THE THEORY OF MOTION OF A DOUBLE MATHEMATICAL PENDULUM

A. A. Martynyuk and N. V. Nikitina UDC 531.36

It is proved that conditionally periodical and chaotic trajectories of a double mathematical pendulumexist when the ratio of the pendulum masses is not small.

The identification of chaotic trajectories is sometimes associated with the problem of integrability [4] and applicationof classical Mel’nikov’s method for small parameters [3]. It is difficult to expand Mel’nikov’s method to other constraints.This method does not include identification of chaos but allows us to find the distance between stable and unstable solutionmanifolds of the system. A trajectory belonging to the unstable manifold may behave differently [1, 2], including chaotically.

There exists an approach based on the ideas of the theories of vibrations and stability [1, 2]. Irregular motions occurin both conservative and dissipative systems due to the appearance of aperiodic solutions in the trajectory and loss ofsymmetry. The trajectory falls into the domain of aperiodic solutions and then departs from it only due to the interactionbetween the oscillatory subsystems; therefore, the number of degrees of freedom of the system generating irregular motionsis more than 2.

1. Symmetry in Two-Frequency Systems. Let us consider a system of the fourth order describing the motion oftwo nonlinear oscillators in phase variables

d x ⁄ d t = X ( x ), x ∈ R 4. (1)

Here, x1 and x3 are displacements from the equilibrium position of the system of interacting oscillators (1). Let the origin ofcoordinates for the linear system

d x~ ⁄ d t = A x~ (2)

corresponding to system (1) be its center; therefore, the characteristic equation of system (2) has only imaginary roots

λ1 = ± i Ω1, λ2 = ± i Ω2,

where Ω1 and Ω2 are aliquant but close frequencies or frequencies of the same order of smallness. Let us represent system (1)

by two coupled subsystems,

d xj ⁄ d t = Pj ( xj, xj + 1 ) + Sj ( x1, x2, x3, x4 ),

d xj + 1 ⁄ d t = Pj + 1 ( xj, xj + 1 ) + Sj + 1 ( x1, x2, x3, x4 )

( j = 1, 3 ). (3)

Here, Sj ( x1, x2, x3, x4 ) and Sj + 1 ( x1, x2, x3, x4 ) are functions of constraints. When the conditions

Sj ( x1, x2, x3, x4 ) = Sj + 1 ( x1, x2, x3, x4 ) = 0 ( j = 1, 3 ) (4)

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from PrikladnayaMekhanika, Vol. 36, No. 9, pp. 138–143, September, 2000. Original article submitted August 19, 1999.

1252 1063-7095/00/3609-1252$25.00 ©2001 Plenum Publishing Corporation

International Applied Mechanics, Vol. 36, No. 9, 2000

are satisfied, system (3) breaks up into two uncoupled subsystems, for which some criterion or principle can be used to provethat a closed trajectory exists (see [5]). Regular trajectories may exist in system (1). We apply the principle of symmetry toestablish the existence of conditionally periodic trajectory (1). A conditionally periodic trajectory must be symmetric about twoaxes. Proving this property for a trajectory is a generalization of the result from [5] for four-dimensional systems.

Let us construct a nondegenerate linear transformation for linear system (1):

yj = ∑ ( k )

βjk xk , y_

j = ∑ ( k )

β__

jk xk ( j = 1, 2 ),

xk = ∑ ( j )

( αkj yj + α__

kj y_

j ) ( k = 1, 2, 3, 4 ). (5)

Here, αkj, βjk, α__

kj, and β__

jk are the constant conversion coefficients that reduce system (1) to a diagonal form:

d yj ⁄ d t = i Ωj yj + Yj ,

d y_

j ⁄ d t = −i Ωj yj + Y

__j ( j = 1, 2 ), (6)

where

Yj = ∑ ( k )

βjk X~

k ( y, y_ ), Y

__j = ∑

( k )

β__

jk X~

k ( y, y_ ).

We will change variables in (6) as follows:

ρj = yj e − i θj , ρj = y_

j e i θj ( j = 1, 2 ). (7)

The system of equations (1) in the variables ρ and θ is transformed into the form

d ρ ⁄ d t = m ( θ ) X~ ( ρ, θ ),

d θ ⁄ d t ρ = Ω ρ + k ( θ ) X~ ( ρ, θ ), (8)

where d θ ⁄ dt and Ω are diagonal matrices, m ( θ ) and k ( θ ) are (2×4) matrices, and X~

( ρ, θ ) is a four-variable vector function.It is convenient to represent system (8) as

d ρj ⁄ d t = Rj ( ρ, θ ),

d θj ⁄ d t = Ωj + Tj ( ρ, θ ) ( j = 1, 2 ). (9)

In Eqs. (6), the component ρ.k sin θk + ρk θ

.k cos θk of the imaginary quantity disappears due to the interaction of the

subsystems. For the moment of bifurcation, we have d ρk ⁄ dt = 0, ρk = const, d θk

⁄ dt = 0, and θk = const. The splitting of the

periodic solution is associated with the condition [1, 2]

d θk ⁄ dt ≤ 0 ( ρk ≠ 0 ). (10)

Equality (10) arises under certain initial conditions. At the moment of bifurcation, the variable ρk , which is equal,

by definition, to the modulus of a complex variable, and the angular variable θk take the values of real quantities ± ρ∗ k .

Let us apply the principle of symmetry to prove that a conditionally periodic trajectory of system (1) exists in thevicinity of zero [5]. The geometrical principle of symmetry was used in [5] to identify the center in a two-dimensional system.This approach can be expanded to multidimensional systems.

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Conditionally periodic trajectories exist in system (1) if(i) system (2) has only imaginary roots,

(ii) the functions Xj ( j = 1, 3 ) are even with respect to x1 and x3 and Xk (k = 2, 4) are odd with respect to x1 and x3,

i.e.,

Xj ( − x1, x2, − x3, x4 ) = Xj ( x1, x2, x3, x4 ) ( j = 1, 3 ),

Xk ( − x1, x2, − x3, x4 ) = − Xk ( x1, x2, x3, x4 ) ( k = 2, 4 ). (11)

The problem of existence of a conditionally periodic trajectory in multifrequency system (1) is to determineconditions under which the trajectory of an autonomous system is a torus in some neighborhood of zero. Theorem 3 from [6,p.91] states that the closure of the conditionally periodic trajectory of system (1) is a smooth N-dimensional torus. Based ongeometrical symmetry, we will specify sufficient conditions for existence of the center of system (1) in a neighborhood ofzero. Let the coordinates x1 and x3 form a manifold and an almost Euclidean basis correspond to them in a small neighborhoodof the origin of coordinates of system (1). First, let condition (4) be satisfied (constraints are disabled); then, the evennessand oddness conditions, which provide the symmetry of the integral curve, are

X^

j ( − xj , xj + 1 ) = X^

j ( xj , xj + 1 ) ( j = 1, 3 ),

X^

k ( − xj , xj + 1 ) = − X^

k ( xk − 1, xk ) ( k = 2, 4 ). (12)

The projection of the vector field onto the phase plane O xi xi + 1 ( i = 1, 3 ) will be symmetric about the axis O xk

( k = 2, 4 ). Any integral curve emerging from a point Mk( 0, − ξk ), ( k = 2, 4 ), ξk > 0, located in the lower half-plane rather

close to the origin of coordinates will intersect again the axis O xk ( k = 2, 4 ) at the point Nk ( 0, ηk ) ( k = 2, 4 ), ηk > 0, of the

upper half-plane. By virtue of the symmetry of field (11), the integral curve on the left of the axis O xk of each phase section

is the mirror image of the right curve, i.e., is symmetric about the axes O xk ( k = 2, 4 ). Thus, the system without constraints

(2) closes the trajectory. When the constraints are enabled, each subsystem of (1) participates in the motions of the othersubsystems. The evenness and oddness conditions for the right-hand sides of Cauchy system (11) provide closure of the

conditionally periodic trajectory, which is a smooth two-dimensional torus. The condition d θj ⁄ d t > 0 ( j = 1, 2) is typical of

small vibrations. Thus, the point O is the center, which is the required result.Let us analyze the behavior of the solutions on the boundary determined by condition (10). We will expand the

functions Xj ( x ) ( j = 1, 2) in Eqs. (1) into power series and retain several first terms. Then, the equations in ρ will take the

form

d ρ~j ⁄ d t = R

~j ( ρ

~, θ ) ( j = 1, 2 ). (13)

We denote deviations (variations) of the aperiodic variables by δ ρk 1 and δ ρk2 and introduce ρ∗ k + δ ρk1,

− ρ∗ k + δ ρk2 into the kth equation of system (13)

d ρ~∗ k ⁄ d t + d δ ρk1

⁄ d t = R~

k1 ( ρ~∗ k + δ ρk1, θ∗ k , ρ~

i, θi ),

d ρ~∗ k ⁄ d t + d δ ρk2

⁄ d t = R~

k2 ( − ρ∗ k + δ ρk2, θ∗ k , ρ~

i, θi )

( i, k = 1, 2; i ≠ k ).

Here, R~

k1 ( ρ~∗ k + δ ρk1, θ∗ k , ρ~

i , θi ) and R~

k2 ( − ρ∗ k + δ ρk2, θ∗ k , ρ~

i , θi ) are the right-hand sides of the kth equation of system

(13) into which the corresponding variables ρ~∗ k + δ ρk1 and − ρ∗ k + δ ρk2 are introduced. Since

d ρ~∗ k ⁄ d t = R

~k1 ( ρ~∗ k , θ∗ k , ρ

~i , θi ),

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− d ρ~∗ k ⁄ d t = R

~k2 ( − ρ~∗ k , θ∗ k , ρ

~i , θi ),

the equations of disturbed motion will take the form

d δ ρk1 ⁄ d t = λk1

∗ δ ρk1 + Rk1∗ ( δ ρk1 , ρ~∗ k , θ∗ k ),

d δ ρk2 ⁄ d t = λk2

∗ δ ρk2 + Rk2∗ ( δ ρk2 , ρ~∗ k , θ∗ k ).

2. Example. Let us consider a double mathematical pendulum. A particle P1 of mass m1 moves in a vertical planealong a circle of radius l1 and center O. A particle P2 of mass m2 moves in a vertical plane and is coupled with the former

point at a distance l2. Lagrange coordinates are ϕ1 (the angle between the vertical and the interval OP1) and ϕ2 (the angle

between the vertical and the interval P1P2). The kinetic and potential energies of the system are

T = ( m1 + m2 ) l12 ϕ.

12 + m2 l1 l2 ϕ

.1 ϕ.

2 cos ( ϕ2 − ϕ1 ) + m2 l22 ϕ.

22,

Π = − ( m1 + m2 ) g l1 cos ϕ1 − m2 g l2 cos ϕ2.

The Lagrange equations reduced to the Cauchy form are written as

d x1 ⁄ d t = x2, d x2

⁄ d t = X2 ( x1, x2, x3, x4 ),

d x3 ⁄ d t = x4, d x4

⁄ d t = X4 ( x1, x2, x3, x4 ), (14)

where

x1 = ϕ1, x2 = ϕ.

1, x3 = ϕ2, x4 = ϕ.

2,

X2 ( x1, x2, x3, x4 ) = − ( 1 + µ ) g sin x1

l1 A +

µ g sin x3 cos ( x3 − x1 )l1 A

+ µ x2

2 sin ( x3 − x1 ) cos ( x3 − x1 )A

+ µ l x4

2 sin ( x3 − x1 )A

,

X4 ( x1 , x2 , x3 , x4 ) = − ( 1 + µ ) g sin x3

l2 A +

( 1 + µ ) g sin x1 cos ( x3 − x1 )l2 A

−

− ( 1 − µ ) x2

2 sin ( x3 − x1 )A

− µ x4

2 sin ( x3 − x1 ) cos ( x3 − x1 )A

,

A = 1 + µ sin2 ( x3 − x1 ), µ = m2 ⁄ m1, l = l2

⁄ l1.

The characteristic equation of the corresponding linear system (14) has only imaginary roots

λ1, 2 = ± i Ω1, λ3, 4 = ± i Ω2,

where

Ω1 =

g ( 1 + µ ) ( 1 + l )

2 l2

1 + √ 1 −

4 l

( 1 + µ ) ( 1 + l )2

1 ⁄ 2

,

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Ω2 =

g ( 1 + µ ) ( 1 + l )

2 l2

1 − √ 1 −

4 l

( 1 + µ ) ( 1 + l )2

1 ⁄ 2

since the parameters of the system satisfy the condition

( 1 + µ ) ( 1 + l )2 > 4 l.

Conditions (i) and (ii) of Section 1 are satisfied for (14); hence, a conditionally periodic trajectory exists in system(14). As is seen from the equations of motion, regular vibrations correspond to small initial conditions. In this case, theinteraction of the subsystems is insignificant. As the initial conditions change, aperiodic solutions appear in the trajectoryand the symmetry is disturbed.

3. Characteristic Measures of Aperiodic Solutions. The following approximate polynomial system corresponds

to system (14) under the condition | µ sin 2 ( x3 − x1 ) | < 1:

d z1 ⁄ d t = z2, d z2

⁄ d t = − ( 1 + µ ) g

l1 z1 + Z2 ( z1, z2, z3, z4 ),

d z3 ⁄ d t = z4, d z4

⁄ d t = ( 1 + µ ) g

l2 z3 + Z4 ( z1, z2, z3, z4 ), (15)

where

Z2 ( z1, z2, z3, z4 ) = − ( 1 + µ ) g

l1 z13

6 + µ z1 ( z3 − z1 )2

+ µ gl1

z3 −

z33

6 − ( 1 + 2 µ )

2 z3 ( z3 − z1 )2

+ µ ( z3 − z1 ) ( z2

2 + l z42 ),

Z4 ( z1, z2, z3, z4 ) = − ( 1 + µ ) g

l2 µ z3 ( z3 − z1 )2 +

z33

6

+ ( 1 + µ ) g

l2 z1 −

( 1 + 2 µ )2

z1 ( z3 − z1 )2 − z13

6 − ( z3 − z1 )

( 1 + µ )

l z2

2 − µ z42 .

The matrices of the conversion coefficients in (6) have the form

α =

1

iω1

0

0

1

− iω1

0

0

0

0

1

iω2

0

0

1

− iω2

,

β =

1 ⁄ 21 ⁄ 2

0

0

− i ⁄ (2ω1)i ⁄ (2ω1)

0

0

0

0

1 ⁄ 21 ⁄ 2

0

0

− i ⁄ (2ω2)i ⁄ (2ω2)

.

Equations (15) in polar coordinates are

d ρk ⁄ d t = −

sin θk

2 ωk Z2k ( ρ, θ ), d θk

⁄ d t = ωk − cos θk

2 ωk ρk Z2k ( ρ, θ ) ( k = 1, 2 ).

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Here,

ω1 = √ g ( 1 + µ )l1

, ω2 = √ g ( 1 + µ )l2

,

z1 = 2 ρ1 cos θ1, z2 = − 2 ω1 ρ1 sin θ1, z3 = 2 ρ2 cos θ2, z4 = − 2 ω2 ρ2 sin θ2.

We will write expressions for characteristic measures with k = 1 using the equations in ρ1. We fix the values of

ρ∗ 1, θ∗ 1, ρ∗ 2, and θ∗ 2 at the moment of bifurcation. The variable ρ1 splits into ± ρ∗ 1. We introduce ρ∗ 1+ δ ρ11 and − ρ∗ 1 + δ ρ12

into the equation and write out the characteristic measures of the aperiodic solutions

λ11 ∗ = −

4 sin θ∗ 1

ω1 ( 1 + µ ) g

l1 cos θ∗ 1

ρ∗ 12

2 cos 2 θ∗ 1

+ µ ( − 4 ρ∗ 1 ρ∗ 2 cos θ∗ 1 cos θ∗ 2 + 3 ρ∗ 1

2 cos2 θ∗ 1 + ρ∗ 22 cos 2 θ∗ 2

)

+ µ gl1

( 1 + 2 µ ) ρ∗ 2 cos θ∗ 2 ρ∗ 2 cos θ∗ 1 cos θ∗ 2 − ρ∗ 1 cos2 θ∗ 1

+ µ ( 2 ρ∗ 1 ρ∗ 2 ω1

2 sin2 θ∗ 1 cos θ∗ 2 − 3 ω12 ρ∗ 1

2 sin2 θ∗ 1 cos θ∗ 1 − l ω22 ρ∗ 2

2 sin2 θ∗ 2 cos θ∗ 1

) ,

λ12 ∗ = −

4 sin θ∗ 1

ω1 ( 1 + µ ) g

l1 cos θ∗ 1

ρ∗ 12

2 cos 2 θ∗ 1

+ µ ( 4 ρ∗ 1 ρ∗ 2 cos θ∗ 1 cos θ∗ 2 + 3 ρ∗ 1

2 cos2 θ∗ 1 + ρ∗ 22 cos2 θ∗ 2

)

+ µ gl1

( 1 + 2 µ ) ρ∗ 2 cos θ∗ 2 ρ∗ 2 cos θ∗ 1 cos θ∗ 2 + ρ∗ 1 cos2 θ∗ 1

+ µ (−2 ρ∗ 1 ρ∗ 2 ω1

2 sin2 θ∗ 1 cos θ∗ 2 − 3 ω12 ρ∗ 1

2 sin2 θ∗ 1 cos θ∗ 1 − l ω22 ρ∗ 2

2 sin2 θ∗ 2 cos θ∗ 1

) .

The characteristic measures λ11∗ and λ12

∗ may be of different signs.

4. Discussion. Thus, irregular vibrations of a double mathematical pendulum are identified if the ratio of its massesis not small. The criterion of occurrence of complex vibrations or chaotic trajectories is the absence of the imaginarycomponent in Eqs. (6). Both irregular and regular trajectories exist in the system, the latter being determined by the smalleffect of the subsystems. The Poisson stability of complex vibrations is due to the boundedness of the domain of aperiodicsolutions or negative signs of the characteristic measures of aperiodic solutions [1, 2]. Of importance for many appliedproblems is not only falling into the bifurcation state but also the validity of the model for those or other values of theparameters. The model cannot always reflect physical boundary processes. In particular, the question arises whetherbifurcation processes correctly represent reality. This remark does not addresses the application of the present paper. Theproblem considered here is a classical model of many processes in engineering. The model of a double mathematical pendulumitself implies that motion is Poisson stable; therefore, analyzing the signs of the characteristic measures of aperiodic solutionson the boundary of the periodicity domain of the system has no much physical meaning. The domain of unstable manifoldshas been called here the domain of aperiodic solutions.

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REFERENCES

1. A. A. Martynyuk and N. V. Nikitina, “Estimating the boundary of the domain of aperiodic motions,” Prikl. Mekh., 33,No. 12, 89–95 (1997).

2. A. A. Martynyuk and N. V. Nikitina, “The theory of the principle of dynamic symmetry,” Prikl. Mekh., 34, No. 11, 97–103(1998).

3. V. K. Mel’nikov, “The stability of the center under periodic perturbations,” Tr. Mosk. Mat. Obshch., No. 12, 3–52 (1963).4. V. Moauro and P. Negrini, “Chaotic trajectories of a double mathematical pendulum,” Prikl. Mat. Mekh., 62, No. 5,

892–895 (1998).5. V. V. Nemytskii and V. V. Stepanov, The Qualitative Theory of Differential Equations [in Russian], Gostekhteorizdat,

Moscow (1949).6. A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Vibrations. Invariant Tori [in Russian],

Nauka, Moscow (1987).

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