Stability of Underwater Periodic Locomotion - Springer Link

c Pleiades Publishing, Ltd., 2013. ISSN 1560-3547, Regular and Chaotic Dynamics, 2013, Vol. 18, No. 4, pp. 380–393.

Stability of Underwater Periodic Locomotion Fangxu Jing* and

Eva Kanso**

Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, California 90089, USA Received April 10, 2013; accepted June 20, 2013

Abstract—Most aquatic vertebrates swim by lateral flapping of their bodies and caudal fins. While much effort has been devoted to understanding the flapping kinematics and its influence on the swimming efficiency, little is known about the stability (or lack of) of periodic swimming. It is believed that stability limits maneuverability and body designs/flapping motions that are adapted for stable swimming are not suitable for high maneuverability and vice versa. In this paper, we consider a simplified model of a planar elliptic body undergoing prescribed periodic heaving and pitching in potential flow. We show that periodic locomotion can be achieved due to the resulting hydrodynamic forces, and its value depends on several parameters including the aspect ratio of the body, the amplitudes and phases of the prescribed flapping. We obtain closedform solutions for the locomotion and efficiency for small flapping amplitudes, and numerical results for finite flapping amplitudes. This efficiency analysis results in optimal parameter values that are in agreement with values reported for some carangiform fish. We then study the stability of the (finite amplitude flapping) periodic locomotion using Floquet theory. We find that stability depends nonlinearly on all parameters. Interesting trends of switching between stable and unstable motions emerge and evolve as we continuously vary the parameter values. This suggests that, for live organisms that control their flapping motion, maneuverability and stability need not be thought of as disjoint properties, rather the organism may manipulate its motion in favor of one or the other depending on the task at hand. MSC2010 numbers: 76Z10, 37N99 DOI: 10.1134/S1560354713040059 Keywords: biolocomotion, solid-fluid interactions, efficiency, motion stability

1. INTRODUCTION A large proportion of fish species are characterized by elongated bodies that swim forward by flapping sideways. These sideways oscillations produce periodic propulsive forces that cause the fish to swim along time-periodic trajectories, [1]. The kinematics of the flapping motion and the resulting swimming performance, as well as their relationship to the swimmer’s morphology, have been the subject of numerous studies, see, for example, [2, 3]. However, little attention has been given to the stability of underwater locomotion. The importance of motion stability and its mutual influence on body morphology and behavior is noted in the work of Weihs, see [4] and references therein. Weihs uses clever arguments and simplifying approximations founded on a deep understanding of the equations governing underwater locomotion to obtain “educated estimates” of the stability of swimming fish without ever solving the complicated set of equations. The swimming motion is said to be unstable if a perturbation in the conditions surrounding the swimmer’s body results in forces and moments that tend to increase the perturbation, and it is stable if these emerging forces tend to reduce such perturbations or keep them bounded so that the fish returns to or stays near its original periodic swimming. Stability may be achieved actively or passively. Active stabilization requires neurological control that activates musculo-skeletal components to compensate for external perturbations acting against * **

E-mail: [email protected] E-mail: [email protected]

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stability. On the other hand, passive stability of the locomotion gaits requires no additional energy input by the fish. In this sense, one can argue that stability reduces the energetic cost of locomotion. Therefore, from an evolutionary perspective, it seems reasonable to conjecture that stability would have a positive selection value in behaviors such as migration over prolonged distances and time. However, stability limits maneuverability and body designs/flapping motions that are adapted for stable swimming are not suitable for high maneuverability and vice versa, [4, 5]. In this work, we study stability of periodic swimming using a simple model consisting of a planar elliptic body undergoing prescribed flapping motion in unbounded potential flow. By flapping motion, we mean periodic heaving and pitching of the body as shown in Fig. 1. We formulate the equations of motion governing the resulting locomotion and examine its efficiency. We then investigate the stability of this motion using Floquet theory (see [6]). We find that stability depends in a non-trivial way on the body geometry (aspect ratio of the ellipse) as well as on the amplitudes and phases of the flapping motion. Most remarkable is the ability of the system to transition from stability to instability and back to stability as we vary some of these parameters.

Fig. 1. Model of flapping fish: An ellipse with semi-axes a and b is submerged in an unbounded potential fluid. Motion is observed in an inertial frame with the position of the mass center given by (x, y) and orientation by θ. Fish flaps in the y- and θ-directions, and propels in the x-direction.

This model is reminiscent of the three-link swimmer used by Kanso et al. (2005) to examine periodic locomotion in potential flow, see [7]. The three-link swimmer undergoes periodic shape deformations that result in coupled heaving, pitching and locomotion. Here, we ignore body deformations for the sake of simplicity and prescribe heaving and pitching directly. Note that the three-link swimmer was also used by Jing & Kanso (2011) to study the effect of body elasticity on the stability of the coast motion of fish (motion at constant speed). They found that elasticity of the body may lead to passive stabilization of the (otherwise unstable) coast motion, see [8, 9]. The present model consisting of a single elliptic body is mostly similar to the system studied by Spagnolie et al. (2010) both experimentally and numerically, see [10]. In the latter, an elliptic body undergoes passive pitching (via a torsional spring) subject to prescribed periodic heaving in a viscous fluid, whereas in our model both pitching and heaving motions are prescribed and the fluid medium is inviscid. Despite these differences, the two models exhibit qualitatively similar behavior as discussed in Section 6. The paper is organized as follows. In Section 2, we formulate the equations of motion governing the locomotion of a periodically flapping body in unbounded potential flow. We analyze the body’s locomotion and efficiency when subject to small amplitude flapping motion in Section 3, and consider the more general case of finite amplitude flapping in Section 4. In Section 5, we assess the stability of the periodic locomotion using Floquet theory. The main findings and their relevance to biolocomotion are discussed in Section 6. 2. PROBLEM FORMULATION Consider a planar elliptic body with semi-major axis a and semi-minor axis b, submerged in unbounded potential flow that is at rest at infinity. The elliptic body is neutrally buoyant, that is to say, the body and fluid densities are equal to ρ. Its mass is given by mb = ρπab, and its moment of inertia about the center of mass C is Jb = mb (a2 + b2 )/4. Let (x, y) denote the position of the mass center with respect to a fixed inertial frame and let θ denote the orientation angle of the ellipse measured from the positive x-direction to the ellipse’s major axis, see Fig. 1. The linear ˙ corresponds to the ˙ respectively, where the dot () and angular velocities are given by (x, ˙ y) ˙ and θ, derivative with respect to time t. REGULAR AND CHAOTIC DYNAMICS

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In order to emulate the flapping motion of a swimming body, we assume that y and θ vary periodically in time due to some periodic flapping force F flap and flapping moment τ flap generated by the swimming body. Note that in the case of a body swimming by deforming itself, y and θ are a result of the body deformation. Here, we do not account for the body deformation but rather prescribe y(t) and θ(t) as periodic functions of time. Namely, we set y(t) = Ay sin(ωt + φy ),

θ(t) = Aθ sin(ωt + φθ ).

(2.1)

and solve for the resulting locomotion in the x-direction. The equations governing the motion of the flapping body are basically Kirchhoff’s equations expressed in the inertial frame and subject to forcing F flap and τ flap in the y- and θ-directions, that is, ¨ = Fx , mb x

(2.2)

and mb y¨ = Fy + F flap ,

Jb θ¨ = τ + τ flap ,

(2.3)

where Fx , Fy and τ are the hydrodynamic forces and moment acting on the body. For motions in potential flow, Fx , Fy and τ can be obtained using a classic procedure (see, e.g., [18–20]) 1 1 ˙ [−(m1 + m2 ) + (m2 − m1 ) cos 2θ] x ¨ + (m2 − m1 )¨ y sin 2θ − (m2 − m1 )(x˙ sin 2θ − y˙ cos 2θ)θ, 2 2 1 1 ˙ x sin 2θ + (m2 − m1 )(x˙ cos 2θ + y˙ sin 2θ)θ, Fy = [−(m1 + m2 ) − (m2 − m1 ) cos 2θ] y¨ + (m2 − m1 )¨ 2 2 1 τ = −J θ¨ + (m2 − m1 ) x˙ 2 sin 2θ − y˙ 2 sin 2θ − 2x˙ y˙ cos 2θ . 2 (2.4)

Fx =

Here, m1 = ρπb2 , m2 = ρπa2 are, respectively, the added mass of the elliptic body along its major and minor directions and J = ρπ(a2 − b2 )2 /8 is the added moment of inertia, see, e.g., [11]. Substituting (2.4) into (2.2) and (2.3), one can use (2.2) to solve for x(t), and (2.3) to compute the forcing F flap and τ flap needed in order for the body to achieve the prescribed flapping motion in (2.1). The total angular momentum h of the body-fluid system is given by h = (Jb + J)θ˙ whereas the total linear momentum can be written as ⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎝px ⎠ = mb ⎝x˙ ⎠ + ⎝cos θ − sin θ⎠ ⎝m1 0 ⎠ ⎝ cos θ sin θ ⎠ ⎝x˙ ⎠ . (2.5) py 0 m2 y˙ sin θ cos θ − sin θ cos θ y˙ The momentum px is conserved since there is no external forcing applied in the x-direction. Therefore, one has px (t) = px (t = 0), which yields x(t) ˙ =

m1 − m2 [(y˙ sin 2θ)t=0 − (y˙ sin 2θ)] . 2 mb + m1 cos2 θ + m2 sin2 θ

(2.6)

That is to say, Eq. (2.2) admits an integral of motion whose value is given by the above equation. The distance traveled by the body’s center of mass in one period of flapping, T = 2π/ω, is given by T x˙ dt = |x(T ) − x(0)|. (2.7) d= 0

The total kinetic energy E of the body-fluid system is given by 1 1 E = (x˙ px + y˙ py ) + (Jb + J)θ˙2 . 2 2 REGULAR AND CHAOTIC DYNAMICS

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By the work-energy theorem, the time derivative of the kinetic energy is equal to the total power input by the flapping force F flap and moment τ flap . Thus, the work done by flapping is equivalent to the kinetic energy E. To this end, the average work done in one period is given by T 1 ¯= E(t)dt. (2.9) E T 0 We define the cost of locomotion e as the average work divided by the average distance over one period, namely ¯ E (2.10) e= . d Smaller e means less energy expenditure for a fixed distance traveled. It is convenient to introduce ¯ an efficiency measure η as the inverse of the cost of locomotion, that is η ≡ 1/e = d/E. Before we proceed to examine the locomotion and efficiency√ of such a swimmer, we nondimensionalize the system by scaling time with T , length with ab and mass with mb = ρπab. All variables are subsequently written in dimensionless form. The important parameters for this system are: the body’s aspect ratio γ ≡ a/b as well as the flapping amplitudes Ay , Aθ and phases φy , φθ . 3. SMALL FLAPPING AMPLITUDES Consider the case of small flapping amplitudes Ay and Aθ . Let Ay ≡ y 1 and Aθ ≡ θ 1 ˙ θ¨ ∼ O( θ ), ˙ y¨ ∼ O( y ) and θ, θ, where both y and θ are of the same order of magnitude. One gets y, y, but ω, φy and φθ are not necessarily small. Use the approximation cos θ ≈ 1 and sin θ ≈ θ and substitute into (2.6) to obtain 1 (γ − 1) ω y θ [sin(2ωt + φy + φθ ) − sin(φy + φθ )] . (3.1) 2 Clearly, the velocity in the x-direction depends on the aspect ratio γ and φy + φθ . Its magnitude is of order ∼ O( y θ ). In other words, for small flapping amplitudes, the motion in the x-direction is small compared to the flapping motion in y and θ. Approximate expressions for F flap and τ flap are obtained by substituting (2.1) and (3.1) into (2.3), x˙ ≈

y, F flap ≈ (mb + m2 )¨

¨ τ flap ≈ (Jb + J)θ.

For small flapping amplitude, we can express the cost of locomotion in closed form

π γ(γ + 1) 2y + 2(γ 2 + 1) 2θ . e≈ 2γ(γ − 1) y θ | sin(φy + φθ )| To minimize e (or, equivalently, to maximize efficiency η), one needs

1 π, n = 0, ±1, ±2, . . . γ → ∞, and φy + φθ = n + 2

(3.2)

(3.3)

(3.4)

These closed-form expressions do not hold for large amplitudes Ay and Aθ where the efficiency needs to be analyzed numerically, as done in the next section. 4. LOCOMOTION AND EFFICIENCY We examine the swimming trajectories and their dependence on the following parameters: aspect ratio γ, amplitudes Ay and Aθ , and phases φy and φθ . The swimming motion is given by (2.1) and (2.6), where the latter is integrated numerically to get x(t). Consider the case where Ay = 1, Aθ = π4 , φy = − π2 , φθ = 0 and consider various aspect ratios γ = 1.01, 4, 8, 1000, as shown in Fig. 2. Note that as we vary the aspect ratio, the total area of the elliptic body remains constant (this is guaranteed by the way we non-dimensionalize length using REGULAR AND CHAOTIC DYNAMICS

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Fig. 2. (color online) Left: Prescribed flapping in the (y, θ) plane. Initial points are marked by ◦. Right: Trajectories of mass center in the (x, y) plane with snapshots of the body in motion overlaid. Simulations are for Ay = 1, Aθ = π4 , φy = − π2 , φθ = 0 and various aspect ratios: (a) γ = 1.01, (b) γ = 4, (c) γ = 8, (d) γ = 1000.

√

ab). As expected, the net locomotion is almost zero when the elliptic body is close to a circular shape (γ = 1.01) and it reaches a maximum as the elliptic body approaches a flat plate (γ = 1000). In Fig. 3, γ is set to 4 and Ay is varied. One can see that the net locomotion d depends linearly on Ay , which is also evident from (2.6). In Fig. 4, different values of Aθ are shown. The net locomotion depends nonlinearly on Aθ . Interestingly, the trajectories that correspond to Aθ = π/4 and Aθ = π/8 are almost identical, whereas for Aθ = 3π/4 the locomotion is in the negative xdirection.

Fig. 3. (color online) Left: Prescribed flapping in the (y, θ) plane. Initial points are marked by ◦. Right: Trajectories of mass center in the (x, y) plane. Simulations are for Aθ = π4 , γ = 4, φy = − π2 , φθ = 0 and various heaving amplitudes: (a) Ay = 0.01, (b) Ay = 0.5, (c) Ay = 1, (d) Ay = 2.

Fig. 4. (color online) Left: Prescribed flapping in the (y, θ) plane. Initial points are marked by ◦. Right: Trajectories of mass center in the (x, y) plane. Simulations are for Ay = 1, γ = 4, φy = − π2 , φθ = 0 and various pitching amplitudes: (a) Aθ = π/8, (b) Aθ = π/4, (c) Aθ = π/2, (d) Aθ = 3π/4.

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Fig. 5. (color online) Left: Prescribed flapping in the (y, θ) plane. Initial points are marked by ◦. Middle and right: Trajectories of mass center in the (x, y) plane. Simulations are for Ay = 1, Aθ = π4 , γ = 4 and various combinations of phases: (a) (φy , φθ ) = − π4 , 0 , (b) (φy , φθ ) = − π2 , − π4 , (c) (φy , φθ ) = π4 , 0 , (d) (φy , φθ ) = − π2 , − 3π , (e) (φy , φθ ) = − π2 , 0 , (f ) (φy , φθ ) = (0, 0), (g) (φy , φθ ) = − π2 , − π2 , (h) (φy , φθ ) = π4 , − 3π . 4 4

Motions for various phases φy and φθ are shown in Figs. 5. Notice that the flapping motion depicted in (y, θ) plane depends on the difference in phase φy − φθ . This can be readily verified by eliminating t from (2.1) and expressing the flapping motion in (y, θ) plane as 2 2 y θ θ y −2 cos(φy − φθ ) + = sin2 (φy − φθ ). (4.1) Ay Ay Aθ Aθ As φy − φθ varies, the path in the (y, θ) plane is elliptic, except for φy − φθ = nπ (n = 0, ±1, ±2, . . .) in which case it is a segment of the straight line given by θ = (−1)n (Aθ /Ay )y. From (2.6), one has that the locomotion x(y, θ) written as a function of (y, θ) possesses the following symmetries x(−y, −θ) = x(y, θ),

x(−y, θ) = −x(y, θ),

x(y, −θ) = −x(y, θ),

(4.2)

whereas the flapping motion in (2.1) has the symmetries y(t; φy + π) = −y(t; φy ), θ(t; φθ + π) = −θ(t; φθ ),

y(−t; φy ) = −y(t; −φy ), θ(−t; φθ ) = −θ(t; −φθ ).

(4.3)

Based on these symmetries, one can immediately conclude that, when all other parameters are held fixed, motions that correspond to (φy , φθ ) and (−φy , −φθ ) are mirror images of each other: their distances and energies are the same, as seen from (4.2). For (φy , φθ ) with φy − φθ = 2nπ + constant, one gets the same path in the (y, θ) space. When tracing the same path in the (y, θ)-plane (but starting at different initial points), the resulting trajectories in the (x, y) plane are similar (with different initial positions). Note that, in general, flapping motions that trace a straight line in the (y, θ)-plane do not correspond to zero net locomotion in the (x, y)-plane, except for φy = φθ = 0 and φy = φθ = −π/2. This is evident from the example of φy = π/4, φθ = −3π/4 in Fig. 5h. The locomotion here is not a result of a geometric phase only but a dynamic phase, see [7]. ¯ and cost of locomotion e = E/d. ¯ We now compute the average work E Ideally, one would like to find optimal parameter values that minimize e (maximize efficiency η) and/or maximize d (see, for example [3, 12]). Instead of minimizing e over the five dimensional parameter space, we study the dependence of e on the system’s parameters by varying one parameter at a time. In Fig. 6, we set φy = − π2 , φθ = 0 and vary γ, Ay and Aθ , respectively. Solid lines correspond to the numerical nonlinear solutions and dashed lines are obtained from the linear approximation in (3.3). Figure 6a shows that, for Ay = 1, Aθ = π/4, there exists an optimal value of γ ≈ 2.9, whereas the small amplitude approximation predicts that e is a decreasing function of γ. Figure 6b shows an optimal value of Ay ≈ 1.3 and that the small amplitude results qualitatively follow the nonlinear behavior of e. This is because the work E depends quadratically on Ay , the displacement d depends linearly on Ay and the small amplitude approximation in (3.3) preserves the form of dependence on this parameter. However, Fig. 6c shows that when varying Aθ , the small amplitude results provide good approximation of the nonlinear efficiency only up to Aθ ≈ 3π 16 . REGULAR AND CHAOTIC DYNAMICS

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(a) e vs. γ

(b) e vs. Ay

(c) e vs. Aθ Fig. 6. Cost of locomotion e as a function of: (a) aspect ratio γ, (b) heaving amplitude Ay , (c) pitching amplitude Aθ . The base parameter values are set to φy = −π/2, φθ = 0, γ = 4, Aθ = π/4, Ay = 1. Solid lines are nonlinear numerical solutions, while dashed lines are based on small amplitude approximation given in (3.3).

(a) γ = 1.01

(b) γ = 2

(c) γ = 4

(d) γ = 8

(e) γ = 16

Fig. 7. Contour plots of cost of locomotion e for the cases Ay = 1, Aθ = π/4 and various aspect ratio γ. Each plot is evaluated on a 201 × 201 mesh in [−π , π] × [−π , π] in the (φy , φθ ) plane. Lower values of e correspond to higher efficiency.

In Figs. 7 to 9, we examine the dependence of e on (φy , φθ ) by discretizing the domain [−π , π] × [−π , π] using a 201 × 201 mesh. Contours of e as a function of (φy , φθ ) are depicted for various γ, Ay and Aθ . It follows from (4.2) and (4.3) that e has a reflection symmetry about the origin e(φy , φθ ) = e(−φy , −φθ ) and the periodic property e(φy + π, φθ + π) = e(φy , φθ ). Therefore, it would have been sufficient to show the dependence of e only on one quarter of the shown domain, say, [0 , π] × [0 , π]. Note that the parameters that minimize e, thus maximize efficiency η, are approximately 3 γ 4, Ay ≈ 1.3, Aθ ≈ 3π/16 and (φy , φθ ) ≈ ((m + 12 )π, nπ), where m, n = 0, ±1, ±2, . . . One example of the optimal phases is φy = π2 , φθ = 0, with corresponding locomotion shown in Fig. 2. For this optimal motion, the pitching angle is zero when the heaving motion is maximum (90◦ out of phase), which qualitatively agrees with the results in [12]. The optimal aspect ratio 3 γ 4 agrees REGULAR AND CHAOTIC DYNAMICS

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(a) Ay = 0.01

(b) Ay = 0.5

(c) Ay = 1

(d) Ay = 2

387

(e) Ay = 6

Fig. 8. Contour plots of cost of locomotion e for the cases Aθ = π/4, γ = 4 and various heaving amplitude Ay . Each plot is evaluated on a 201 × 201 mesh in [−π , π] × [−π , π] in the (φy , φθ ) plane. Lower values of e correspond to higher efficiency.

(a) Aθ = 0.01

(b) Aθ = π/8

(c) Aθ = π/4

(d) Aθ = 3π/8

(e) Aθ = π/2

Fig. 9. Contour plots of cost of locomotion e for the cases Ay = 1, γ = 4 and various pitching amplitude Aθ . Each plot is evaluated on a 201 × 201 mesh in [−π , π] × [−π , π] in the (φy , φθ ) plane. Lower values of e correspond to higher efficiency.

with the optimal shape aspect ratio obtained in the comprehensive optimization study in [3], and is representative of the aspect ratio of various Carangiform swimmers such as Bass (γ = 3.8) in [13], Tuna (γ = 3.5) in [14] and Saithe (γ = 4.1) in [15]. The optimal heave to cord ratio Ay /a ≈ 0.75 √ √ (where a = γ ≈ 3) and maximum angle Aθ ≈ 3π/16 = 16.875◦ both agree with the optimal motions for the rigid flapping body (0.75 Ay /a 1, Aθ ≈ 16◦ ) given by [16]. This is remarkable given the simplicity of our model in comparison to the models in [3, 16]. 5. STABILITY OF PERIODIC LOCOMOTION We examine the stability of periodic motion subject to arbitrary perturbations in the surrounding ˙ T , and rewriting (2.2) and (2.3) as follows fluid. We begin by introducing q = [x, ˙ y, ˙ θ, θ] M(θ)q˙ = f (q) + Fflap ,

(5.1)

where detailed expressions for M, f and Fflap are listed in the Appendix. In Sections 2–4, we prescribed the flapping motion y(t) and θ(t) according to (2.1) and used (2.6) to solve for x(t) and (2.3) to solve for F flap and τ flap . The resulting motion x(t), y(t), and θ(t) as well as the forcing F flap and τ flap are periodic with period T . We let qp denote the q corresponding to such periodic motion. We study the stability of qp by introducing a small perturbation δq such that q = qp + δq while keeping F flap and τ flap the same as that producing the periodic solution. In other words, we account for arbitrary perturbations in the fluid environment while keeping the same flapping forces to check if such perturbations destabilize the periodic trajectory. We linearize Eq. (5.1) about the periodic trajectory qp (t) to get δq˙ = J(t)δq.

(5.2)

The Jacobian J(t) is a 4 × 4 periodic matrix (J(0) = J(T )) given by (see the Appendix for details) ∂g , where g = M−1 (f + Fflap ). (5.3) J(t) = ∂q flap (qp ,F


)

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Fig. 10. Left: stability of the case Ay = 1, Aθ = π/4 and γ = 4, with phases (φy , φθ ) varied on a 201 × 201 mesh in [−π , π] × [−π , π]. Stable cases correspond to shaded areas, unstable cases are white areas. Middle and right: (top) (φy , φθ ) = (−π/2, 0) is stable. Solid lines correspond to unperturbed periodic solutions, dashed lines correspond to perturbed solutions; (bottom) (φy , φθ ) = (−π/4, −π/4) is unstable.

Let Φ(t) denote the fundamental solution matrix of (5.2). The eigenvalues λi of the timeindependent matrix, B = Φ(0)−1 Φ(T ), are referred to as the characteristic multipliers. Their locations in the complex plane indicate the stability of the periodic solution qp : if at least one characteristic multiplier lies outside the unit circle, qp is unstable; if all λi ’s (i = 1, . . . , 4) are on the unit circle, then qp is regarded as marginally stable. For a non dissipative system as in our model, these two are the only possible scenarios, namely, unstable or marginally stable (simply referred to as “stable” hereafter). One eigenvalue λ1 is always 1, reflecting the fact that qp is periodic. The remaining eigenvalues may be complex. Complex eigenvalues come in conjugate pairs. For the case Ay = 1, Aθ = π4 , γ = 4, the stability results are plotted in Fig. 10 as a function of the phases (φy , φθ ), again evaluated on a 201 × 201 mesh discretizing the domain [−π , π] × [−π , π]. Shaded regions are stable. Notice the reflection symmetry about (0, 0) and periodicity of π in both φy and φθ that we observed in the efficiency analysis is again seen in the stability plot. Two examples with different stability characteristics are shown. The solid lines are unperturbed periodic solutions qp , and dashed lines are solutions with random initial perturbations with magnitude |δq(t = 0)| ∼ O(10−3 ). Clearly, the trajectory corresponding to parameters in the stable region remains close to the periodic trajectory for the integration time whereas that corresponding to parameters in the unstable region does not. In Fig. 11, we examine the behavior of the real and imaginary parts of the characteristic multipliers as a function of φθ for Ay = 1, Aθ = π/4, γ = 4 and φy = 0. In other words, we explore the behavior of λi ’s for φy = 0, along the dashed line in the left plot of Fig. 10. One can see that two characteristic multipliers are always located at (1, 0) (represented by ). The other two are represented by . The dynamics changes from stable to unstable when the two conjugates collide at (1, 0) and split onto the real axis. For the parameters considered, when φθ varies from −π to π, stability changes from unstable to stable and stable to unstable four times in total. We now examine the stability behavior as we change γ, Ay and Aθ , respectively. Figure 12 shows stability regions for Ay = 1, Aθ = π4 while varying γ. For bodies closer to circular shape (γ = 1.01), the motion is stable for all (φy , φθ ) (but this stability property is not very useful since the net displacement is almost zero). When γ > 1.43, unstable regions start to appear. As γ increases, unstable regions grow while stable regions shrink. The total area of the stable regions becomes minimum when γ ≈ 2, and the stable areas around (φy , φθ ) = ((m + 1/2)π, nπ) persist. Interestingly, as γ continues to increase, new stable regions start to emerge and grow from the previous unstable areas around (φy , φθ ) = (nπ, (m + 1/2)π). Then, at these spots, unstable regions emerge and grow from the newly formed stable regions, and so on and so forth. The boundaries between stable and unstable regions around (nπ, (m + 1/2)π) become blurry as γ becomes larger, and ((m + 1/2)π, nπ) remain stable. This trend is reminiscent of the phenomenon observed in Spagnolie et al. [10], in which the authors noticed the motion of an elliptic body subject to prescribed heaving and passive pitching goes through states from “coherence to incoherence, and back again” REGULAR AND CHAOTIC DYNAMICS

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(a) Real vs. Imaginary parts of λi

(b) Real parts of λi vs. φθ

(c) Imaginary parts of λi vs. φθ

Fig. 11. (Color online) Characteristic multipliers of the cases Ay = 1, Aθ = π/4, γ = 4, φy = 0, and φθ varied from −π to π: (a) Real and Imaginary parts in the complex plane. Two characteristic multipliers always located at (1, 0) are represented by . The two complex conjugates are represented by . (b) Real and (c) Imaginary parts of characteristic multipliers. Motion becomes unstable when the two complex conjugates collide at (1, 0) and split on the real axis.

(a) γ = 1.01

(e) γ = 4

(b) γ = 1.45

(c) γ = 1.75

(d) γ = 2.5

(f) γ = 8

(g) γ = 16

(h) γ = 32

Fig. 12. Stability regions for the cases Ay = 1, Aθ = π/4 and various aspect ratio γ. Each plot is evaluated on a 201 × 201 mesh in [−π , π] × [−π , π] in the (φy , φθ ) plane. Shaded areas correspond to stable cases, white areas correspond to unstable cases.

as the aspect ratio changes. Note that the latter studies are in viscous fluid whereas the analysis here is for an inviscid fluid model. Interestingly, this simplified model is able to capture, at least qualitatively, the behavior observed in [10]. Figure 13 shows stability regions for Aθ = π/4, γ = 4 while varying Ay . For small Ay the motion is stable for all (φy , φθ ) but with no net locomotion. As Ay increases, unstable regions start to form around (nπ, (m + 1/2)π). As Ay continues to increase, unstable regions grow while stable regions shrink. Then, layers of stable/unstable regions start to form around (nπ, (m + 1/2)π). REGULAR AND CHAOTIC DYNAMICS

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(a) Ay = 0.01

(b) Ay = 0.6

(c) Ay = 0.625

(f) Ay = 1.25

(g) Ay = 1.5

(h) Ay = 1.75

(d) Ay = 0.75

(e) Ay = 1

(i) Ay = 2

(j) Ay = 6

Fig. 13. Stability regions for the cases Aθ = π/4, γ = 4 and various heaving amplitude Ay . Each plot is evaluated on a 201 × 201 mesh in [−π , π] × [−π , π] in the (φy , φθ ) plane. Shaded areas correspond to stable cases, white areas correspond to unstable cases.

(a) Aθ = 0.01

(b) Aθ = π/8

(e) Aθ = 5π/16

(f) Aθ = 3π/8

(c) Aθ = 3π/16

(g) Aθ = 7π/16

(d) Aθ = π/4

(h) Aθ = π/2

Fig. 14. Stability regions for the cases Ay = 1, γ = 4 and various pitching amplitude Aθ . Each plot is evaluated on a 201 × 201 mesh in [−π , π] × [−π , π] in the (φy , φθ ) plane. Shaded areas correspond to stable cases, white areas correspond to unstable cases.

In Fig. 14, we vary Aθ while keeping Ay = 1 and γ = 4. When Aθ is small, the whole plane in (φy , φθ ) is stable. As Aθ increases, unstable regions start to emerge and grow, while stable regions shrink but persist around (nπ, (m + 1/2)π), and as Aθ increases further unstable regions start to emerge within the stable strips. This trend of switching from stability to instability and back to stability when varying parameters suggests that such swimmers can change their stability behavior by changing their flapping motion. That is, they can switch from stable periodic swimming to an unstable motion (more maneuverable). Based on this, one can conjecture that when it comes to live organisms, maneuverability and stability need not be thought of as disjoint properties, rather the organism may manipulate its flapping in favor of one or the other depending on the task at hand. For example, a fish may transition from stable periodic swimming to an unstable maneuver when evading a predator. REGULAR AND CHAOTIC DYNAMICS

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6. CONCLUSIONS We studied the locomotion, efficiency and stability of periodic swimming of fish using a simple planar model of an elliptic swimmer undergoing prescribed sinusoidal heaving and pitching in potential flow. We obtained expressions for the locomotion velocity for both small and finite flapping amplitudes, and showed how trajectories depend on key parameters, namely, aspect ratio γ, amplitudes Ay and Aθ and phases φy and φθ . Efficiency is defined as the inverse of the cost of locomotion e. We studied the dependence of e on the parameters shown for both small and finite amplitudes. We observed that the efficiency maximizing parameters are approximately 1 3 γ 4, Ay ≈ 1.3, Aθ ≈ 3π 16 , φy ≈ (m + 2 )π and φθ ≈ nπ, where n, m = 0, 1, 2, . . ., whose values are in excellent agreement with values based on experimental and computational motions of flapping fish, see [3, 12, 16] and references therein. We then studied the stability of periodic locomotion using Floquet theory. To our best knowledge, besides the work of Weihs which uses approximate arguments, this is the first work that rigorously studies the stability of periodic locomotion albeit in a simplified model. We focused on evaluating stability on the whole (φy , φθ ) parameter space, and examined the effect of varying γ, Ay and Aθ . We observed that stable and unstable regions in the (φy , φθ ) plane evolve as these parameters change. Particularly noteworthy is the back and forth switching between stability and instability around ((m + 12 )π, nπ) and (nπ, (m + 12 )π). This switching is reminiscent of the observation in [10] that the motion of a heaving and pitching foil switches from coherence to incoherence and back to coherence when varying the aspect ratio of foil. In our study, we found a similar behavior when varying not only the aspect ratio but also the flapping parameters. This indicates that such a swimmer can change its stability behavior by changing its flapping motion, and thus can easily switch from stable periodic swimming to an unstable, yet more maneuverable, state. Based on this, one could conjecture that, when it comes to live organisms, maneuverability and stability are not disjoint properties but may be manipulated depending on the needs of the organism. Clearly, this statement is speculative until verified by experimental evidence. To date, little is known experimentally on the stability of underwater periodic motions, let alone the stability of biological swimmers. Future extensions of this work will include the effects of body deformation and body elasticity, vortex shedding, and frequency of flapping on the observed stability of periodic swimming, as well as on motion efficiency such as in [17].

APPENDIX For completeness, we rewrite equation (5.1) as M(θ)q˙ = f (q) + Fflap , where

r2 ⎜ c2 − cos 2θ ⎜ ⎜ ⎜ − sin 2θ M(θ) = ⎜ ⎜ ⎜ 0 ⎜ ⎝ 0 and

⎞

⎛

⎛

− sin 2θ r2 c2

0

+ cos 2θ 0 0

1

0

0

−2θ˙x˙ sin 2θ + 2θ˙y˙ cos 2θ

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⎟ ⎟ ⎟ ⎟ 0 ⎟, ⎟ ⎟ 0 ⎟ 8 8 4 4 r − c + 4r c ⎠

No. 4

(A.1)

4r 4 c2

⎞

⎛

⎞ 0

⎟ ⎜ ⎟ ⎜ ⎜ 2θ˙x˙ cos 2θ + 2θ˙y˙ sin 2θ ⎟ ⎟ ⎜ ⎟, f (q) = ⎜ ⎟ ⎜ ⎟ ⎜ θ˙ ⎟ ⎜ ⎠ ⎝ 2 2 (x˙ − y˙ ) sin 2θ − 2x˙ y˙ cos 2θ REGULAR AND CHAOTIC DYNAMICS

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Fflap

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⎜ ⎟ ⎜ ⎟ ⎜F flap ⎟ ⎟ 1 ⎜ ⎜ ⎟. = ⎜ ⎟ 2 2ρπc ⎜ ⎟ ⎜ 0 ⎟ ⎝ ⎠ flap τ

(A.2)

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These equations can be rewritten as q˙ = g ≡ M−1 (f + Fflap ). One can linearize the above equation to obtain δq˙ = J(t)δq,

where

(A.3)

∂g . J(t) = ∂q (qp ,Fflap )

The entries of the Jacobian J(t) are given by ⎛ ˙ 2 + r 2 cos 2θ) 2c2 θ(c 2r 2 c2 θ˙ sin 2θ ⎜ −r 4 + c4 r 4 − c4 ⎜ ⎜ 2˙ 2 ⎜ 2c θ(c − r 2 cos 2θ) 2r 2 c2 θ˙ sin 2θ ⎜ 4 4 −r + c r 4 − c4 J(t) = ⎜ ⎜ ⎜ ⎜ 0 0 ⎜ ⎝ μα −μβ

⎞ J13

J14 ⎟ ⎟ ⎟ ⎟ J23 J24 ⎟ ⎟, ⎟ ⎟ 0 1 ⎟ ⎟ ⎠

2 0 μ (x˙ − y˙ 2 ) cos 2θ + 2x˙ y˙ sin 2θ (A.4)

where α = x˙ sin 2θ − y˙ cos 2θ, β = y˙ sin 2θ + x˙ cos 2θ, μ = 8r 4 c2 /(r 8 − c8 + 4r 4 c4 ), 4

c2 flap 2 ˙ , J14 = 2c F cos 2θ − 4r ρπβ θ ˙ 2 + r 2 cos 2θ) , −xr ˙ 2 sin 2θ + y(c J13 = 4 4 4 4 ρπ(r − c ) r −c 2 4

2 c flap 2 ˙ , J24 = 2c F sin 2θ − 4r ρπα θ − r 2 cos 2θ) − yr ˙ 2 sin 2θ . x(c ˙ J23 = 4 4 4 4 ρπ(r − c ) r −c ACKNOWLEDGMENTS The authors would like to thank Dr. Andrew A. Tchieu and Professor Paul K. Newton for the enlightening discussions. The work of EK is partially supported by the National Science Foundation through the CAREER award CMMI 06-44925 and the grant CCF 08-11480. REFERENCES 1. Lighthill, M. J., Aquatic Animal Propulsion of High Hydromechanical Efficiency, J. Fluid Mech., 1970, vol. 44, no. 2, pp. 265–301. 2. Wu, T. Y., Fish Swimming and Bird/Insect Flight, Annu. Rev. Fluid Mech., 2011, vol. 43, no. 1, pp. 25– 58. 3. Eloy, C., On the Best Design for Undulatory Swimming, J. Fluid Mech., 2013, vol. 717, pp. 48–89. 4. Weihs, D., Stability Versus Maneuverability in Aquatic Locomotion, Integ. and Comp. Biol., 2002, vol. 42, no. 1, pp. 127–134. 5. Weihs, D., Stability of Aquatic Animal Locomotion, Cont. Math., 1993, vol. 141, pp. 443–461. 6. Jordan, D. W. and Smith, P., Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, 4th ed., New York: Oxford Univ. Press, 2007. 7. Kanso, E., Marsden, J. E., Rowley, C. W., and Melli-Huber, J. B., Locomotion of Articulated Bodies in a Perfect Fluid, J. Nonlinear Sci., 2005, vol. 15, pp. 255–289. 8. Jing, F., Part 1. Viscous Evolution of Point Vortex Equilibria, Part 2. Effects of Body Elasticity on Stability of Fish Motion, PhD Thesis, University of Southern California, Los Angeles, 2011. 9. Jing, F. and Kanso, E., Effects of Body Elasticity on Stability of Underwater Locomotion, J. Fluid Mech., 2012, vol. 690, pp. 461–473. 10. Spagnolie, S. E., Moret, L., Shelley, M. J., and Zhang, J., Surprising Behaviors in Flapping Locomotion with Passive Pitching, Phys. Fluids, 2010, vol. 22, 041903, 20 pp. 11. Newman, J. N., Marine Hydrodynamics, Cambridge, MA: MIT Press, 1977. 12. Kern, S. and Koumoutsakos, P., Simulations of Optimized Anguilliform Swimming, J. Exp. Biol., 2006, vol. 209, pp. 4841–4857. 13. Jayne, B. C. and Lauder, G. V., Red Muscle Motor Patterns during Steady Swimming in Largemouth Bass: Effects of Speed and Correlations with Axial Kinematics, J. Exp. Biol., 1995, vol. 198, pp. 1575– 1587. REGULAR AND CHAOTIC DYNAMICS

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14. Donley, J. M. and Dickson, K. A., Swimming Kinematics of Juvenile Kawakawa Tuna (Euthynnus affinis) and Chub Mackerel (Scomber japonicus), J. Exp. Biol., 2000, vol. 203, pp. 3103–3116. 15. Videler, J. J. and Hess, F., Fast Continuous Swimming of Two Pelagic Predators, Saithe (Pollachius virens) and Mackerel (Scomber scombrus): A Kinematic Analysis, J. Exp. Biol., 1984, vol. 109, pp. 209– 228. 16. Triantafyllou, M. S., Hover, F. S., Techet, A. H., and Yue, D. K. P., Review of Hydrodynamic Scaling Laws in Aquatic Locomotion and Fishlike Swimming, Appl. Mech. Rev., 2005, vol. 58, no. 4, pp. 226–237. 17. Jing, F. and Alben, S., Optimization of Two- and Three-Link Snakelike Locomotion, Phys. Rev. E, 2013, vol. 87, 022711, 15 pp. 18. Lamb, H., Hydrodynamics, 6th ed., Cambridge: Cambridge Univ. Press, 1932. 19. Sedov, L. I., Two-Dimensional Problems in Hydrodynamics and Aerodynamics, C. K. Chu, H. Cohen, B. Seckler, J. Gillis (Eds.), New York: Intersci. Publ., 1965. 20. Milne-Thomson, L. M., Theoretical Hydrodynamics, New York: Dover Publications, 1968.


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