}\mathrm{p}$$\mathrm{E}\mathrm{x}\mathrm{p}$