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The effects of non-stationary processes on chaotic and regular responses of the duffing oscillator
Inr. , Non-Luwor Mrchantcs. Pnnlcd in Great Bntam
Vol. 26. No. I. pp. 61 71. 1991
0020-7462,91 53.00 + .a, C 1990 Pergamoa Press plc
THE EFFECTS OF NON-STATIONARY PROCESSES ON CHAOTIC AND REGULAR RESPONSES OF THE DUFFING OSCILLATOR F. A. MOSLEHY and R. M. EVAN-IWANOWSKI University of Central Florida, Orlando, FL 32816, U.S.A. (Received
2 December
1988; in
revisedform 2 February
1990)
Abstract-This paper deals with the effects of non-stationary regimes on stationary chaotic motions and non-linear attractors: (1) linear variations of the excitation frequency, v = v,, + a,~, and the amplitude B = B, + rat; (2) cyclic variations of the excitation frequency, Y = Y,,+ 7 sin xct. It was shown in (I) that for very small values of x, or z a, i.e. for slow sweeps, the non-stationary responses initially coincide with the stationary chaotic, but then they depart. The faster the sweep, the earlier is the departure from the stationary and from other non-stationary responses. An observation is made that for sufficiently fast sweeps, the initially chaotic motion may be changed into a structured one. In (2) initially stationary chaotic motion is changed instantaneously to another type of motion. The stationary attractors transit into different attractors. Many dynamic phenomena in the real world are modelled mathematically by the non-stationary Duffing differential equation. This paper presents the first attempt to apply non-stationary processes to chaotic motion. It is the objective of this study to contribute to the theory of dynamics and technical design.
1. INTRODUCTION
The Dufing non-stationary driven differential equation dealt with in this paper has the following form: 2 + w’x + ki + px3 = B(r)cos O(t), where B(t) and O(r) are explicit functions of time. Two classes of non-stationary regimes may be discerned: (1) robust, when time-varying parameters are given as explicit functions of time with extended ranges of their variations, and (2) evolving, when these parameters vary very slowly with time, or they never need be defined as given functions of time. The former finds its applications in relatively early felt effects of the non-stationary regimes, particularly in structural mechanics subject to combustive propulsions, air and water waves, earth movements, etc. The qualitative and quantitative effects of robust non-stationary processes may be readily observed from the state variables or perturbation plots, such as lowering of the maximum response amplitudes and elimination of the fold discontinuities [l-7]. They are, however, difficult to analyse. The latter regime allows detailed analysis of such phenomena as cyclic folds (jumps), bifurcations, attractors, etc. It is useful in the analysis of systems with relatively slow but long time of accumulation of non-stationary effects [g-24]. The most commonly used robust non-stationary modes are: (1) linear variations of the frequencies v(t) and amplitudes B(t) of the external excitation (ee): namely, v(t) = v,, + r,t and B(t) = B, + z,t, called transition through resonances (TtR); and (2) cyclic variations of the frequencies V(C)= tlo + ysin sCt, where v,,, B,, xv, ~a, rC, and 7 are constants. The parameters z, and ra are called rates of sweep or simply sweeps. The first publication dealing with robust non-stationary responses is due to Lewis in 1932 [I]. The effects of linear transitions through resonances were applied to a linear singledegree-of-freedom system. The results show the characteristic robust non-stationary manifestation mentioned above. A well-known contribution to the literature on the robust nonstationary vibrations was made by Mitropolskii [2]. The Krylov-Bogolyubov asymptotic stationary methods were expanded to include non-stationary processes, resulting in what is known as the KBM method. Applications of special functions and asymptotic methods for solutions of non-stationary technical problems are found in the book by Goloskokow and Filippow [3]. It constitutes a valuable contribution to the literature on mechanics of TtR. Contributed by A. H. Nayfeh. 61
F. A. MOSLEHY
62
and R. M. EYAN-IW~NOWSKI
A review of non-stationary publications up to 1969 was prepared by Evan-Iwanowski [4]. A book containing work done at the Applied Mechanics Laboratory of Syracuse University was published by Evan-Iwanowski [S]. The KBM method has been expanded to include systems with multiple-degrees-of-freedom and the attendant combination resonances consisting of additive, differential and internal or autoparametric resonances. Lately. the main focus of research work on robust non-stationary processes was directed toward parametrically excited systems, i.e. toward the problems of dynamic stability [6,7]. The early developments on evolving non-stationary processes can be found in a paper by Kevorkian [S] for a linear undamped oscillator, followed by Ablowitz et al. [9] for the non-linear system with damping. It was found in [9] that non-stationary jumps may occur for positive sweeps for some values of the excitation amplitudes. The method of multiple scales was used. By appropriately selecting smallness of the magnitude of the involved parameters and the time scales, the authors succeeded in analysing the mechanisms of the non-stationary jumps very near the stationary discontinuity in detail. A similar approach was used by Collinge [lo] and Collinge and Ockendon [ll], with additional refinements to uncover details of the evolving jump phenomenon. Numerous extensions of this technique and the introduction of others were applied to solve some specific non-stationary problems governed by the Duffing differential equation. They cover many areas: physics (nuclear physics). fluid mechanics (sloshing), combustion (flames), chemistry, biology, and others [ 12-241. The results regarding prediction of the incipient jump are presented in a paper by Thompson and Virgin [2.5]. TtR was used to determine the plot of B(t) vs frequencies of the response relative to the stationary jump response, which tend to zero near the fold bifurcation. Topics of a genera1 nature on non-stationary processes may be found in the article by Kronauer [26] and in the text by Nayfeh and Mook [27]. From the available rest&s, it may be concluded that the non-stationary processes affect strongly the stationary manifestations, such as bifurcation, including Hopf bifurcation, attractors, including strange attractors (chaotic motion), domains of attraction, etc. Most, but not all, of the non-stationary effects are desirable. Moreover, the non-stationary processes may introduce strictly non-stationary phenomena which may be undesirable.
2. PRELIMINARIES
The Duffing equation is as follows: d2.u, mdtt + 2~2
+ OS, + d.x: = B, cos&t); b(t) = v. L
Introducing
non-dimensional
(1)
coordinates x=
d -x1
J 0
and
t=
Co
-t, J m
we obtain d”.u 2 + k$
+ .Y+ .X3= B cosB(t);
8(r) = ”
(2)
where
We consider a special Duffing equation introduced by Hayashi [28], which is often referred to as the Duffing-Hayashi equation, by dropping the linear term from (2): d2x dt2 + kg Ueda [24], followed by Thompson
+ .x3 = B costI(
8(t) = v.
and Stewart [29], found a specific set of initial
The Duffing oscillator
63
conditions and parameter values, k = 0.05;
B = 7.5; x,, = 3; i,, = 4;
v= 1
(4)
for which the response is a typical stationary chaotic motion. Finally, we cast (3) in a non-stationary form: j;_+kzi+x3=B(t)cosB(t),
(.)=-$
(5)
where k = 0.05, and B(t) and 8(t) = v(t) are now functions of time, specified in the text. It should be noted that the set of values in (4) is not unique. It is used here to minimize start-up transients and to determine the non-stationary effects on the known chaotic stationary motion. The following non-stationary processes are used in this paper: (i) linear variation of the frequency of the external excitation (TtR) v(t) = 1 + a,t;
(6)
(ii) linear variation of the amplitude of the external excitation (TtR) B(t) = 7.5 + cr,t;
(7)
(iii) cyclic variation of the frequency of the external excitation v(t) = 1.05 + ysin a,~; and
(8)
(iv) cyclic variation of the frequency of external excitations was also used for regular cases of different values of k and B, and initial conditions x0 and x0 are specified in Section 3, Results; c(,, LY.~, aC, and y are constants. Equation (5) written in terms of state variable representation i=y was used together calculations.
and
j=B(t)cosB(t)-ky-x3
with the fourth
3. RESULTS
order
Runge-Kutta
(9)
method
in the numerical
AND DISCUSSIONS
3.1. Efects of sweep rates a, and ug on chaotic motion To observe the effects of the non-stationary regimes on chaotic responses, the stationary chaotic response for CI,= 0 is presented first (Fig. 1). This figure agrees with the result of ref. [29]. The corresponding phase plane plot is shown in Fig. 2. Note the appearance of the first kinks (inflection points) in Fig. 1 at t = 32.5 and the corresponding near cusp in the phase plane (for discussion, see Section 3.4). The non-stationary results are presented next. For low sweep rates cr, = 10e5 or zR = 10V5, the non-stationary responses coincide with the stationary chaotic for a considerable time (Figs 3 and 4). The departure times between these two responses are given in Table 1. Detailed calculations reveal a slow build-up of the differences. The maxima of the responses correspond to the maxima of the excitation. There are additional smaller peaks. one, two, or three, between the larger peaks, indicating the existence of two or three different waveforms (patterns) (see Section 3.5). Similarly, the behavior close to the chaotic stationary motion was observed for u, = 10e4 or CQ,= 10s4. The departure, however, is somewhat earlier from the stationary or non-stationary slower sweep responses (Table 1). The non-stationary motion for CX,= lo-’ coincides initially with the stationary and previously described non-stationary cases (Fig. 5). For t = 37, however, it departs from the stationary, and at t between 152 and 200, it shows well-structured motion (Fig. 6). 3.2. EfSects of cyclic variations on stationary chaotic motion Non-stationary cyclic variations of the excitation frequencies have the following form: v(t) = v. + ysin a,t
(IO)
F. A. M~SLEHYand R. M. Eva%lwa~oww
64
Fig. I. Solutions
of
the
Dufiing
equation
(t) exhibiting Ueda &?4j.
stationary
CL,= 0
chaotic
motion,
after
6 -i
5-i 4
3 j 2 f 0
rx
-1 -2 -3 -4 -5 -6 i -4
-2
0
2
4
X Fig. 2. Phase portrait 3, = 0. Note the near-cusps, C, corr~~ondi~~
-4
0
I
20
/
40
,
60
I
80
I
100
/
120
/
140
to the kinks, K. in the response pfot.
,
160
/
180
t
2co
220
Fig. 3. Time history of the response, I, = 10-s. The non-stationary pfots are close to the chaotic responses for 1, = 4 but eventually they deparf (Tab,Ie I).
The DutIing oscillator
3 :
2
c
40
20
60
80
TOC
120
160
140
220
IEm
220
t Fig. 4. Time history ol the response, ~a = IO-s, has similar characteristics with 1, = 10 -5. See Table 1.
Table 1. Departure times for non-stationary chaotic responses from stationary for different z, and q,
I
-51 0
20
40
60
/ 80
I 1CO
, 120
I 140
I 160
I 200
180
220
t
Fig. 5. Non-stationary time history of the response, z, = 10-L. Initial chaotic motion shows early departures from the responses for z, = 0 or z, = lo-‘, Table 1, eventually exhibiting a structured motion for t 3 152.
Two cases are considered in this paper, see equation (5): (i) v0 = 1.05; 7 = 0.1; xc = 0.04; x0 = 3; i, = 4, and (ii) vg = 1.05; y = 0.1; zy, = 0.16; x0 = 3; -i-, = 4. For sl, = 0.04, after initial beat t < 10, the motion is different from the chaotic stationary or
66
Fig. 6, Segment
ofthe non-stationary
phase portrait for 2, = IO-‘. for 151.7 i t c 209, showing structured. close to harmonic motion with slight@ modulated displacement and velocity.
Fig. 7. Elrects of non-stationary cyclic regime xc = 0.04, ;’ = 0. t. on the stationary chaotic motion. Note an early and considerable departure from the stationary and linear non-stationary responses.
linear non-stationary motions for most of the time, with the peaks coinciding with those of the external excitation (Fig. 7). For LX,= 0.16, the non-stationary motion remains unspecified to the end of the recording period {Fig. 8).
Five cases, (l)-(5), of the regutar stable non-linear responses were selected by Ueda [24] to demonstrate the effect of initial conditions on the responses and the co-existence of various attractors in the Duffing oscillator: j; + ks + x3 = B cost: k = 0.08; B = 0.2. Below are presented the effects of the non-stationary regimes on these responses and on the corresponding attractors; v = ve + f-’ sina,t; ;’ = 0.1; xc = 0.04; ra = 1.05 for all cases (Figs 9-13). Case 1. Figure 9, x0 = 0.21; 1, = 0.02, shows the non-stationary effects on the stationary motion, small amplitude n = 1 attractor. For 20 c I < 40 and JO CCr < 60, two cases of *Cyclic non-srationary
regimes
are observed, lor instance. in non-ideal systems (see [Z) and [43).
The Duffing oscillator
Fig. 8. Effects of non-stationary
cyclic mode z, = 0.16, ;’ = 0.1. Almost instantaneous departure from the stationary chaotic and other non-stationary responses.
\I-_- ~_ ._ ._______._...-_13
67
._..
--------A
V
4
1
I
0
20
40
60
/
80
100
I
120
I
140
/
153
I
200
150
220
t Fig. 9. Case 1. x0 = 0.21, ;i, = 0.02; non-stationary,
n = 1 attractor,
with decreasing amplitude.
----
X
-4
/
0
/
20
1
40
t
60
/
a0
/
100
Fig. 10. Case 2. x0 = 1.05, k0 = 0.77; non-stationary,
ringing waveforms (see Section quency v(t). Eventually, for 80 c nized motion with the excitation n = 1 attractor with modulated
/
120
/
140
n = 1 attractor
/
160
/
180
zoo
I 220
with modulated amplitude.
3.4), appear with frequencies equal to the excitation fret < 200, symmetric slightly modulated harmonic synchrofrequency takes place with time lag = TI,terminating in an amplitude.
Case 2. Figure 10, x0 = 1.05, ;Ce= 0.77. The stationary motion is a simple harmonic, n = 1 oscillation with small amplitude. The non-stationary cyclic regime modifies this
6%
Fig. 1f. Case 3. x0 = 0.67, TQ=
0.02;non-stationery, n = I attractor with decreasing amplitude.
t Fig. $2. &se 4. x0 = 0.45, f, = &3Q after a brief chaotic motion non-stationary, with increasing ampiitudc
J 1
0
n = 2 attractor
..- ...-----._---..~_
20
40
60
80
i
700
120
140
160
2co
180
220
t Fig. 13. Case 5. x0 = - 0.43, & = 0.12; eon-statjonafy,
n = 1 attractor.
response considerably. At t = 22, 32 and 43, three kinks are observed. After that an asymmetrical stable motion appears with a three-peak pattern. Each peak coincides with the maximum of the excitation and is shifted by 71relative to it, terminating in an n = 3 attractor with moduIated amplitudes. Case 3. Figure II, x0 = 0.67, 2, = OB2. The stationary response is of tt = 2 type. The non-stationary motion exhibits two consecutive ringings at 20 c t < 42 and 42 < r c 70,
69
The Duffing osciiIator
with the frequencies of the excitation and different background waveforms. This is followed by asymmetric motion, n = 1 with time lag = K, terminating in an n = 1 attractor with slightly modulated amplitude. Case 4. Figure 12, _yO= - 0.46,5, = 0.30. The stationary response is of n = 2 type. For 25 < t < 200, an n = 3 attractor is appearing with modulated amplitude with the peaks corresponding to the frequency of excitation v(t).
Case 5. Figure 13, .x0 = 0.43, -ito= 0.12. The stationary motion is of n = 3 type. The initial non-stationary motion is followed by ringings for 44 < t < 70 and 70 < t < 100. Finally, for 100 < c < 200 an n = 1 response is settling in with slightly modulated amplitude. From the available sample, it is possible to conclude that the cyclic non-stationary mode may considerably modify the corresponding stationary (regular) responses, and no resemblance is observed from the very beginning between these motions; that is to say, the non-stationary responses form their specifically non-stationary attractors, different from the corresponding stationary attractors. 3.4. Structure of the chnotic responses One of the common waveforms (patterns) appearing in the responses of the Duffing osciflator are points of inflection or kinks. The appearance of kinks is very signi~cant as confirmed analytically by Novak and Frehlich [303 and observed experimentally by Robinson [12], because the kinks indicate the entry of even harmonics in addition to the previous single odd harmonics. The even- and odd-harmonic responses are then followed by period doubling which eventually leads to chaos. The kinks are easily identified as nearcusps or cusps in the phase plane plots (Figs 1 and 2). Another frequency appearing pattern in the phase plane is the closed loops corresponding to the two-hump waveforms in the response. The appearance in the Duffing oscillator of the ringings, i.e. high frequency oscillations superimposed on a background response could be seen, for instance, in Fig. 10 for 0 < t c 40. The ringings were also reported by Hijhler [ 133 and again by Robinson [12] for the stationary cases. A brief catalog of regular and chaotic stationary waveforms is found, for example, in the textbook by Kreuzer [31]. 3.5. ON-OFF chaotic responses ON-OFF or interrupted non-stationary regimes present a realistic situation in the real world. They introduce modifications to the continuous non-stationary response. This is demonstrated in Figs 14 and 15. The responses for the specific values of c(s within the
!
-1 X
0
20
40
60
80
100
120
140
160
~E)O
200
220
t QY - t_
0 +
m-3
Fig. 14. ON-OFF
-b+---
10-f--+--
non-stationary
w-2 -----q
0 ---+---
regimes as specified
in the
plot; k = 0.05;
B = 7.5.
Fig. IS. ON-OFF
corresponding responses.
non-statianary
regimes as specified in the plot; k = 0.05; B = 7.5.
time intervals are markedly different from the continuous non-stationary
4. CONCLUSiONS
From the results obtained in this investigation, some reveating and puzzling conclusions may be derived as foliows: (I) The stationary chaotic motion, sY = rR = 0, may be modified by introducing nonstationary linear processes; namely, by linear variations of the excitation frequencies and amplitudes in two ways: (i) Initially, for relatively small values of X”or xB of order IO-’ or less, the non-stationary and stationary chaotic motion are identical, then they depart after an extended period of time shown in Table 1. (ii) For xv or ~a approximately equal to lo-‘, the non-stationary responses remain initially close to stationary chaotic. They depart from them after some time {Table 1). That is, chaotic motion is being annihilated by means of a non-stationary process. This fact suggests that for a,,, rs z IO-j or below the nonstationary processes are evolving, and for zV, zil 2 10W2 or above they are robust. An important question is raised: what are the transition mechanisms from non-stationary chaotic to structured motions? More generally, what are the mechanisms of the onsets and terminations of chaotic motions in con-stationary processes? (2) Cyclic non-stationary processes investigated introduce at t = 0 different motions from the stationary chaotic and other non-stationary responses. They also strongly change the regular stationary non-linear responses and non-linear attractors. (3) The phase plane plots, particularly the segmented (detailed) plots, may be used to identify the patterns (waveforms) of the motion. Thus, kinks in the response plots correspond to the near-cusps in the phase plane plots, the two-hump waves, with loops, ringings with winding loops, etc. Appearances of some waveforms may forecast the incipient bifurcated motions. For instance, the appearance of the kinks may read eventually to chaos. (4) It is surprising indeed that on the one hand chaotic motions are quiet durable, that is they maintain their chaoticity inde~nitely~ and on the other hand they succumb relatively fast to non-stationary processes. An obvious question is posed: what are the other factors which one may introduce into a chaotic motion to eliminate chaos? (5) The ON-OFF regimes may be helpful to uncover the effects of initial conditions on the dynamic responses. They present a very puzzling picture, and need further study. (6) It is anticipated that by the accumulation of the results on the effects of the mm-stationary processes on the stationary regular and chaotic dynamical responses, it will be possible to detect new formulations and approaches for explanations of non-stationary behavior, in particular, to identify different waveforms (patterns) with the corresponding bifurcations or limit sets. This is significant from the basic mechanics point of view and applications as well. One of the basic questions may be, for example, stated as follows:
The Duffing osciliator
71
determine the thresholds between the evolving and robust non-stationary processes, i.e. determination of the thresholds of zV, rB or q. In application, non-stationary processes offer numerous and substantial mechanical advantages such as elimination of catastrophic jumps, reductions of the maximum stationary response amplitudes, annihilation of chaos, control of the responses, limitation of maxima of the response amplitudes, prediction of the corresponding CX,,rB or “C for which the maxima are attained. ~~~~~~~~~~~e~~~[-T~eauthors are grateful to Mr Shahram Etmi for his assistance in the computer work.
REFERENCES 1. F. M. Lewis, Vibration during acceleration through a critical speed. From. ASiME 54,253 (f932). 2. Y. A. Mitropolskii. Probie~s offhe Asymptotic Theory o~No~srufionary Vibrutions. Daniel Davey, New York (1965). 3. E. G. Goloskokow and A. P. Filippow, Nonscarionarp Vibrations o~Mechanical Sysrrms. Academic. Berlin (1971). 4. R. M. Evan-Iwanowski, Non-stationary vibrations of mechanical systems. Appl. Mech. Rev. 22, 213 (1969). 5. R. M. Evan-Iwanowski, Resonance Oscillarions in Mechanical S_vstems. Elsevier, Amsterdam (1976). 6. G. L. Ostiguy, Etfects of aspect ratio on response of non-linear rectangular plates-analysis and experiment. Ph.D. Dissertation, Syracuse University (1970). 7. G. L. Ostiguy and R. M. Evan-iwanowski. Influence ofthe aspect ratio on the dynamic stability and non-linear response of rectangular plates. ASME J. Mech. Design 104,417 (1982). 8. J. Kevorkian, Passage through resonance for a one-dimensional oscillator with slowly varying frequency. SIAM J. appt. ~~a?h. 20, 364 (f971). 9. M. J. Ablowitt 3. A. Funk and A. C. Newell, Semi-resonant interactions and frequency dividers. Stud. appf. .tfarh, 52, 5 1 ( 1973). IO. 1. R. Collinge. Resonant oscillations. Doctoral Dissertation, Oxford University (1974). Il. I. R. Colfinge and J. R. Ockendon, Transition through resonance of a Dulling oscitlator. SIAM J. appl. Mnrh. 37, 350 (1979). 12. F. N. H. Robinson, Experimental observation of the large amplitude solutions of Duffing’s and related equations. To appear in IMAJ. appl. Math. 13. G. Hijhler and 1. Sabba - Stefanescu. Schwingungen Dulling oszillator. TKP 87-9 (preprint). 14. M. C. Cross. P. C. Hohenberg and M. Lucke. Forcing of convection due to time-dependent heating near threshold. .I. F/aid Mech. 136, 269 (1982). 15. G. Alhers, M. f. Cross, P. C. ~ohenber~ and S. Safran, The amplitude equation near the convection threshold: application to timedependent heating experiments. $. Fluid fife&, No. 110, 297 (t981). IS. I. Ishida, I. Takashi and Y, Toshio, Transient vibration of rotating shaft with non-linear spring characteristics during acceleration through a major critical speed. JSME fnt. J. 38, 458 61987). 17. E. Jakobsson and R. Guttman. continuous stimulation and threshold of axons: the other legacy of Kenneth Cole, in ~~op~ysicu~Approach to Excitable Systems {Edited by W. Adelman and D. Goldman). Plenum, New York (1981). 18. R. Haberman, Slowly varying jump and transition phenomena associated with algebraic bifurcation problem. SIAilil ,r. appl. ~~fafh.37, 69 (1979). 19. T. Erneux and P. Mandel, imperfect bifurcation with a slowlv-varvinn controi parameter. SIA&f f. anpi. Mach. 46, I (1986). 20. W. Scharf, M. Squicciarini, D. Bromley. C. Green, J. R. Tredicce and L. M. Narducci, Experimental observation of a delayed bifurcation at the threshold of an argon laser. Opt. Comm. 63, 344 (1987). 21. R. M. Evan-Iwanowski and G. L. Ostiguy, Non-linear, non-stationary process, Proc. Second Con& on Non-linear Vibrarions, &ability, and Dynamics of Srructures and Mechanisms, Vol. 1, Virginia Polytechnic Institute and State University (1988). 22. I-l. Ockendon, J. R. Ockendon and A. D. Johnson, Resonant sloshing in shallow water. f. F&id Meek. 167,465 f 1986). 23. R. M. Evan-Iwanowski and G. L. Ostiguy, Transitions through resonances. Israel J. Technot. 21, Proc. Seventh Israel Cant Mech. Engrs, p. 163 (1983). 24. Y. Ueda, Steady motions exhibited by Doffing’s equation: a picture book of regular and chaotic motions, in New Approaches to eon-linear Problems in Dynamics [Edited by P. J. Holmes), p. 31 I. SIAM, Phiiadelphia j1980). 25. J. M. 1: Thompson and L. N. Virgin, Predicting a jump to resonance using transient maps and beats. Ins. J. Non-linear Mech. 21, 205 (1986). 26. R. E. Kronauer, Handbook ofAppliedMathematics: Selected Results and Methods, 2nd edn (Edited by Carl E. Pearson), Chapter 13. Van Nostrand Reinhold Company, New York (1982). 27. A. H. Nayfeh and D. T. Mook, Non-linear Oscillations. Wiley, New York (1979). 28. C. Hayashi, Non-Linear Oscillations in Physical Sysrems. McGraw-Hill, New York (1964). 29. J. M. T. Thompson and K. B. Stewart, Non-linear Dynamics and Chaos. Wiley, New York (1986). 30. S. Novak and R. D. Frehlich, Transition to chaos in the DuRing oscillator. Phys. Rec. A 26, 3660 (1982). 31. E. Kreuzer. Numerisclte Unrersuc~u~g ~ichrlinearer ~y~am~scher Systeme. Springer, Berlin (1982).
Vol. 26. No. I. pp. 61 71. 1991
0020-7462,91 53.00 + .a, C 1990 Pergamoa Press plc
THE EFFECTS OF NON-STATIONARY PROCESSES ON CHAOTIC AND REGULAR RESPONSES OF THE DUFFING OSCILLATOR F. A. MOSLEHY and R. M. EVAN-IWANOWSKI University of Central Florida, Orlando, FL 32816, U.S.A. (Received
2 December
1988; in
revisedform 2 February
1990)
Abstract-This paper deals with the effects of non-stationary regimes on stationary chaotic motions and non-linear attractors: (1) linear variations of the excitation frequency, v = v,, + a,~, and the amplitude B = B, + rat; (2) cyclic variations of the excitation frequency, Y = Y,,+ 7 sin xct. It was shown in (I) that for very small values of x, or z a, i.e. for slow sweeps, the non-stationary responses initially coincide with the stationary chaotic, but then they depart. The faster the sweep, the earlier is the departure from the stationary and from other non-stationary responses. An observation is made that for sufficiently fast sweeps, the initially chaotic motion may be changed into a structured one. In (2) initially stationary chaotic motion is changed instantaneously to another type of motion. The stationary attractors transit into different attractors. Many dynamic phenomena in the real world are modelled mathematically by the non-stationary Duffing differential equation. This paper presents the first attempt to apply non-stationary processes to chaotic motion. It is the objective of this study to contribute to the theory of dynamics and technical design.
1. INTRODUCTION
The Dufing non-stationary driven differential equation dealt with in this paper has the following form: 2 + w’x + ki + px3 = B(r)cos O(t), where B(t) and O(r) are explicit functions of time. Two classes of non-stationary regimes may be discerned: (1) robust, when time-varying parameters are given as explicit functions of time with extended ranges of their variations, and (2) evolving, when these parameters vary very slowly with time, or they never need be defined as given functions of time. The former finds its applications in relatively early felt effects of the non-stationary regimes, particularly in structural mechanics subject to combustive propulsions, air and water waves, earth movements, etc. The qualitative and quantitative effects of robust non-stationary processes may be readily observed from the state variables or perturbation plots, such as lowering of the maximum response amplitudes and elimination of the fold discontinuities [l-7]. They are, however, difficult to analyse. The latter regime allows detailed analysis of such phenomena as cyclic folds (jumps), bifurcations, attractors, etc. It is useful in the analysis of systems with relatively slow but long time of accumulation of non-stationary effects [g-24]. The most commonly used robust non-stationary modes are: (1) linear variations of the frequencies v(t) and amplitudes B(t) of the external excitation (ee): namely, v(t) = v,, + r,t and B(t) = B, + z,t, called transition through resonances (TtR); and (2) cyclic variations of the frequencies V(C)= tlo + ysin sCt, where v,,, B,, xv, ~a, rC, and 7 are constants. The parameters z, and ra are called rates of sweep or simply sweeps. The first publication dealing with robust non-stationary responses is due to Lewis in 1932 [I]. The effects of linear transitions through resonances were applied to a linear singledegree-of-freedom system. The results show the characteristic robust non-stationary manifestation mentioned above. A well-known contribution to the literature on the robust nonstationary vibrations was made by Mitropolskii [2]. The Krylov-Bogolyubov asymptotic stationary methods were expanded to include non-stationary processes, resulting in what is known as the KBM method. Applications of special functions and asymptotic methods for solutions of non-stationary technical problems are found in the book by Goloskokow and Filippow [3]. It constitutes a valuable contribution to the literature on mechanics of TtR. Contributed by A. H. Nayfeh. 61
F. A. MOSLEHY
62
and R. M. EYAN-IW~NOWSKI
A review of non-stationary publications up to 1969 was prepared by Evan-Iwanowski [4]. A book containing work done at the Applied Mechanics Laboratory of Syracuse University was published by Evan-Iwanowski [S]. The KBM method has been expanded to include systems with multiple-degrees-of-freedom and the attendant combination resonances consisting of additive, differential and internal or autoparametric resonances. Lately. the main focus of research work on robust non-stationary processes was directed toward parametrically excited systems, i.e. toward the problems of dynamic stability [6,7]. The early developments on evolving non-stationary processes can be found in a paper by Kevorkian [S] for a linear undamped oscillator, followed by Ablowitz et al. [9] for the non-linear system with damping. It was found in [9] that non-stationary jumps may occur for positive sweeps for some values of the excitation amplitudes. The method of multiple scales was used. By appropriately selecting smallness of the magnitude of the involved parameters and the time scales, the authors succeeded in analysing the mechanisms of the non-stationary jumps very near the stationary discontinuity in detail. A similar approach was used by Collinge [lo] and Collinge and Ockendon [ll], with additional refinements to uncover details of the evolving jump phenomenon. Numerous extensions of this technique and the introduction of others were applied to solve some specific non-stationary problems governed by the Duffing differential equation. They cover many areas: physics (nuclear physics). fluid mechanics (sloshing), combustion (flames), chemistry, biology, and others [ 12-241. The results regarding prediction of the incipient jump are presented in a paper by Thompson and Virgin [2.5]. TtR was used to determine the plot of B(t) vs frequencies of the response relative to the stationary jump response, which tend to zero near the fold bifurcation. Topics of a genera1 nature on non-stationary processes may be found in the article by Kronauer [26] and in the text by Nayfeh and Mook [27]. From the available rest&s, it may be concluded that the non-stationary processes affect strongly the stationary manifestations, such as bifurcation, including Hopf bifurcation, attractors, including strange attractors (chaotic motion), domains of attraction, etc. Most, but not all, of the non-stationary effects are desirable. Moreover, the non-stationary processes may introduce strictly non-stationary phenomena which may be undesirable.
2. PRELIMINARIES
The Duffing equation is as follows: d2.u, mdtt + 2~2
+ OS, + d.x: = B, cos&t); b(t) = v. L
Introducing
non-dimensional
(1)
coordinates x=
d -x1
J 0
and
t=
Co
-t, J m
we obtain d”.u 2 + k$
+ .Y+ .X3= B cosB(t);
8(r) = ”
(2)
where
We consider a special Duffing equation introduced by Hayashi [28], which is often referred to as the Duffing-Hayashi equation, by dropping the linear term from (2): d2x dt2 + kg Ueda [24], followed by Thompson
+ .x3 = B costI(
8(t) = v.
and Stewart [29], found a specific set of initial
The Duffing oscillator
63
conditions and parameter values, k = 0.05;
B = 7.5; x,, = 3; i,, = 4;
v= 1
(4)
for which the response is a typical stationary chaotic motion. Finally, we cast (3) in a non-stationary form: j;_+kzi+x3=B(t)cosB(t),
(.)=-$
(5)
where k = 0.05, and B(t) and 8(t) = v(t) are now functions of time, specified in the text. It should be noted that the set of values in (4) is not unique. It is used here to minimize start-up transients and to determine the non-stationary effects on the known chaotic stationary motion. The following non-stationary processes are used in this paper: (i) linear variation of the frequency of the external excitation (TtR) v(t) = 1 + a,t;
(6)
(ii) linear variation of the amplitude of the external excitation (TtR) B(t) = 7.5 + cr,t;
(7)
(iii) cyclic variation of the frequency of the external excitation v(t) = 1.05 + ysin a,~; and
(8)
(iv) cyclic variation of the frequency of external excitations was also used for regular cases of different values of k and B, and initial conditions x0 and x0 are specified in Section 3, Results; c(,, LY.~, aC, and y are constants. Equation (5) written in terms of state variable representation i=y was used together calculations.
and
j=B(t)cosB(t)-ky-x3
with the fourth
3. RESULTS
order
Runge-Kutta
(9)
method
in the numerical
AND DISCUSSIONS
3.1. Efects of sweep rates a, and ug on chaotic motion To observe the effects of the non-stationary regimes on chaotic responses, the stationary chaotic response for CI,= 0 is presented first (Fig. 1). This figure agrees with the result of ref. [29]. The corresponding phase plane plot is shown in Fig. 2. Note the appearance of the first kinks (inflection points) in Fig. 1 at t = 32.5 and the corresponding near cusp in the phase plane (for discussion, see Section 3.4). The non-stationary results are presented next. For low sweep rates cr, = 10e5 or zR = 10V5, the non-stationary responses coincide with the stationary chaotic for a considerable time (Figs 3 and 4). The departure times between these two responses are given in Table 1. Detailed calculations reveal a slow build-up of the differences. The maxima of the responses correspond to the maxima of the excitation. There are additional smaller peaks. one, two, or three, between the larger peaks, indicating the existence of two or three different waveforms (patterns) (see Section 3.5). Similarly, the behavior close to the chaotic stationary motion was observed for u, = 10e4 or CQ,= 10s4. The departure, however, is somewhat earlier from the stationary or non-stationary slower sweep responses (Table 1). The non-stationary motion for CX,= lo-’ coincides initially with the stationary and previously described non-stationary cases (Fig. 5). For t = 37, however, it departs from the stationary, and at t between 152 and 200, it shows well-structured motion (Fig. 6). 3.2. EfSects of cyclic variations on stationary chaotic motion Non-stationary cyclic variations of the excitation frequencies have the following form: v(t) = v. + ysin a,t
(IO)
F. A. M~SLEHYand R. M. Eva%lwa~oww
64
Fig. I. Solutions
of
the
Dufiing
equation
(t) exhibiting Ueda &?4j.
stationary
CL,= 0
chaotic
motion,
after
6 -i
5-i 4
3 j 2 f 0
rx
-1 -2 -3 -4 -5 -6 i -4
-2
0
2
4
X Fig. 2. Phase portrait 3, = 0. Note the near-cusps, C, corr~~ondi~~
-4
0
I
20
/
40
,
60
I
80
I
100
/
120
/
140
to the kinks, K. in the response pfot.
,
160
/
180
t
2co
220
Fig. 3. Time history of the response, I, = 10-s. The non-stationary pfots are close to the chaotic responses for 1, = 4 but eventually they deparf (Tab,Ie I).
The DutIing oscillator
3 :
2
c
40
20
60
80
TOC
120
160
140
220
IEm
220
t Fig. 4. Time history ol the response, ~a = IO-s, has similar characteristics with 1, = 10 -5. See Table 1.
Table 1. Departure times for non-stationary chaotic responses from stationary for different z, and q,
I
-51 0
20
40
60
/ 80
I 1CO
, 120
I 140
I 160
I 200
180
220
t
Fig. 5. Non-stationary time history of the response, z, = 10-L. Initial chaotic motion shows early departures from the responses for z, = 0 or z, = lo-‘, Table 1, eventually exhibiting a structured motion for t 3 152.
Two cases are considered in this paper, see equation (5): (i) v0 = 1.05; 7 = 0.1; xc = 0.04; x0 = 3; i, = 4, and (ii) vg = 1.05; y = 0.1; zy, = 0.16; x0 = 3; -i-, = 4. For sl, = 0.04, after initial beat t < 10, the motion is different from the chaotic stationary or
66
Fig. 6, Segment
ofthe non-stationary
phase portrait for 2, = IO-‘. for 151.7 i t c 209, showing structured. close to harmonic motion with slight@ modulated displacement and velocity.
Fig. 7. Elrects of non-stationary cyclic regime xc = 0.04, ;’ = 0. t. on the stationary chaotic motion. Note an early and considerable departure from the stationary and linear non-stationary responses.
linear non-stationary motions for most of the time, with the peaks coinciding with those of the external excitation (Fig. 7). For LX,= 0.16, the non-stationary motion remains unspecified to the end of the recording period {Fig. 8).
Five cases, (l)-(5), of the regutar stable non-linear responses were selected by Ueda [24] to demonstrate the effect of initial conditions on the responses and the co-existence of various attractors in the Duffing oscillator: j; + ks + x3 = B cost: k = 0.08; B = 0.2. Below are presented the effects of the non-stationary regimes on these responses and on the corresponding attractors; v = ve + f-’ sina,t; ;’ = 0.1; xc = 0.04; ra = 1.05 for all cases (Figs 9-13). Case 1. Figure 9, x0 = 0.21; 1, = 0.02, shows the non-stationary effects on the stationary motion, small amplitude n = 1 attractor. For 20 c I < 40 and JO CCr < 60, two cases of *Cyclic non-srationary
regimes
are observed, lor instance. in non-ideal systems (see [Z) and [43).
The Duffing oscillator
Fig. 8. Effects of non-stationary
cyclic mode z, = 0.16, ;’ = 0.1. Almost instantaneous departure from the stationary chaotic and other non-stationary responses.
\I-_- ~_ ._ ._______._...-_13
67
._..
--------A
V
4
1
I
0
20
40
60
/
80
100
I
120
I
140
/
153
I
200
150
220
t Fig. 9. Case 1. x0 = 0.21, ;i, = 0.02; non-stationary,
n = 1 attractor,
with decreasing amplitude.
----
X
-4
/
0
/
20
1
40
t
60
/
a0
/
100
Fig. 10. Case 2. x0 = 1.05, k0 = 0.77; non-stationary,
ringing waveforms (see Section quency v(t). Eventually, for 80 c nized motion with the excitation n = 1 attractor with modulated
/
120
/
140
n = 1 attractor
/
160
/
180
zoo
I 220
with modulated amplitude.
3.4), appear with frequencies equal to the excitation fret < 200, symmetric slightly modulated harmonic synchrofrequency takes place with time lag = TI,terminating in an amplitude.
Case 2. Figure 10, x0 = 1.05, ;Ce= 0.77. The stationary motion is a simple harmonic, n = 1 oscillation with small amplitude. The non-stationary cyclic regime modifies this
6%
Fig. 1f. Case 3. x0 = 0.67, TQ=
0.02;non-stationery, n = I attractor with decreasing amplitude.
t Fig. $2. &se 4. x0 = 0.45, f, = &3Q after a brief chaotic motion non-stationary, with increasing ampiitudc
J 1
0
n = 2 attractor
..- ...-----._---..~_
20
40
60
80
i
700
120
140
160
2co
180
220
t Fig. 13. Case 5. x0 = - 0.43, & = 0.12; eon-statjonafy,
n = 1 attractor.
response considerably. At t = 22, 32 and 43, three kinks are observed. After that an asymmetrical stable motion appears with a three-peak pattern. Each peak coincides with the maximum of the excitation and is shifted by 71relative to it, terminating in an n = 3 attractor with moduIated amplitudes. Case 3. Figure II, x0 = 0.67, 2, = OB2. The stationary response is of tt = 2 type. The non-stationary motion exhibits two consecutive ringings at 20 c t < 42 and 42 < r c 70,
69
The Duffing osciiIator
with the frequencies of the excitation and different background waveforms. This is followed by asymmetric motion, n = 1 with time lag = K, terminating in an n = 1 attractor with slightly modulated amplitude. Case 4. Figure 12, _yO= - 0.46,5, = 0.30. The stationary response is of n = 2 type. For 25 < t < 200, an n = 3 attractor is appearing with modulated amplitude with the peaks corresponding to the frequency of excitation v(t).
Case 5. Figure 13, .x0 = 0.43, -ito= 0.12. The stationary motion is of n = 3 type. The initial non-stationary motion is followed by ringings for 44 < t < 70 and 70 < t < 100. Finally, for 100 < c < 200 an n = 1 response is settling in with slightly modulated amplitude. From the available sample, it is possible to conclude that the cyclic non-stationary mode may considerably modify the corresponding stationary (regular) responses, and no resemblance is observed from the very beginning between these motions; that is to say, the non-stationary responses form their specifically non-stationary attractors, different from the corresponding stationary attractors. 3.4. Structure of the chnotic responses One of the common waveforms (patterns) appearing in the responses of the Duffing osciflator are points of inflection or kinks. The appearance of kinks is very signi~cant as confirmed analytically by Novak and Frehlich [303 and observed experimentally by Robinson [12], because the kinks indicate the entry of even harmonics in addition to the previous single odd harmonics. The even- and odd-harmonic responses are then followed by period doubling which eventually leads to chaos. The kinks are easily identified as nearcusps or cusps in the phase plane plots (Figs 1 and 2). Another frequency appearing pattern in the phase plane is the closed loops corresponding to the two-hump waveforms in the response. The appearance in the Duffing oscillator of the ringings, i.e. high frequency oscillations superimposed on a background response could be seen, for instance, in Fig. 10 for 0 < t c 40. The ringings were also reported by Hijhler [ 133 and again by Robinson [12] for the stationary cases. A brief catalog of regular and chaotic stationary waveforms is found, for example, in the textbook by Kreuzer [31]. 3.5. ON-OFF chaotic responses ON-OFF or interrupted non-stationary regimes present a realistic situation in the real world. They introduce modifications to the continuous non-stationary response. This is demonstrated in Figs 14 and 15. The responses for the specific values of c(s within the
!
-1 X
0
20
40
60
80
100
120
140
160
~E)O
200
220
t QY - t_
0 +
m-3
Fig. 14. ON-OFF
-b+---
10-f--+--
non-stationary
w-2 -----q
0 ---+---
regimes as specified
in the
plot; k = 0.05;
B = 7.5.
Fig. IS. ON-OFF
corresponding responses.
non-statianary
regimes as specified in the plot; k = 0.05; B = 7.5.
time intervals are markedly different from the continuous non-stationary
4. CONCLUSiONS
From the results obtained in this investigation, some reveating and puzzling conclusions may be derived as foliows: (I) The stationary chaotic motion, sY = rR = 0, may be modified by introducing nonstationary linear processes; namely, by linear variations of the excitation frequencies and amplitudes in two ways: (i) Initially, for relatively small values of X”or xB of order IO-’ or less, the non-stationary and stationary chaotic motion are identical, then they depart after an extended period of time shown in Table 1. (ii) For xv or ~a approximately equal to lo-‘, the non-stationary responses remain initially close to stationary chaotic. They depart from them after some time {Table 1). That is, chaotic motion is being annihilated by means of a non-stationary process. This fact suggests that for a,,, rs z IO-j or below the nonstationary processes are evolving, and for zV, zil 2 10W2 or above they are robust. An important question is raised: what are the transition mechanisms from non-stationary chaotic to structured motions? More generally, what are the mechanisms of the onsets and terminations of chaotic motions in con-stationary processes? (2) Cyclic non-stationary processes investigated introduce at t = 0 different motions from the stationary chaotic and other non-stationary responses. They also strongly change the regular stationary non-linear responses and non-linear attractors. (3) The phase plane plots, particularly the segmented (detailed) plots, may be used to identify the patterns (waveforms) of the motion. Thus, kinks in the response plots correspond to the near-cusps in the phase plane plots, the two-hump waves, with loops, ringings with winding loops, etc. Appearances of some waveforms may forecast the incipient bifurcated motions. For instance, the appearance of the kinks may read eventually to chaos. (4) It is surprising indeed that on the one hand chaotic motions are quiet durable, that is they maintain their chaoticity inde~nitely~ and on the other hand they succumb relatively fast to non-stationary processes. An obvious question is posed: what are the other factors which one may introduce into a chaotic motion to eliminate chaos? (5) The ON-OFF regimes may be helpful to uncover the effects of initial conditions on the dynamic responses. They present a very puzzling picture, and need further study. (6) It is anticipated that by the accumulation of the results on the effects of the mm-stationary processes on the stationary regular and chaotic dynamical responses, it will be possible to detect new formulations and approaches for explanations of non-stationary behavior, in particular, to identify different waveforms (patterns) with the corresponding bifurcations or limit sets. This is significant from the basic mechanics point of view and applications as well. One of the basic questions may be, for example, stated as follows:
The Duffing osciliator
71
determine the thresholds between the evolving and robust non-stationary processes, i.e. determination of the thresholds of zV, rB or q. In application, non-stationary processes offer numerous and substantial mechanical advantages such as elimination of catastrophic jumps, reductions of the maximum stationary response amplitudes, annihilation of chaos, control of the responses, limitation of maxima of the response amplitudes, prediction of the corresponding CX,,rB or “C for which the maxima are attained. ~~~~~~~~~~~e~~~[-T~eauthors are grateful to Mr Shahram Etmi for his assistance in the computer work.
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