The nonstationary effects on a softening duffing oscillator

Medianics Reseatd, Communication.~,Vol. 21, No. 6, pp. 555-564, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/94 $6.00 + .00

Pergamon

0093-6413(94)00018-2

The Nonstationary Effects on A Softening Duffing Oscillator

C.-H. Lu Assistant Professor Department of Mechanical Engineering The University of Memphis Memphis, TN 38152

R.M. Evan-Iwanowski Professor Dept. of Mechanical & Aerospace Engineering University of Central Florida Orlando, Florida 32816

(Received 11 March 1994; accepted for print I July 1994)

Abstract

This paper presents a numerical study for the bifurcations of a softening Duffing oscillator subjected to stationary and nonstationary excitation. The nonstationary inputs used are linear functions of time. The bifurcations are the resuRs of either a single control parameter or two control parameters that are constrained to vary in a selected direction on the plane of forcing amplitude and forcing frequency. The results indicate: 1. Delay (memory, penetration) of nonstationary bifurcations relative to stationary bifurcations may occur. 2. The nonstationary trajectories jump into the neighboring stationary trajectories with possible overshoots, while the stationary trajectories transit smoothly. 3. The nonstationary penetrations (delays) are compressed to zero with an increasing number of iterations. 4. The nonstationary responses converge through a period-doubling sequence to a nonstationary limit motion that has the characteristics of chaotic motion. The Duffmg oscillator has been used as an example of the existence of broad effects of nonstationary (time dependent) and codimensional (control parameter variations in the bifurcation region) inputs which markedly modify the dynamical behavior of dynamical systems.

555

556

C.-H. LU and R.M. EVAN-IWANOWSKI 1. Introduction

One of the main objectives in the study of a dynamical system is the determination of bifurcations and their topological characteristics. The bifurcations are defined as a set of values of the control parameters, appearing in the governing equations, for which marked changes in trajectories take place. A widely used technique in the investigation of this behavior is to plot the response of a dynamical system via the Poincar6 section with respect to the variations of a control parameter, resulting in the bifurcation diagrams. If some control parameters are functions of time, the dynamical system is said to be nonstationary. In some differential or difference equations containing one or more control parameters, period-doubling bifurcations may appear. This phenomenon has been observed in many engineering and physical systems and confirmed by numerous experiments (Dowell and Pezeshki, 1988; Anderson and Tadjbakhsh, 1989; Moon, 1992). The period-doubling bifurcation cascade in stationary systems is known as the most common route to stationary chaos; it has received considerable attention by many researchers of dynamics and chaos (Feigenbaum, 1980; Linsay, 1981; McLaughlin, 1981; Novak and Frehlich, 1982; Perez and Glass, 1982; Swift and Wissenfeld, 1984; Erneux and Mandel, 1986; Byatt-Smith, 1987; Kapitaniak, 1988; Szemplinska-Stupnicka, 1987; Van Dooren and Janssen, 1990). Exhaustive information on this subject at the level of theoretical mechanics and engineering has also been presented in a number of monographs (Guckenheimer and Holmes, 1983; Berg6 et al., 1984; Thompson and Stewart, 1986; Seydel, 1988; Wiggins, 1988; Tabor, 1989; Rasband, 1990; Moon, 1992). In nonstationary systems, control parameters continuously change with time. Two classes of nonstationary regimes are discernible: (i) evolving or slowly varying (imperceptible), whereby the control parameters need never be given as explicit functions of time; and (ii) robust, when the nonstationary control parameters are explicit functions of time defined over extended ranges. The general form of a nonstationary robust process is as follows: CPNs = CPo + W(t) where CPNs are nonstationary control parameters, CP o are the initial values (usually stationary values) and ~(t) are the nonstationary time-dependent or process-dependent arbitrary functions, which mostly are linear, cyclic, exponential, or impulsive. The codimension 4)( CP1, CP2,..., CPn) = 0 describes the relationship that may exist among the control parameters. Nonstationary processes occur naturally in the engineering and physical world: in the movements of water, earth, and air; in start-up and shut-down of engines; in laser beams and nuclear reactors; in thermodynamics-combustion; in fluid mechanics-turbulence; in chemical and biochemical systems; in rheostats, control, and switching devices; in charges passing through a nerve membrane; in the brain neural network, etc. Often, nonstationary inputs are introduced intentionally into experimental work in order to uncover possible changes that may offer more satisfactory explanations of the observed physical phenomena. Most of the results on the nonstationary responses of the regular, non-chaotic, mechanical systems are found in two monographs (Mitropolskii, 1965; Evan-Iwanowski,

SOFTENINGDUFFINGOSCILLATOR

557

1976). Characteristic manifestations of nonstationary regular responses are: (1) the phenomena of penetration (delay or memory); (2) the elimination of stationary response discontinuities (jumps); (3) the appearance of drag-in and drag-out phenomena; (4) reduction of maxima of the stationary amplitudes; (5) the existence of various separatrices that diverge from finite responses. Relatively little has been accomplished in nonstationary regimes related to recent developments in nonlinear dynamics and chaos. Recent research indicates that chaotic behavior can be generated by linear nonstationary inputs in relaxation systems (Tran and Evan-Iwanowsld, 1990); that nonstationary processes affect the strange and regular stationary attractors in Duffing oscillators (Moslehy and Evan-Iwanowski, 1991); and that the transitions through period-doubling to chaos in nonstationary processes are quite different from those in stationary cases for Duffing oscillators (Evan-Iwanowski and Abhyanker, 1990; Evan-Iwanowski and Ostiguy, 1990). In this paper, a study has been made for single- and two-control parameter codimension-one bifurcations in the ~ D)-plane of a softening Duffing oscillator with nonstationary excitation: ~i(t)+ax(t)+bYc(t)-flx3(t)= f(t)cosO(t), O(t)=D(t) (1) where a = ,8 = 1.0 and b = 0.4. The stationary form of equation (1), i.e., when f and £2 are constants, has been investigated by Nayfeh and Sanchez (1989) for modeling the capsizing of a vessel in beam seas. They showed the existence of stationary period-two (2T) bifurcation in ~ f~)parameter space, together with other types of bifurcations along a Y-shaped line. The Yline contains a line segment, called the L-line, defined a s f = -0.414 + [2 forf~(0.361, 0.386) and f1~(0.775, 0.800). For nonstationary processes along the L-line, j(t) = -0.414 + fl(t). The phenomena of nonstationary bifurcations, penetrations, and other nonstationary behavior have been studied by Lu and Evan-Iwanowski (1994). The objective of this study is to determine nonstationary effects on the eodimensionone stationary bifurcations in the ~, f2)-parameter space of equation (1). Single-control parameter bifurcations along the D-line, fl(t) = D o ± tzt, f = constant and two-control parameter bifurcations along the E-line for f~(t) = f2o + o~t and f(t) = fo:l=ott, which traverses the L-line, will be used in equation (1). The stationary period-doubling (T, 2T, 4T, etc.) bifurcation regions of the Duffing softening oscillator is presented in Section 2. Section 3 includes the nonstationary bifin'eations along the fl-line for various sweep rates ix. The nonstationary processes for the two-control parameters codimension-one are described in Section 4, with conclusions in Section 5.

2. Stationary Bifurcation Diagrams in the Softening Duffing System The stationary form of equation (1) is: £(t) + ax(t) + bYe(t) - f i x 3(t) = f cos D.t

(2)

where a = ff = 1.0, b = 0.4,f= constant and f~= constant. Integrating this equation by the

558

C.-H. LU m~d R.M. EVAN-IWANOWSKI Adams-Moulton method with f = 0.361 and taking Poincar6 sections by sampling x at tn ~: 2nx/~,

n = 1, 2, ..., we obtain a bifurcation diagram along the fl-line, as shown in Fig. I.

For increasing ~ the stationary chaos transits toward the T-region, and for decreasing stationary T transits toward the chaotic region. This diagram shows that the perioddoubling bifurcation sequence may exist on the two branches: the upper branch and the lower branch of the (x, fl) curve -- a mirror image or flip-flop response. Fig. 2 shows the stationary T, 2T, 4T, .... chaos bifurcation regions in the ~ fl)parameter space of equation (2). It contains the L-line, along which a simple analytical relationship betweenfand ~ exists f = - 0.414 + ~

forfc(0.361, 0.386) and fl~(0.775, 0.800)

(3)

The L-line is in essence the locus of the points of 2Tbifurcations. To study bifurcations related to the time variations of the two control parameters, we define an E-line, which is perpendicular to the L-line, in the following form: f = I. 166- F2

(4)

Along the E-line, the control parameters f and f~ move transversely across the stationary T, 2T, 4T, .... chaos regions. Whenfincreases and fl decreases simultaneously so as to satisfy equation (4), the bifurcations are moving toward chaos. Conversely, when f decreases and f2 increases, the bifurcations move toward T. The bifurcation diagram along the E-line is similar to that shown in Fig. 1, with the exception that the region of chaotic motion appears for lower values of ~1. This is due to the joint effect of decreasing f~ and increasing f, each of them makes the bifurcations move toward chaos.

3. Nonstationary Bifurcations in the Softening Duffing System along the ~l-line We now study nonstationary effects on the softening Duffing system using equation (1) along the f2-1ine: fl(t) = f2o + cxt a n d f = 0.361, for sweep rates: ot = 10 -7, l0 -6, and 10-5, with initial conditions x(0) = -0.7222, £(0) = 0.4419. The nonstationary responses x are evaluated at the values t =

tn

corresponding to the maxima of the forcing cosD(tn) = 1.

For f2(t) = ~o + out, the values o f t

n

are:

SOFTENING OF DUFFING OSCILLATOR

t,-

,

2a

559

c t ~ 0 a n d n = 1,2 ....

(5)

ct ~ 0 a n d n = 1, 2 ....

(6)

and for fl(t) = ~1o - at, tn =

Do

- 8,.ra

2a

,

For ot = 0, we have a stationary case and

tn =

2mr./f~o.

In the case that El(t) = f~o " at, fl o = 0.805 and f = 0.361, Fig. 3 shows the nonstationary responses and part of the corresponding stationary bifurcation diagram. For <~= 10-7 the nonstationary response starts with T, proceeds to the upper branch of the (x, ~ ) curve, transits to 2T, 4/', ..., and ends by forming a wide band of points that is termed the nonstationary limit motion.

The case for ct = 10-6 presents a similar picture.

However, for (x = 10"5, the nonstationary T response drops to the lower branch of the (x, ~ ) curve. This figure also show that once a nonstationary bifurcation diagram jumps or drops to a branch, it converges through the period-doubling sequence to the nonstationary limit motion on the same branch.

The nonstationary trajectories jump or drop to the

neighboring stationary trajectories with overshoots, while the stationary branching transits smoothly from T to 2T, from 2T to 4T, .... etc.

The transitions of nonstationary

bifurcations from T t o 2T, from 2 T t o 4T, etc., occur earlier than those of the stationary, exhibiting the phenomenon of anticipation.

This behavior is different fi'om the results

obtained by Lu and Evan-Iwanowski (1994) forf(t) = f o ± a t and ~1 = constant. They showed that there are no overshoots of the jumps or drops of nonstationary branching to neighboring stationary ones and that the transitions of nonstationary bifurcations always occur later than those of the stationary (the delays or penetrations). Comparing the three nonstationary responses in Fig. 3, it can be observed that the nonstationary bifurcations appear later for larger sweep rate, a typical penetration phenomenon. The larger the value of the sweep the deeper the nonstationary penetration. The number of successive penetrations, however, decreases with increasing sweeps, and eventually tends to zero. This fact allows the nonstationary bifurcation to converge to a

560

C.-H. LU and R.M. EVAN-IWANOWSKI limit motion.

Some penetrations, e.g., for ot = 10 -5, are so vigorous that their

nonstationary sequences T directly transit to the nonstationary limit motion. For f2(t) = ~ o + at, the nonstationary responses are extremely sensitive to the stationary values o f f 2 o. With f~o = 0.771 and cx = 10 -7, 10 -6, 10 -5, Fig. 4 demonstrates the following four points. bifurcations.

1. There are no jumps or drops in the nonstationary

2. The nonstationary limit motion (chaotic region) in the nonstationary

bifurcations is narrower than the stationary one.

3. The escape from chaos can be

accomplished by means of a nonstationary process proceeding toward periodic motion (reversed sweep).

4. The phenomenon of nonstationary penetration from 2T to T,

although not clearly seen for ~t = 10"Tand 10-6, is distinctly visible for ¢x = 10 -5

4. Nonstationary Bifurcations Along the E-line: ~l(t) = f2 o ± oJ, f(t) = fo-T-o.t in the Softening Duffing System According to the stationary relation between f and ~ on the E-line, equation (4), we determine the nonstationary form as follows: f(t) = 1.166 - ~(t)

(8)

where f2(t) = ~ o + at. The initial conditions (x(0) and ~ (0)) and the sweep rates et are the same as those used in Section 3. Figs. 5 and 6 show the nonstationary bifurcations for the three sweep rates for ~(t) = f2o-e~t with f2 o = 0.805 and ~(t) = f2 o + o t t with f2 o = 0.7883, respectively.

In these

figures, part of the stationary bifurcation diagram of the E-line is also included.

The

general nonstationary dynamical characteristics shown in these figures are similar to what has been observed in Figs. 3 and 4. For decreasing f~(t) the nonstationary bifurcations transit through period-doubling with characteristic penetrations and jumps (drops) to nonstationary limit motion.

For increasing f2(t), the system escapes from the

nonstationary limit motion to nonstationary periodic motion with no jumps (drops). The nonstationary solutions may live on the upper branch or the lower branch depending on the values of initial conditions (x(0) and ~ (0)), sweep rates and f2 o.

SOFTENINGDUPING OSCILLATOR 5. Conclusions

One- and two-control parameter codimension-one bifurcations of a so~ening Duffing oscillator is studied. The following points have been found. 1. Nonstationary perioddoubling bifurcations exhibit a sequence of successive penetrations from nonstationary T to 2T, from nonstationary 2T to 4T, etc. They disappear eventually in a maze of nonstationary 2nT bifurcations leading to a nonstationary limit motion. 2. In some cases, the penetrations are sufficiently vigorous that the nonstationary T may directly transit to a nonstationary limit motion. 3. The jumps (or drops) appear in nonstationary bifurcation branching when the bifurcations proceed toward chaos. 4. An escape from chaos can be accomplished when a reversed nonstationary process moves toward the periodic motion, in which case no jumps (drops) have been observed. Further progress on nonstationary processes hinges on future developments in the mathematical theory of differential equations with variable coefficients. In the meantime, the authors found that direct numerical integration allows one to uncover a wealth of new and diverse nonstationary dynamical behavior. It should also be noted that the nonstationary processes, although essentially transient, are different from the common transient phenomena. The former preserve some basic long-term characteristics of the associated stationary processes; the latter are usually ignored or rejected. A cardinal question concerning nonstationary behavior remains open; namely, do stationary dynamics constitute a limiting case of nonstationary dynamics?

References

Anderson, G.L,. and Tadjbakhsh, I.G., 1989, "Stabilization of Ziegler's Pendulum by Means of the Method of Vibration Control," £ Math. Anal. ,4pplic. 143, 198-223. Bergr, P., Pomeau, Y., and Vidal, C., 1984, Order Within Chaos, John Wiley and Sons, New York. Byatt-Smith, J.G., 1987, "2x Periodic Solutions of Duffing's Equation with Negative Stiffness," SIAM J. Appl. Math. 47, 60-91. Dowell, E.H., and Pezeshld, C., 1988, "On the Understanding of Chaos in Duffing's Equation Including a Comparison with Experiment," £ Appl. Mechanics, 53, 5-9. Emeux, T., and Mandel, P., 1986, "Imperfect Bifurcation with a Slowly Varying Control Parameter," SIAM £ Appl. Math. 46, 1.

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C.-H. LU and R.M EVAN-IWANOWSKI Evan-Iwanowski, R.M., 1976, Resonance Oscillations in Mechanical Systems, Elsevier, Amsterdam. Evan-Iwanowski, R.M., and Abhyanker, NS., 1990, "Non-Stationary Effects on the Chaotic Duffing Oscillator," Proceedings of CSME Forum, 1-6. Evan-Iwanowski, R.M., and Ostiguy, G.L., 1990, "Recent Developments and an Agenda for Nonstationary Chaotic Motions," The VPI Conference on Nonlinear Vibrat., 1-10. Feigenbaum, M.J., 1980, "Universal Behavior of Nonlinear Systems," Los Alamos Science 1, 4-27. Guckenheimer, J., and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York. Kapitaniak, T., 1988, "Combined Bifurcations and Transition to Chaos in a Non-Linear Oscillator with Two External Periodic Forces," J. Sound Vibrat. 121, 259. Linsay, P.S., 1981, "Period Doubling and Chaotic Behavior in a Driven Anharmonic Oscillator," PhysicalReview Letters 47, 1349. Lu, C.-H., and Evan-Iwanowski, R.M., 1994, "Period Doubling Bifurcation Problems in the Softening Duffing Oscillator with Nonstationary Excitation," Int. J. of Nonlinear Dynamics (in press). McLaughlin, J.B., 1981, "Period-Doubling Bifurcations and Chaotic Motion for a Parametrically Forced Pendulum," J. Statistical Physics 24, 375. Mitropolskii, Y.A., 1965, Problems of the Asymptotic Theory of Non-Stationary Vibrations, Daniel Davey, New York. Moon, F.C., 1992, Chaotic and Fractal Dynamics, John Wiley and Sons, New York. Moslehy, F.A., and Evan-Iwanowski, R.M., 1991, "The Effects of Non-Stationary Processes on Chaotic and Regular Responses of the Duffing Oscillator," Int. J. Nonlinear Mechanics 26, 61-71. Nayfeh, A.H., and Sanchez, N., 1989, "Bifurcations in the Forced Softening Duffing Oscillators," International J. Non-Linear Mechanics 24, 483-497. Novak, S., and Frehlich, R.G., 1982, "Transition to Chaos in the Duffing Oscillator," Physical Review Letters A26, 3660. Perez, R., and Glass, L., 1982, "Bistability, Period-Doubling Bifurcations and Chaos in a Periodically Forced Oscillator," Physics Letter A. 90A, 441. Rasband, S.N., 1990, Chaotic Dynamics of Nonlinear Systems, John Wiley and Sons, New York. Seydel, R., 1988, From Equilibrium to Chaos, Springer-Verlag, New York. Szemplinska-Stupnicka, W., 1987, "Secondary Resonances and Approximate Models of Routes to Chaotic Motions in Non-Linear Oscillators," J. Sound Vibrat. 113, 155. Tabor, M., 1989, Chaos and Integrability in Nonlinear Dynamics, John Wiley and Sons, New York. Thompson, J.M.T., and Stewart, H.B., 1986, Nonlinear Dynamics and Chaos, John Wiley and Sons, New York. Tran, M.H., and Evan-Iwanowski, R.M., 1990, "Non-Stationary Response of Self-Excited Driven System," Int. J. Nonlinear Mechanics 25, 285-297. Van Dooren, R., and Janssen, H., 1990, "Period-Doubling in the Duffing Oscillator: A Galerkin Approach," J. Computational Physics 52, 161. Wiggins, S., 1988, Global Bifurcation and Chaos, Springer-Verlag, New York.

SOFTENING OF DUFFING OSCILLATOR

563

-0.2

-0.4

x

-0.6

i

-0.8

0

-I

i

J

0.773

0.765

0.7~1

,

L

i

0.789

0.797

0.805

0

Figure 1.

Stationary bifurcation diagram along the Q-line.

0.39

•..

~ .:t:~4T "". '-.

0.38

f 0.37

. o ~ , ¢/2 T x,~-'," •. c,<;,~:/ / ~ ..~?;:;2" / T

.....~ h /

'~ . re';.. ~...

.%,~>," , -,~ ~;,'<,,e:

,~" 0.36 /

/

0.3 . 7

Figure 2.

%;3.. T Response 2T Response 4T Response

0.78

0.79 Q

0.81

Stationary period-doublin 8 region s and their boundaries.

-0.2

Stotionary / j./~/, limit motion . 11ii~

(choos)

-o.~

#-

Nonstatlonory limit motion

i:"

:~

"

c¢=10-7

•...~f x

0.80

-0.6

a=O

~=, o - ~ , ~ , ~

'A

/

-0.8 ~:10-5

."~ -1.0

Figure 3.

i

~

=

r

f

0.77

0.78

0.79 0

0.80

0.81

Bifurcations alon8 the Q-line, f = 0.361, Q = 0.805-~t for nonstationary cases and ~ = 0 for the stationary case in which Qe[0.771, 0.805].

564

C.-}I.

LU and R.M.

EVAN-IWANOWSKI

o=0 Stotionclrv limit m o t i o n

-0.2

(,:taos)

I

:~,.> . . . .

-0.4

,,o~,o

1.

,3 J

e ×

'q

/ Nonsl(dion~]~y ~

-0.6

, 1 limit

rrlolior;

~-V

-

/

-

/"

~

/

c, " 1 1:3 ' ~ /

-0.8

--1 0

I 0.77

I 0 78

I 079

, 080

I 0.81

0

Figure 4.

Bifurcations along the f~-line, f = 0.361, ~ = 0.771+m for nonstationary cases and ~ = 0 for the stationary case in which ~ e [ 0 . 7 7 1 , 0.805]_

--02

./IXdonsb)tlonc~ry bruit m o t i o n

'::.,~

x

-0.6

Stationory ; ~-5,~ ." ",:~

limit m o t i o n

i (chaos)

r

. . ~

::~..~::

•:.i~ 0~=0 :):""

-1 0

~ = 1 O-

i'

0.787

i

0 292

0

~

29]

0.802

, 0.807

/)

Figure 5.

Bifurcations along the E-line, f (t) = 1.166-&Q(0 , &Q(I) = 0.805-ott for nonstationary cases and cc = 0 for the stationary case in which ~ e [ 0 . 7 8 8 3 , 0.805].

-0.2

~ 0,4

......

¢~=10 .5

i~i'~ ~\i'.......... ~">~' ~::'2~ ~

ir

j

~

~=o

~ 0 ~~

S

~

{~=lO-7
-

\

-------/

//

\ Nonstalionmy --1 .0

0.787

Figure 6.

I

0792

lirHi[ m o t i o n I

I

I

() 797 0

0.802

0.807

Bifurcations along the E-line, f (t) = 1.166-~(0, ~(t) = 0.7833+ott for nonstationary cases and ¢t = 0 for the stationary case in which ~ e [ 0 . 7 8 8 3 , 0.805].