- Email: [email protected]
Stability of the 4-vortex system
ht. 1. Engng Sci. Vol. 30, No. 9, pp. 1233-1236, Printed in Great Britain. All rightsreserved
LETTERS
1992
IN APPLIED
C020-7225/92 $5.00+ 0.00 Copyright@ 1992PergamonPress Ltd
AND ENGINEERING
STABILITY OF THE 4-VORTEX I. A. KUNIN,
F. HUSSAIN
SCIENCES
SYSTEM
and X. ZHOU
Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, U.S.A Abstract-The stability of a symmetric system of 4-point vortices has been studied in a special rotating frame. New results reveal domains in the parameter space of periodic and chaotic motion, some in contradiction with prior conjectures.
The dynamics of point vortices has attracted the interest of researchers in hydrodynamics, physics, and mathematics for a long time (see review papers [l-3]). This Hamiltonian system reveals a variety of integrable and chaotic motions. In particular, it was shown that the motion of four or more vortices is in general chaotic but the motion of four vortices with central symmetry is integrable. This letter studies linear and nonlinear stabilities of these integrable motions for all energies. We consider a system of four identical point vortices with unit circulations and coordinates .zA=xA+iyA,)c=l,..., 4. The Hamiltonian of the system is [3]
(1) where the prime indicates that A # p. The equations of motion are (d, = a/ az,) if
= -2 d,X(z,
f).
(2)
Without loss of generality we fix values of two integrals of motion
Using the first integral we introduce new variables w = Uz where U is a matrix such that w.,= 0 and the second integral is invariant (unitary transformation). We have
(4 and the corresponding
equations of motion are iGa = -d,H(w)
= v,(w)
where the “complex Hamiltonian” H(w)
= -
& ln[( w: -
w;>
has the Hamiltonian X(w, ti) as its real part. Following [7] we consider motion in a special rotating frame (SRF) with the angular velocity Q(w, I?) =
$J*(w)viA(ni). 1233
(7)
I. A. KUNIN
1234
et al.
The equations of motion in the SRF are i& = V,(W) - Q(w, l?)&
(8)
One of the advantages of the SRF is that now not only X’= Re H = constant, Im H = constant. Thus we have the integrals of (8) H(w) = constant,
C wAtiA= 1. A
but also
(9)
Let us consider z-configurations with the central symmetry (parallelograms), for example It is easy to check that the corresponding w-configurations satisfy 2, = -23, 22 = -24. IwIw~w~I = 0. It follows from (8) that if at t = 0, ~~(0) = 0, then w3(t) = 0. Thus motions of z-configurations with central symmetry correspond to orbits in the (wr, w,)-plane. The equations of motion in the SRF for these orbits are i&Z+ = V,(W) - szi+r,
ii+*= tJz(w) - G!Cz,
(10)
where _I
1
w1
-++
1 Jr
(4
w:-w:
v2=-
>’
1
1
--+A
Jr ( w2
w:-w:
> ’
(11)
It was indicated in [7] that these motions are quasi-periodic in the fixed frame and periodic in the SRF (one more advantage of the latter). Our goal is to investigate stabilities of these orbits with respect to small and finite perturbations in w3. Let us consider first the case, E = Iw,l <<1 (linear stability). It follows from (8) that the equations for wl, w2 coincide with (10) up to terms O(E*) and iG3=
-i($+--$) w3-S2ti3.
(12)
This equation may be rewritten in the form
(;:)=A(4 (;j
9
(13)
where A(t) is a periodic matrix defined by the (wr, w,)-orbit in the SRF. Stability of motion given by (13) can be investigated using standard methods of Floquet multipliers [8] or Lyapunov exponents [9]. Both methods give the same results but in this case, the first method proved to be computationally more efficient. To apply the Floquet method one has to integrate the system (13) for two initial data: (0,l) and (l,O). The corresponding solutions taken at t = T, where T is the period, define the fundamental matrix M of the system (13). Let pr, p2 be eigenvalues of M (Floquet multipliers) and the characteristic exponents o = +max(ln It is known that o stable (unstable) if It is convenient 0 c Ad 1 uniquely
lhl, In 1~~1).
(14)
is a non-negative number and the unperturbed motion (wr, w2) is linearly u = O(a > 0). to characterize the unperturbed motions by a normalized parameter related to the energy [6]. In w-variables A = 4 Iw,121w,121w: - w:1 .
(13
Notice some characteristic values of A. The square z-configuration rotates with a constant angular velocity and corresponds to A = 1. In w-variables, we have a fixed point (w, = l/fi,
Letters in Applied and Engineering Sciences
1235
5.0 4.5 4.0 3.5 3.0 c c
2.5
I2.0 1.5
1.0
0.5 0
0.5
1.0
1.5
2.0
2.5
h/A,
Fig. 1. The period T of w-orbits in the SFW as a function of A.
w,= i/e) with respect to the SRF. The value A, = 2/(3ti) corresponds to a separatrix. Finally, the case A + 0 corresponds to z-configurations having two points at small distances. The period T of w-orbits in the SRF is shown as function of A in Fig. 1. We see that T + Q) when A-+ AC as it should be for the separatrix. There are two different families (1 and 2) of orbits for 0 < A < A,. Figure 2 shows CTas functions of A. The unperturbed motion is linearly stable in three regions: A-* 0, 1.08 A, C A < 1.15 AC and A > 1.74 AC. It would be interesting to interpret the small window of stability (second region). Maximum instability in the vicinity of A = AC is characteristic for a separatrix. Let us compare these results on linear stability with direct integrations of the system (8). Power spectra, Poincare sections, and trajectory plotting permit one to distinguish between quasi-periodic and chaotic motions. The region A > 1.74 AC is stable both linearly and nonlinearly. Perturbations lead to quasi-periodic motions. The window 1.08 AC < A < 1.15 A, is linearly stable but unstable for finite initial perturbations E = lw,l. Trajectories are quasiperiodic for E = 0.01, but become chaotic for E = 0.05. The corresponding z-trajectories in the SRF are shown in Fig. 3 (a, b). 2.00
1.75
1.50
1.25
b
1.00
0.75
0.50
0.25
L 0
0.5
1.5
1.0
2.0
h/h,
Fig. 2. Characteristic exponent u as a function of A.
2.5
I. A. KUNIN et al.
1236
I (a)
II(b)
I
II Fig. 3. The perturbed z-orbits in the SRF for A = 1.084 AC. (a) E = 0.01, (b) E = 0.05.
To summarize, quasi-periodic and chaotic perturbed motions may exist in all regions of A though their measures vary significantly. The region A > 1.74 AC is characterized by linear and nonlinear stabilities of unperturbed motions as well as high measure of quasi-periodic motions. For A < 1.74 &, unperturbed motions are nonlinearly unstable though there exists a small window of linear stability. Perturbed motions are typically chaotic. These conclusions differ somewhat from a conjecture in [lo] that motions are chaotic for AC
REFERENCES [l] E. A. NOVIKOV, Ann. NY Acad. Sci. 357, 47 (1980). [2] A. LEONARD, 1. Comput. Phys. 37, 289 (1980). [3] H. AREF, Theorericol and Applied Mechanics (Edited by F. I. NIORDSON and N. OLHOFF), pp. 43-68. Elsevier, Amsterdam (1985). [4] E. A. NOVIKOV, Zh. Eksp. Teor. Fiz. 68, 1868 (1975). [5] S. L. ZIGLIN, Soviet Moth. Dokf. 21, 2% (1980). [6] H. AREF and N. POMPHREY, Proc. R. Sot. Lond. A380,359 (1982). [7] I. A. KUNIN, F. HUSSAIN, X. ZHOU and D. KOVICH, Int. J. Engng Sci. 28, 965 (1990). [8] H. LEIPHOLZ, Stability Theory, 2nd edn, Chap. 1.3. Teubner, Stuttgart; Wiley, New York (1987). [9] P. BERGE, Y. POMEAU and C. VIDAL, Order Within Chaos, Appendix B. Hermann, Paris (1984). [lo] E. A. NOVIKOV and YU. B. SEDOV, Zh. Eksp. Teor. Hz. 77, 588 (1979). (Received 7 January 1992; accep!ed 20 January 1992)
LETTERS
1992
IN APPLIED
C020-7225/92 $5.00+ 0.00 Copyright@ 1992PergamonPress Ltd
AND ENGINEERING
STABILITY OF THE 4-VORTEX I. A. KUNIN,
F. HUSSAIN
SCIENCES
SYSTEM
and X. ZHOU
Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, U.S.A Abstract-The stability of a symmetric system of 4-point vortices has been studied in a special rotating frame. New results reveal domains in the parameter space of periodic and chaotic motion, some in contradiction with prior conjectures.
The dynamics of point vortices has attracted the interest of researchers in hydrodynamics, physics, and mathematics for a long time (see review papers [l-3]). This Hamiltonian system reveals a variety of integrable and chaotic motions. In particular, it was shown that the motion of four or more vortices is in general chaotic but the motion of four vortices with central symmetry is integrable. This letter studies linear and nonlinear stabilities of these integrable motions for all energies. We consider a system of four identical point vortices with unit circulations and coordinates .zA=xA+iyA,)c=l,..., 4. The Hamiltonian of the system is [3]
(1) where the prime indicates that A # p. The equations of motion are (d, = a/ az,) if
= -2 d,X(z,
f).
(2)
Without loss of generality we fix values of two integrals of motion
Using the first integral we introduce new variables w = Uz where U is a matrix such that w.,= 0 and the second integral is invariant (unitary transformation). We have
(4 and the corresponding
equations of motion are iGa = -d,H(w)
= v,(w)
where the “complex Hamiltonian” H(w)
= -
& ln[( w: -
w;>
has the Hamiltonian X(w, ti) as its real part. Following [7] we consider motion in a special rotating frame (SRF) with the angular velocity Q(w, I?) =
$J*(w)viA(ni). 1233
(7)
I. A. KUNIN
1234
et al.
The equations of motion in the SRF are i& = V,(W) - Q(w, l?)&
(8)
One of the advantages of the SRF is that now not only X’= Re H = constant, Im H = constant. Thus we have the integrals of (8) H(w) = constant,
C wAtiA= 1. A
but also
(9)
Let us consider z-configurations with the central symmetry (parallelograms), for example It is easy to check that the corresponding w-configurations satisfy 2, = -23, 22 = -24. IwIw~w~I = 0. It follows from (8) that if at t = 0, ~~(0) = 0, then w3(t) = 0. Thus motions of z-configurations with central symmetry correspond to orbits in the (wr, w,)-plane. The equations of motion in the SRF for these orbits are i&Z+ = V,(W) - szi+r,
ii+*= tJz(w) - G!Cz,
(10)
where _I
1
w1
-++
1 Jr
(4
w:-w:
v2=-
>’
1
1
--+A
Jr ( w2
w:-w:
> ’
(11)
It was indicated in [7] that these motions are quasi-periodic in the fixed frame and periodic in the SRF (one more advantage of the latter). Our goal is to investigate stabilities of these orbits with respect to small and finite perturbations in w3. Let us consider first the case, E = Iw,l <<1 (linear stability). It follows from (8) that the equations for wl, w2 coincide with (10) up to terms O(E*) and iG3=
-i($+--$) w3-S2ti3.
(12)
This equation may be rewritten in the form
(;:)=A(4 (;j
9
(13)
where A(t) is a periodic matrix defined by the (wr, w,)-orbit in the SRF. Stability of motion given by (13) can be investigated using standard methods of Floquet multipliers [8] or Lyapunov exponents [9]. Both methods give the same results but in this case, the first method proved to be computationally more efficient. To apply the Floquet method one has to integrate the system (13) for two initial data: (0,l) and (l,O). The corresponding solutions taken at t = T, where T is the period, define the fundamental matrix M of the system (13). Let pr, p2 be eigenvalues of M (Floquet multipliers) and the characteristic exponents o = +max(ln It is known that o stable (unstable) if It is convenient 0 c Ad 1 uniquely
lhl, In 1~~1).
(14)
is a non-negative number and the unperturbed motion (wr, w2) is linearly u = O(a > 0). to characterize the unperturbed motions by a normalized parameter related to the energy [6]. In w-variables A = 4 Iw,121w,121w: - w:1 .
(13
Notice some characteristic values of A. The square z-configuration rotates with a constant angular velocity and corresponds to A = 1. In w-variables, we have a fixed point (w, = l/fi,
Letters in Applied and Engineering Sciences
1235
5.0 4.5 4.0 3.5 3.0 c c
2.5
I2.0 1.5
1.0
0.5 0
0.5
1.0
1.5
2.0
2.5
h/A,
Fig. 1. The period T of w-orbits in the SFW as a function of A.
w,= i/e) with respect to the SRF. The value A, = 2/(3ti) corresponds to a separatrix. Finally, the case A + 0 corresponds to z-configurations having two points at small distances. The period T of w-orbits in the SRF is shown as function of A in Fig. 1. We see that T + Q) when A-+ AC as it should be for the separatrix. There are two different families (1 and 2) of orbits for 0 < A < A,. Figure 2 shows CTas functions of A. The unperturbed motion is linearly stable in three regions: A-* 0, 1.08 A, C A < 1.15 AC and A > 1.74 AC. It would be interesting to interpret the small window of stability (second region). Maximum instability in the vicinity of A = AC is characteristic for a separatrix. Let us compare these results on linear stability with direct integrations of the system (8). Power spectra, Poincare sections, and trajectory plotting permit one to distinguish between quasi-periodic and chaotic motions. The region A > 1.74 AC is stable both linearly and nonlinearly. Perturbations lead to quasi-periodic motions. The window 1.08 AC < A < 1.15 A, is linearly stable but unstable for finite initial perturbations E = lw,l. Trajectories are quasiperiodic for E = 0.01, but become chaotic for E = 0.05. The corresponding z-trajectories in the SRF are shown in Fig. 3 (a, b). 2.00
1.75
1.50
1.25
b
1.00
0.75
0.50
0.25
L 0
0.5
1.5
1.0
2.0
h/h,
Fig. 2. Characteristic exponent u as a function of A.
2.5
I. A. KUNIN et al.
1236
I (a)
II(b)
I
II Fig. 3. The perturbed z-orbits in the SRF for A = 1.084 AC. (a) E = 0.01, (b) E = 0.05.
To summarize, quasi-periodic and chaotic perturbed motions may exist in all regions of A though their measures vary significantly. The region A > 1.74 AC is characterized by linear and nonlinear stabilities of unperturbed motions as well as high measure of quasi-periodic motions. For A < 1.74 &, unperturbed motions are nonlinearly unstable though there exists a small window of linear stability. Perturbed motions are typically chaotic. These conclusions differ somewhat from a conjecture in [lo] that motions are chaotic for AC
REFERENCES [l] E. A. NOVIKOV, Ann. NY Acad. Sci. 357, 47 (1980). [2] A. LEONARD, 1. Comput. Phys. 37, 289 (1980). [3] H. AREF, Theorericol and Applied Mechanics (Edited by F. I. NIORDSON and N. OLHOFF), pp. 43-68. Elsevier, Amsterdam (1985). [4] E. A. NOVIKOV, Zh. Eksp. Teor. Fiz. 68, 1868 (1975). [5] S. L. ZIGLIN, Soviet Moth. Dokf. 21, 2% (1980). [6] H. AREF and N. POMPHREY, Proc. R. Sot. Lond. A380,359 (1982). [7] I. A. KUNIN, F. HUSSAIN, X. ZHOU and D. KOVICH, Int. J. Engng Sci. 28, 965 (1990). [8] H. LEIPHOLZ, Stability Theory, 2nd edn, Chap. 1.3. Teubner, Stuttgart; Wiley, New York (1987). [9] P. BERGE, Y. POMEAU and C. VIDAL, Order Within Chaos, Appendix B. Hermann, Paris (1984). [lo] E. A. NOVIKOV and YU. B. SEDOV, Zh. Eksp. Teor. Hz. 77, 588 (1979). (Received 7 January 1992; accep!ed 20 January 1992)