Chaotic motions in the resonant separatrix band of a parametrically excited pendulum

Vol. 5. No. 4

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Chaotic motions in the resonant seoaratrix

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band

Chaotic motions in the resonant separatrix of a parametrically excited pendulum

band

Albert C. 3. LUO Department Edwardsville, (Received

of Mechanical and Industrial Engineering, Southern IL 68026-1805, USA; e-mail: [email protected] September

Iliinois

University

at Edwardsville,

23, 2000)

Abstract: The energy increment spectrum method is developed for the numerical prediction of the appearance and disappearance of a primary (2M : 1)-librational or (M : l)rotational resonant separatrix band in a parametrically excited pendulum. The analytical conditions for the presence of such separatrix bands are also obtained. Illustrations of the analytical and numerical results for the appearance and destruction of the resonant bands are given for a comparison. Keywords: energy spectrum method, resonant separatrix band, parametric pendulum

Introduction For the resonant layer, in 1964, the numerical investigation was given by Henon and Heile#l through the internal resonant layer in a two-degrees-of-freedom, nonlinear system. Since then, the qualitative description of the resonant layer has been discussed (see [2]). Because it is difficult to determine the location of resonant layers, it seams th.at the analytical and numerical predictions are impossible. In 1998, Han and Luor31 developed an accurate standard-map approach for prediction of the onset of resonant layers in nonlinear dynamical systems. In 1999, Luo and HanL41 also modified the resonant overlap criterion of ChirikovL51 to predict of the appearance of the resonant layer. However, it is still difficult to verify the analytical predictions of resonant layers in nonlinear Hamiltonian systems numerically and experimentally. In this paper, a numerical approach will be developed for the prediction of the presence of resonant layers. The pendulum oscillator possesses complicated dynamic (characteristics existing in other nonlinear dynamic systems (e. g., [2, 5, 61). In 1999, Luo and ~an[~l investigated the stochastic and resonant layers in the periodically driven pendulum. However, the dynamic behaviors in the parametrically excited pendulum are different from the ones in the periodically forced one. In 2000, Luo[*] investigated the stochastic layer of the parametrically excited pendulum. In this paper, chaotic motions in the resonant separatrix bands (resonant layers) of the parametrically excited pendulum oscillator will be investigated. The numerical and analytical results for the presence of such chaotic motions in such an oscillator will be given.

1 Problem

statement

Consider a parametrically

excited pendulum Z+(o+Q,cosRt)sina:=O

where Qo and R are excitation H(TY,t)

= Ho&Y)

(1)

strength and frequency. The Hamiltonian

+ fhb,y,t),

Ho = $ - cxcosz,

of Eq. (1) is

H1 = -Qocosfltcos~

For the conservative energy Ho = EL (0 2 EL < Q), the t:ime-dependent

Hamiltonian

(2) H1

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in (2) is approximated

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December 2000

by HF z - Q. cos(Rt) cos x;

where the subscript (or superscript) L denotes the libration. solution xi and the rotational resonant condition are: x: = f2jr

f 2 arcsin ICLsn 2WLL)cpL ) kL lr 1 (

where sn is the Jacobi elliptic the elliptic integral of the first where cpfi is the initial phase the (m : n)-resonant band. In a like fashion, for HO rotation is approximated by

)I

As in [2, 4, 5, 81, the rotational

and

~MWL = R

functions, LL = ,/m/a is the elliptic modulus, K(kr,) is kind, WL = 27r/T~ = 7r&‘/2K(lc~), and the phase (pL = wLt+&, angle. Note that a resonant band satisfying mw = nR is termed = ER > cy, the time-dependent H:M-

x; = f2am

K(~RPR

(

Hamiltonian

HI in (2) for the

Q. cos Rt cos xi

where the subscript (or superscript) R denotes the rotation. the rotational resonant condition [2, 5, 81 are: 7r

,kR

and

The rotational

solution xi and

i,!f‘,& = fl

)

where am is the amplitude of Jacobi elliptic function, WR = 2n/T~ = ?r&/kRK(kR) the natural frequency of the unperturbed pendulum, the modulus kR = a/dphase (Pi = wRt + cpt.

2 Energy

increment

(3)

denotes and

spectrum

The chaotic motions in the resonant bands for the parametrically presented through the Poincare mapping section given by 1 satisfying Eq. (1)) s(t~)

excited pendulum

are

2Nn mod 27r and tN = - R

+to, N=O,l;** (4) 1 1 where z(tN) = XN, ?(tN) = ?N and x(to) = x0, iT(to) = SOat t = to are initial conditions. The Poincare map is: P : C -+ C. The energy spectrum approach developed in [9] was used for the numerical prediction of the onset of resonance in the stochastic layer. In the energy spectrum, the maximum and minimum conservative energies are computed through the Poincare mapping section. Using the Poincare map section in (4), the conservative energy for each Poincard mapping point of the parametrically excited pendulum is

c =

(x(tlv), 4hJ)

HtN) = % _ acosxN 0 2 However, in the computation of conservative energy for specified resonant band, the changes in energy spectrums cannot be observed clearly. To observe the energy changes by the sub-resonance for the specified resonant band, the minimum and maximum increments with respect to the unperturbed resonant orbit are introduced herein. That

for the (2M : 1)-librational

resonant band, and

energy caused energy is,

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for the (M : 1)-rotational resonant band. Such an energy increment versus excitation frequency is termed the energy increment spectrum. For Qc = 0.05 and cy = 1.0 in Eq. (l), the maximum and minimum energy increment spectra are computed. Fig. 1 gives the energy increment spectrum for the (4 : 1)-librational, resonant bands. Note that the maximum and minimum energy increments are computed from 10,000 iterations of Poincard map for each specified excitation frequency and strength.

3 Analytical

conditions

3.1

resonant

Ii3

-0.5 1

1.5 Librat

ional

band

I

I

I

J

2.0

2.5

3.0

3.5

Excitation Frequency CJ

Since the resonant separatrix can be approximated by the corresponding two unperturbed orbits in Eq. (3), the energy increment, based on two unperturbed orbits relative to (2M : 1)-librational resonant separatrix, is approximated by rTL+tO AH)=2 (fig2 - fsgi)dt J to and

Fig. 1 T:he energy increment spectrum for the (4 : 1)-librational resonant band at Qo = 0.05 and 0: = 1.0

M 2QcQ~~“‘1) sin fits

(5)

where Pki-I = csch[rr(i

- i)z]

for

i = {m,n}

and f~ = y M y:,

fs = crsinx x asinxf,

gi = 0;

gs = Qecosfltsinx

z QccosRtsinxi

AS in [2, 31, the energy increments AH,$ in Eq. (5) and phase changes (A& an whisker map for the (2M : 1)-librationai resonant band:

= CUL) give

Rti +

cpk, Ei is the conservative energy and AH: = Ei+l - Ei, A& = cpf+i - cp”. From In the second one of Eq. (3), cpf+i - cp” = 2(2M)7r gives the resonant energy Ef"'l). the accurate standard mapping approach in 131,an approximate, critical condition for the onset of the librational resonant band is

where ‘p” =

QoM where R

G(2M:1) _ L

-

(fi)3(pw)5

0.9716354 zQ(2M:l)

GrMzl)

(6)

1

)I

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Letting EF = EfMzl) and Et+1 = Ef”+2’1), the excitation strength Qe for the (2.M : l)librational resonant band destroyed by the (2M + 2 : 1) resonance is approximately predicted from [3] by E(2M+2:1)

&,,=I

3.2

Rotational

L

resonant

_ E(2M:1) 1

2Q(2M:l)L L

(7)

band

The energy increment, based on two unperturbed onant separatrix, is approximated by AH;

w 2QoQk”‘l’

orbits relative to (M : 1)-rotational

res-

sin(Rte)

From the accurate standard mapping approach [3], the condition for the appearance of the (M : 1)-rotational resonant band is

Qo M

0.9716354

R

where

q&M:“)3

GcMzl) = _ R

(8)

G(“:l) IR 1

2QtMz1)

2(&)3

[l

-

(&“:‘))2]

(M:l) E(kR

)

The excitation strength Qe for the (A4 : 1)-rotational resonant band destroyed by the (M + 1 : 1)-resonance is approximately computed from [3] by

Qo =

IE~“+l”) _ ELM:” 1 and R = 2Q(“4

R

4 Numerical

MT+ p:‘qgf:l)) R

(9)

results

The analytical conditions for the appearance and destruction of the resonant band have been developed and the energy spectrum approach has been developed for the numerical prediction of the resonant band. To make a comparison of two predictions, the excitation frequency versus the excitation strength for a specified resonant band will be presented in this section. The conditions for the appearance and destruction of the resonant bands are computed and illustrated through the excitation frequency and strength in Fig. 2. The solid curves give the analytical predictions of excitation strength for the appearance of the resonant band from Eqs. (6) and (8), and the dash-dot curves represent the conditions for the destruction of the resonant bands, computed by Eqs. (7) and (9). The numerical predictions of the appearance and destruction of the resonant bands are given by the circular and triangular symbol-curves, which are generated from the energy increment spectrum approach. For the librational resonant bands, the numerical and analytical results are in good agreement. But the analytical results of the rotational resonant band are different from the numerical predictions. Such a phenomenon may be caused by the sub-resonance. For the higher order resonant band, the analytical prediction becomes poorer and poorer because the energy increments is computed by the unperturbed resonant orbit, instead of the perturbed resonant separatrix which can be obtained by renormalization. For demonstration of chaotic motion in the resonant bands of the parametrically excited pendulum, the Pnd-order symplectic scheme is used herein. The chaotic motion is illustrated

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Chaotic motions in the resonant separatrix band

through the PoincarC mapping section. In the left plot of Fig. 3, the (2 : l)-librational resonant band positioning inside the homoclinic orbit (or separatrix) is clearly observed for parameters R = 1.572099784,

Qo = 0.06,

The right plot of Fig. 3 gives an illustration outside the homoclinic orbit with parameters 0 = 3.987793001,

Qo = 0.1,

zo = 0,

*c, = 1.603121954

of the (3 : 1)-rotational

z. = 1.52663301,

resonant bands located

i. = f1.615022877

0.06

1

2

3

4

2

1

Excitation Frequency R

3

4

Excitation Frequency R

Fig. 2 The (A4 : 1)-rotational resonant conditions (left) and the excitation strength conditions (right) for the appearance (solid) and destruction (dash-dot) of the rotational resonant bands at cy = 1. The circular and triangular symbol curves give the numerical predictions of the appearance and destruction of the resonant bands

3

3

(2: 1 )-resonant layer

2

Homoclinic orbit I I I -3 ’ -3 -2 0 0

-2 I

I

1

2

Displacement x Fig. 3 The (2 : 1)-librational

resonant

c 3

-3 0

‘I

2

3

4

Displacement bands (left) and the (3 : l)-rotational

resonant

5

6

x

bands (right)

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2000

References [l] Henon, M. and Heiles, C., The applicability of the third integral of motion: Some numerical experiments, Astron. J., 1964, 69: 73-79 [2] Lichtenberg, A. J. and Lieberman, M. A., Regular Chaotic Dynamics, Springer-Verlag, New York, 1992 [3] Han, R. P. S. and Luo, A. C. J., Resonant layers in nonlinear dynamics, ASME J. Appl. Mech.: 1998. 65: 727-736 [4] Luo, A. C. J. and Han, R. P. S., Analytical predictions of chaos in a nonlinear rod, J. Sound and Vibrat., 1999, 227(3): 523-544 [5] Chirikov, B. V., A universal instability of many dimensional oscillator systems, Phys. Rep., 1979, 52: 263-379 [6] Zaslavsky, G. M. and Filonenko, N. N., Stochastic instability Iof trapped particles and conditions of application of the quasi-linear approximation, Soviet Phys. JETP, 1968, 27: 851-857 [7] Luo, A. C. J. and Han, R. P. S., The dynamics of stochastic and resonant layers in a periodically driven pendulum, Chaos, Solitons and Fractals, 2000, 11: 2349-2359 [8] Luo, A. C. J., Stochastic layers in a parametrically excited pendulum, Nonlinear Dyn., 2000, in press [9] Luo, A. C. J., Gu, K. and Han, R. P. S., Resonant separatrix webs in the stochastic layers of the twin-well Duffing oscillator, Nonlinear Dyn., 1999, 19: 37-48