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Transitions in Weakly-Coupled Nonlinear Oscillators
COUPLED NONLINEAR OSCILLATORS
1.Chandra and A.C. Scott (eds.)
0North-Holland h blishing Company, 1983
1
TRANSITIONS IN WEAKLY-COUPLED NONLINEAR OSCILLATORS Paul H. Steen(a) and Stephen H. Davis Department of Engineering Sciences and Applied Mathematics Northwestern University Evanston, Illinois 60201
Two oscillators having natural frequencies 6 and y and damping coefficients proportional to - A are weakly coupled through cubic nonlinearities. As A is increased through A E 0, nonlinear solutions of amplitude t bifurcate. To leasing order in E such systems (including the case of coupled van der Pol oscillators) have behaviors governed by the single para2(6-y)/(A-h ) . p has a critical value p determeter p mined by a Josephson-junction equation. For IpI >'p, the oscillations are quasiperiodic; the modulations die out in the far-from-resonance case IpI -i m. For l p ( < p there are phase-locked oscillations and competing stablg states can occur. An increase in A, causing lpl to cross p , results in a type of reverse bifurcation (quasiperioiic to periodic). No chaotic behavior is found.
-
Complex physical systems often display orderly sequences of transitions among states. Simplified models of rudimentary systems can often shed light on the physics of such transitions. Hence, there is great interest [ l ] in the predictions of highly idealized difference equations. Experimental observations in fluid dynamics [ 2 ] , chemical [ 3 ] and laser [ 4 ] systems commonly reveal transitions to quasiperiodic states. The lack of direct theoretical results on such behavior leads us to pose a model system consisting of a pair of nonlinearly coupled oscillators. 2 d x dt2
dx 2 -A+ 6 x = f dt
(x,
dx
,
y
, 2) ,
where 6 and y are the linear-oscillator frequencies at zero damping, and the nonlinear functions f and g vanish for x = y = 0 . The damping coefficient - A is the bifurcation parameter in that the linearized oscillators have decaying solutions for A < 0 and growing solution for A > 0 . At the critical value A = 0 C non-trivial solutions may bifurcate from the null solution. We can generalize system (1) slightly by considering the following system of four first-order equations:
2
P.H. S E E N mid S.H. DAVIS
where the four-vector $ depends on x, u , y and v. All of the above discussed properties remain. A special case of system ( 2 ) is a pair of coupled van der Pol oscillators. In order to bypass certain mathematical technicalities we confine atten n to nonlinearities that are cubic and satisfy a strong-definiteness condition,"' viz. there exists a positive constant K such t6at (x,u,y,v)*C <
-
2 2
K(x2 + u2 + y2 + v )
.
(3)
Condition (3) ensures that all solutions of system (2) will remain in a bounded neighborhood of the origin. We also assume that both 6 and y are bounded away from zero so that the only strong resonance for weakly nonlinear interactions occurs near 6 = y. We seek asymptotic solutions in terms of an amplitude parameter t as follows:
- &iA(T2)e
x(t;&)
y(t;t) -, tiB(T2)e
A(&)
2
+ complex conjugate
2
+ complex conjugate
i6T0
+
O(E )
i6T0
+
O(t )
- Ac + t 2A2 + O ( E 3)
(4c)
by using a two time-scale expansion where
T0 =- t and T2
I
t'E
.
(4d)
In order to describe the strong resonance near 6 = y, we relate [5] the two frequencies as follows: 1
6 - y - ' - k e 2 , k=0(1) 2
& + O .
as
(5)
The forms (4) and (54 are substituted into system (2) and multiple-scale analysis is used. At order t a solvability condition determines equations for the envelIf we use the polar forms A(T ) = a(T2) exp[i$(T2)] and opes A(T ) and B(T2). 2 . B(T2) = g ( T 2 ) exp[iJI(T ) , the resulting envelope equations are as follows: 2
- a2 = a2(A2cl + c a2 + (c3 + c4 cos Z)b 2 ) 2 dT2 d b2 -
= b2(A d
2 1
dT2
d -Z = dT2
-
+
d2b 2 t (d
3
2 (d4a2 + c4b )sin Z
-
+ d4 cos Z)a 2
k
TRANSITIONS IN WEAKLY-COUPLEDNONLINEAR OSCILLATORS
d -$ = dT2
b2c4 sin Z
.
3 (7)
Here we have written the phase-difference as follows: Z
2(JI
-
$)
-
kT2
.
(8)
The c.,d. i = 1 , . . .4 are computable numbers dependent on the nonlinearity C that are ristifcted by condition ( 3 ) . Condition ( 3 ) also implies that A2 > 0. Hence, if A < 0 all solutions decay to zero and nontrivial solutions exist only for A > 0. The dominant behavior of the oscillator amplitudes a and b is coupled to the phase difference through Z. Thus, the governing system is three dimensional. It The individual phases $ and CL can then be obtained by using Eqs. ( 7 ) and ( 8 ) . is easy to see that simple rescaling combines the two parameters k and A2 into a single parameter p p
k/A2
- 2(6-y)/(A-Ac)
.
(9)
p is the entrainment parameter, which is to leading order in
E the ratio of the frequency difference 6 - y to the degree of supercriticality. The behavior of frequencies 6 and y since the oscillators (2) is described by system ( 6 ) for we find that the formal limit I p J + m gives the correct behaviors for 6 and y widely separated i.e. for k m in Eq. ( 5 ) . -f
There are three types of solutions of system ( 6 ) that are found. There are pure mode (a 5 0 or b S O ) constants, mixed-mode (a,b f 0) constants and mixed-mode (a,b f 0) limit cycles. These correspond through Eqs. (4) to the oscillators having pure-mode periodic solutions, mixed-mode periodic solutions and mixedmode quasiperiodic behavior, respectively. No chaotic solutions of system ( 6 ) are found. There is a critical value p of I p I . When I p I < p there are typically four or eight mixed-mode constant sglutions at most half of which are stable. These are phase-locked periodic solutions for the oscillators. There can be competing stable attractors. There coexist pure-mode constant solutions whose stability depends on details of 2 . When l p l > p , there are "0 mixed-mode constant solutions but mixed-mode limit cycles exisf and are stable. There coexist pure-mode constant solutions whose stability depends on details of C. The results are depicted in Fig. 1. Note that the present bifurcation prevalent A is increased, there phase-locked periodic
coupled-oscillator system displays a type of reverse in many physical systems [4]. Here, if 6 - y is fixed and is a transition at p from quasiperiodic oscillations to oscillations.
There is an abrupt change in behavior as l p l crosses p . The change is controlled by the phase-difference equation (6c) whose quslitative nature is captured by the scalar equation
Here p = 1. For I p I > pc there is a large amplitude periodic solution replacing the pa?r of steady states that typically exist for J p ( < pc. Equation (10) governs the phase-difference of the wave functions across a Josephson junction [ 6 ] when the conductance is large; p plays the role of the current. The detailed asymptotic analysis, the proof of existence of periodic amplitudes for I p I > p and the discussion of the numerical experiments will be covered in a forthcomingcpaper.
4
P.H. STEEN and S.H. DAVIS
8 -Y
\
Figure 1 The (6-y) vs. (A-A ) plane for small E . The slopes of the lines are 2 1/2 p Outside the rgys in the horizontally-hatched region only periodic m:xed-mode amplitudes exist. Inside the rays in the verticallyhatched region steady mixed-mode amplitudes exist. To the left in the unhatched region all nontrivial solutions decay to zero.
.
A knowledge of the behaviors of nonlinearly-coupled oscillators gives one an elemental understanding of the behaviors of more complicated discrete systems as well as some continuous systems. These oscillators display transitions to quasiperiodic states, reverse bifurcations to phase-locked periodic states and the possibility of competing stable phase-locked states.
We wish to thank Professor S. Rosenblat for his constructive advice. This work was supported by grants from the Army Research Office, Applied Mathematics Program.
TRANSITIONS IN WEAKLY-COUPLED NONLINEAR OSCILLATORS
(a)
Current address:
Department of Chemical Engineering Stanford University, Stanford, CA 94305
(b)
can be defined from a trilinear form each of whose three arguments is (x,u,y,v). The trilinear form is constructed by replacing each of the 40 nonzero entries in the most general fourth-order isotropic tensor by a negative constant.
REFERENCES 1.
Physics Today, News-Search and Discovery, 34, 17 (March 1 9 8 1 ) .
2.
H . L. Swinney and J. P. Gollub, Physics Today
3.
M . Marek and I. Stuchl, Biophys. Chem.
4.
Bowden, C. M., Ciftan, M. and Robl, H. R., Optical Bistability, Plenum Publ., New York ( 1 9 8 1 ) .
5.
H. Kabakow, Ph.D. Dissertation, Cal. Tech., 1968; Int. J. Nonl. Mech. 1,
6.
C. Kittel, Introduction to Solid State Physics, John Wiley (5th edition) (1976).
2,
31,
41 (August 1 9 7 8 ) .
241 ( 1 9 7 5 ) .
125 ( 1 9 7 2 ) .
5
1.Chandra and A.C. Scott (eds.)
0North-Holland h blishing Company, 1983
1
TRANSITIONS IN WEAKLY-COUPLED NONLINEAR OSCILLATORS Paul H. Steen(a) and Stephen H. Davis Department of Engineering Sciences and Applied Mathematics Northwestern University Evanston, Illinois 60201
Two oscillators having natural frequencies 6 and y and damping coefficients proportional to - A are weakly coupled through cubic nonlinearities. As A is increased through A E 0, nonlinear solutions of amplitude t bifurcate. To leasing order in E such systems (including the case of coupled van der Pol oscillators) have behaviors governed by the single para2(6-y)/(A-h ) . p has a critical value p determeter p mined by a Josephson-junction equation. For IpI >'p, the oscillations are quasiperiodic; the modulations die out in the far-from-resonance case IpI -i m. For l p ( < p there are phase-locked oscillations and competing stablg states can occur. An increase in A, causing lpl to cross p , results in a type of reverse bifurcation (quasiperioiic to periodic). No chaotic behavior is found.
-
Complex physical systems often display orderly sequences of transitions among states. Simplified models of rudimentary systems can often shed light on the physics of such transitions. Hence, there is great interest [ l ] in the predictions of highly idealized difference equations. Experimental observations in fluid dynamics [ 2 ] , chemical [ 3 ] and laser [ 4 ] systems commonly reveal transitions to quasiperiodic states. The lack of direct theoretical results on such behavior leads us to pose a model system consisting of a pair of nonlinearly coupled oscillators. 2 d x dt2
dx 2 -A+ 6 x = f dt
(x,
dx
,
y
, 2) ,
where 6 and y are the linear-oscillator frequencies at zero damping, and the nonlinear functions f and g vanish for x = y = 0 . The damping coefficient - A is the bifurcation parameter in that the linearized oscillators have decaying solutions for A < 0 and growing solution for A > 0 . At the critical value A = 0 C non-trivial solutions may bifurcate from the null solution. We can generalize system (1) slightly by considering the following system of four first-order equations:
2
P.H. S E E N mid S.H. DAVIS
where the four-vector $ depends on x, u , y and v. All of the above discussed properties remain. A special case of system ( 2 ) is a pair of coupled van der Pol oscillators. In order to bypass certain mathematical technicalities we confine atten n to nonlinearities that are cubic and satisfy a strong-definiteness condition,"' viz. there exists a positive constant K such t6at (x,u,y,v)*C <
-
2 2
K(x2 + u2 + y2 + v )
.
(3)
Condition (3) ensures that all solutions of system (2) will remain in a bounded neighborhood of the origin. We also assume that both 6 and y are bounded away from zero so that the only strong resonance for weakly nonlinear interactions occurs near 6 = y. We seek asymptotic solutions in terms of an amplitude parameter t as follows:
- &iA(T2)e
x(t;&)
y(t;t) -, tiB(T2)e
A(&)
2
+ complex conjugate
2
+ complex conjugate
i6T0
+
O(E )
i6T0
+
O(t )
- Ac + t 2A2 + O ( E 3)
(4c)
by using a two time-scale expansion where
T0 =- t and T2
I
t'E
.
(4d)
In order to describe the strong resonance near 6 = y, we relate [5] the two frequencies as follows: 1
6 - y - ' - k e 2 , k=0(1) 2
& + O .
as
(5)
The forms (4) and (54 are substituted into system (2) and multiple-scale analysis is used. At order t a solvability condition determines equations for the envelIf we use the polar forms A(T ) = a(T2) exp[i$(T2)] and opes A(T ) and B(T2). 2 . B(T2) = g ( T 2 ) exp[iJI(T ) , the resulting envelope equations are as follows: 2
- a2 = a2(A2cl + c a2 + (c3 + c4 cos Z)b 2 ) 2 dT2 d b2 -
= b2(A d
2 1
dT2
d -Z = dT2
-
+
d2b 2 t (d
3
2 (d4a2 + c4b )sin Z
-
+ d4 cos Z)a 2
k
TRANSITIONS IN WEAKLY-COUPLEDNONLINEAR OSCILLATORS
d -$ = dT2
b2c4 sin Z
.
3 (7)
Here we have written the phase-difference as follows: Z
2(JI
-
$)
-
kT2
.
(8)
The c.,d. i = 1 , . . .4 are computable numbers dependent on the nonlinearity C that are ristifcted by condition ( 3 ) . Condition ( 3 ) also implies that A2 > 0. Hence, if A < 0 all solutions decay to zero and nontrivial solutions exist only for A > 0. The dominant behavior of the oscillator amplitudes a and b is coupled to the phase difference through Z. Thus, the governing system is three dimensional. It The individual phases $ and CL can then be obtained by using Eqs. ( 7 ) and ( 8 ) . is easy to see that simple rescaling combines the two parameters k and A2 into a single parameter p p
k/A2
- 2(6-y)/(A-Ac)
.
(9)
p is the entrainment parameter, which is to leading order in
E the ratio of the frequency difference 6 - y to the degree of supercriticality. The behavior of frequencies 6 and y since the oscillators (2) is described by system ( 6 ) for we find that the formal limit I p J + m gives the correct behaviors for 6 and y widely separated i.e. for k m in Eq. ( 5 ) . -f
There are three types of solutions of system ( 6 ) that are found. There are pure mode (a 5 0 or b S O ) constants, mixed-mode (a,b f 0) constants and mixed-mode (a,b f 0) limit cycles. These correspond through Eqs. (4) to the oscillators having pure-mode periodic solutions, mixed-mode periodic solutions and mixedmode quasiperiodic behavior, respectively. No chaotic solutions of system ( 6 ) are found. There is a critical value p of I p I . When I p I < p there are typically four or eight mixed-mode constant sglutions at most half of which are stable. These are phase-locked periodic solutions for the oscillators. There can be competing stable attractors. There coexist pure-mode constant solutions whose stability depends on details of 2 . When l p l > p , there are "0 mixed-mode constant solutions but mixed-mode limit cycles exisf and are stable. There coexist pure-mode constant solutions whose stability depends on details of C. The results are depicted in Fig. 1. Note that the present bifurcation prevalent A is increased, there phase-locked periodic
coupled-oscillator system displays a type of reverse in many physical systems [4]. Here, if 6 - y is fixed and is a transition at p from quasiperiodic oscillations to oscillations.
There is an abrupt change in behavior as l p l crosses p . The change is controlled by the phase-difference equation (6c) whose quslitative nature is captured by the scalar equation
Here p = 1. For I p I > pc there is a large amplitude periodic solution replacing the pa?r of steady states that typically exist for J p ( < pc. Equation (10) governs the phase-difference of the wave functions across a Josephson junction [ 6 ] when the conductance is large; p plays the role of the current. The detailed asymptotic analysis, the proof of existence of periodic amplitudes for I p I > p and the discussion of the numerical experiments will be covered in a forthcomingcpaper.
4
P.H. STEEN and S.H. DAVIS
8 -Y
\
Figure 1 The (6-y) vs. (A-A ) plane for small E . The slopes of the lines are 2 1/2 p Outside the rgys in the horizontally-hatched region only periodic m:xed-mode amplitudes exist. Inside the rays in the verticallyhatched region steady mixed-mode amplitudes exist. To the left in the unhatched region all nontrivial solutions decay to zero.
.
A knowledge of the behaviors of nonlinearly-coupled oscillators gives one an elemental understanding of the behaviors of more complicated discrete systems as well as some continuous systems. These oscillators display transitions to quasiperiodic states, reverse bifurcations to phase-locked periodic states and the possibility of competing stable phase-locked states.
We wish to thank Professor S. Rosenblat for his constructive advice. This work was supported by grants from the Army Research Office, Applied Mathematics Program.
TRANSITIONS IN WEAKLY-COUPLED NONLINEAR OSCILLATORS
(a)
Current address:
Department of Chemical Engineering Stanford University, Stanford, CA 94305
(b)
can be defined from a trilinear form each of whose three arguments is (x,u,y,v). The trilinear form is constructed by replacing each of the 40 nonzero entries in the most general fourth-order isotropic tensor by a negative constant.
REFERENCES 1.
Physics Today, News-Search and Discovery, 34, 17 (March 1 9 8 1 ) .
2.
H . L. Swinney and J. P. Gollub, Physics Today
3.
M . Marek and I. Stuchl, Biophys. Chem.
4.
Bowden, C. M., Ciftan, M. and Robl, H. R., Optical Bistability, Plenum Publ., New York ( 1 9 8 1 ) .
5.
H. Kabakow, Ph.D. Dissertation, Cal. Tech., 1968; Int. J. Nonl. Mech. 1,
6.
C. Kittel, Introduction to Solid State Physics, John Wiley (5th edition) (1976).
2,
31,
41 (August 1 9 7 8 ) .
241 ( 1 9 7 5 ) .
125 ( 1 9 7 2 ) .
5