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Periodic attractors of complex damped non-linear systems
International Journal of Non-Linear Mechanics 35 (2000) 309}323
Periodic attractors of complex damped non-linear systems Gamal M. Mahmoud *, S.A. Aly Department of Mathematics, Faculty of Science, University of Assiut, Assiut 71516, Egypt Department of Mathematics, Faculty of Science, University of Al-Azhar, Assiut 71511, Egypt
Abstract The aim of the present paper is to investigate the dynamics of a class of complex damped non-linear systems described by the equation zK #uz#ez f(z, z , z , z )P()t)"0,
(*)
where z(t)"x(t)#iy(t), i"(!1, the bar denotes the complex conjugate and e is a small positive parameter. The periodic attractors of Eq. (*) are important in the study of these systems, since they represent stationary or repeatable behavior. This equation appears in several "elds of physics, mechanics and engineering, for example, in high-energy accelerators, rotor dynamics, robots and shells. In the numerical investigation of this work we used the indicatrix method which has been introduced and extended in our previous studies to study the existence of the periodic attractors of our systems. To illustrate these periodic attractors we constructed Poincare& plots at some of the parameter values which are obtained by the indicatrix method for the case u , f""z" and P()t)"sin 2t as an example. Our recent method which is based on the generalized averaging method is used to obtain approximate analytical solutions of Eq. (*) and investigate the stability properties of the solutions. We compared the analytical results of our example with the numerical results and excellent agreement is found. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Periodic attractors; Complex systems; Non-linear; Damped
1. Introduction Recently, approximate analytical and numerical methods have been applied to conservative complex non-linear dynamical systems, and some of their periodic solutions and their stability properties have been studied in detail in Refs. [1,2]. In the present work we wish to turn to the investigation of the dynamics of complex damped systems, which displays many interesting features, di!erent from the conservative cases. Periodic solutions are important in the study of dynamical systems, since they represent stationary or repeatable behavior. Any practical system always contains dissipation.
* Corresponding author. Tel.: 002-088-312564; fax: 002-088-312564. E-mail address: [email protected] (G.M. Mahmoud) 0020-7462/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 9 ) 0 0 0 1 6 - 5
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Herein we study a class of complex damped non-linear systems described by the equation: zK #uz#ez f(z, z , z , z )P()t)"0, 0(e;1,
(1.1)
where z(t)"x(t)#iy(t) is complex with i"(!1 the imaginary unit, z is the complex conjugate function, the frequency u is a constant, f is non-linear analytical function of its arguments, P()t) is a periodic function of frequency ), and dots denote time derivatives. Eq. (1.1) describes a damped model for which the periodic solutions we seek are attractors (see Fig. 2). This equation appears in several "elds of physics, engineering, electronics and statistical mechanics (eg. in high-energy accelerators, rotor dynamics, robots and shells [1}3]). For example, the connection of Eq. (1.1) to the problem of colliding beams has been discussed in Refs. [3,4], while its connection to rotor dynamics is found in Refs. [5,6]. The mathematical models of robots and shells are stated in Ref. [7]. Interesting examples of damped systems have been the subject of many papers (see for example, Refs. [8}19]). Forces that are functions of the velocity are called damping forces. Many researchers have used damping forces such as x f(x) or x g(x, x ), where f(x) and g(x, x ) are analytical functions of their arguments [13}18]. This paper is divided into two parts. The "rst part deals with the numerical investigations of periodic attractors of Eq. (1.1). In this part the indicatrix method which has been introduced and developed in our pervious work [19,20] is applied to study the existence of the periodic attractors of Eq. (1.1). Periodic attractors with period 4n of Eq. (1.1) are studied for the case u , f""z" and P()t)"sin 2t as an example (other cases of u, f and P can be similarly studied). Using this method which is a numerical technique, we "nd two periodic attractors with period 4n for each value of e, one for u(, and the other for u'. These periodic attractors are studied here on the Poincare& surface of section: R"+x(t ), x (t ), t "t #kn, I I I k"0, 1, 2,2, and R"+y(t ), y (t ), t "t #kn, k"0, 1, 2,2,. The parameter values which are suggested I I I by the indicatrix method are used to study the stability properties of these attractors. The second part is concerned with the analytical investigation of these periodic attractors. Our method which is based on the generalized averaging method [1,2] is used to derive approximate analytical solutions of Eq. (1.1). Expressions for the frequency u, valid to second order in e, are obtained for the above example. These expressions are in very good agreement with the results by the indicatrix method of part 1 (up to e)0.15). Approximate formulas are derived for the solution z( (t)"x( (t)#iy( (t) of our example. These formulas are used to derive the stability regions for our periodic attractors with period 4n which turn out to be accurate when compared with the numerical results. Some concluding remarks are presented in Section 4.
2. Numerical investigations In this part we apply the indicatrix method [19,20] for numerically "nding the periodic attractors of a class of complex damped non-linear systems (1.1). For eO0, the sets of points (x(t #¹), x (t #¹)), (y(t #¹), y (t #¹)) with period ¹"2n/u , u u "n/m, (n, m are positive integers) form closed curves parameterized by t , t 3[0, 2n/)] in the spaces (x, x ), (y, y ), respectively, which may or may not pass through the speci"ed initial conditions. These closed curves are called the indicatrices. By varying slightly u and e, we obtain the periodic components x(t), y(t) when the indicatrices in the spaces (x, x ), (y, y ) are observed to pass through the prescribed initial conditions at some values of t , t . This can happen, of course, for more than one set of values of t , t . In this way we obtained the numerical values of the parameters u, e at which periodic components x(t), y(t) of the solutions of Eq. (1.1) with period ¹"2n/u pass through the speci"ed initial conditions (x(t ), x (t ), y(t ), y (t )).
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To explain this method in detail, consider the following damped non-linear system zK #uz#ez "z" sin 2t"0,
(2.1)
where u u " (other values of u can be similarly studied). It is noted that Eq. (2.1) is a special case of Eq. (1.1) with f""z" and P()t)"sin 2t (other cases of f and P()t) can be similarly treated). Eq. (2.1) is a system of two-coupled non-linear second-order di!erential equations: xK #ux#ex (x#y) sin 2t"0,
yK #uy#ey (x#y) sin 2t"0.
(2.2)
Using the indicatrix method we shall determine the values of the parameters u, e at which periodic attractors of Eq. (2.2) with frequency pass by a speci"ed initial conditions e.g. x(t )"!0.1, x (t )"0, y(t )"0, y (t )"0.3. For eO0 the sets of points (x(t #4n), x (t #4n)), (y(t #4n), y (t #4n)) form closed curves parametrized by t , t 3[0, n] in the spaces (x, x ) and (y, y ), respectively, that may or may not pass through our speci"ed initial conditions. These closed curves (indicatrices) are plotted here in Fig. 1 for e"0.1. By varying slightly u and e, we obtain the desired periodic attractors with period 4n when the indicatrices are observed to pass through our initial conditions ((!0.1, 0), (0, 0.3)) at some values of t , t 3[0, n]. This can happen, of course, for more than one set of values of t and t , see Fig. 1a and Fig. 1b for
Fig. 1. Indicatrix plots of Eq. (2.2) for e"0.1, t , t 3[0, n]: (a) u"0.24776: (!0.1, 0) at t "0 (x, x space); (b) u"0.24776: (0, 0.3) at t "n/2 (y, y space); (c) u"0.2521823: (!0.1, 0) at t "n/2 (x, x space); (d) u"0.2521823: (0, 0.3) at t "0 (y, y space).
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Table 1
e
0.001 0.01 0.02 0.04 0.05 0.06 0.08 0.1 0.12 0.14 0.15
u
t ,t
0.249978 0.2500219 0.24978 0.250219 0.249562 0.2504378 0.249124 0.2508746 0.248905 0.251094 0.2486856 0.251312 0.2482467 0.2517465 0.24776 0.2521823 0.247365 0.2526171 0.2469273 0.253054 0.246706 0.2532679
0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0
t "0, t "n/2, e"0.1 and u"0.24776 and Fig. 1c and Fig. 1d for t "n/2, t "0, e"0.1 and u"0.2521823. We study in this way the existence of the periodic attractors with period 4n. We "nd, with the indicatrix method, two periodic attractors for each value of e, one for u( (t "0 and t "n/2) and the other for u' (t "n/2 and t "0), see Table 1. To illustrate these periodic attractors we use the Poincare& map associated with Eq. (2.2) at the parameter values suggested by the indicatrix method. In fact, using the Poincare& map one can even determine the stability properties of these periodic attractors under small perturbation in the initial condition. Using Poincare& map, or the (x, x ) surface of section R"+[x(t ), x (t )], t "t #kn, k"0, 1, 2,2,, since I I I our system (2.2) is symmetric, we "nd that for several initial conditions the origin is a stable period n orbit and there are two periodic attractors with 4n-period, one of them is stable (marked by 䢇) and the other unstable (marked by ;), see Fig. 2 for e"0.1 and (a) u"0.24776, (b) u"0.2521823. In Fig. 2a we take three initial conditions, (x(0)"0.5, x (0)"0), (x(0)"0.3, x (0)"0), (x(0)"0.25, x (0)"0) and y(t )"y (t )"0, while in Fig. 2b, (x(n/2)"0.5, x (n/2)"0), (x(n/2)"0.35, x (n/2)"0), (x(n/2)"0.25, x (n/2)"0) and y(t )"y (t )"0. The periodic attractors can be viewed here on the Poincare& surface of section as a set of distinct points. We have four points marked by 䢇 for the stable periodic attractor and four points marked by ; for the unstable one since the period of these attractors is 4n. This surface of section shows those attractors in Fig. 2a and Fig 2b. The values of u and e in Fig. 2a correspond to an unstable periodic attractors while in Fig. 2b correspond to the stable one. It is noted that for the initial conditions with y(t )'0, y (t )"0, we obtain a similar "gure as in Fig. 2. In the numerical calculations we used the method of Runge}Kutta of order 4. In the next part we shall present our analytical investigations and a comparison with the results of this part for these periodic attractors.
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313
Fig. 2. Surface of sections of Eq. (2.2) for e"0.1 and: (a) u"0.24776, y(t )"y (t )"0, t "0 (unstable); (b) u"0.251823, y(t )"y (t )"0, t "n/2 (stable). A stable (marked by 䢇) and an unstable (marked by ;) periodic attractors with
4n-period.
3. Analytical investigations Using the approximate method which is based on the generalized averaging method [1,2], we study in this part the periodic attractors of a class of complex damped non-linear systems whose equation of motion is described by Eq. (1.1). This method starts form the so-called generating solutions z(t)"A exp[i(ut#h )]#A exp[!i(ut#h )], z (t)"iuA exp[i(ut#h )]!iuA exp[!i(ut#h )] (3.1) which satisfy Eq. (1.1) to zero-order (e"0). The constants A and h (i"1, 2) can be determined by the initial G G conditions. In order to solve Eq. (1.1), it is assumed that the constants A and h depend on time, so that in Eq. G G
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(3.1) A PA (t), h Ph (t). Then Eq. (3.1) becomes G G G G z(t)"A (t) exp[i (t)]#A (t) exp[!i (t)],
(3.2a)
z (t)"iuA (t) exp[i (t)]!iuA (t) exp[!i (t)]
(3.2b)
with
(t)"ut#h (t), i"1, 2. G G
(3.3)
By di!erentiating Eq. (3.2a) with respect to t and comparing with Eq. (3.2b) and di!erentiating Eq. (3.2b) and substituting into Eq. (1.1) we get 2[AQ (t)#iA (t)hQ (t)]"!efp()t)+A (t)!A (t) exp[!i( (t)# (t))],, 2[AQ (t)!iA (t)hQ (t)]"efp()t)+A (t) exp[i( (t)# (t))]!A (t),.
(3.4a) (3.4b)
Splitting Eqs. (3.4a) and (3.4b) into real and imaginary parts we get a system of equations which we solve for AQ (t) and hQ (t) to obtain G G e AQ (t)"! Re+ fp()t)(A (t)!A (t) exp[!i( # )]),, 2
(3.5a)
e AQ (t)" Re+ fp()t)(A (t) exp[i( # )]!A (t)),, 2
(3.5b)
e hQ (t)"! Im+ fp()t)(A (t)!A (t) exp[!i( # )]),, 2A (t)
(3.5c)
e hQ (t)"! Im+ fp()t)(A (t) exp[i( # )]!A (t)),. 2A (t)
(3.5d)
Eqs. (3.5a), (3.5b), (3.5c) and (3.5d) can now be solved by the extension of the technique of generalized averaging [20] as follows: Casting the original Eqs. (3.5a), (3.5b), (3.5c) and (3.5d) in the form du "e h(u, t), u"[A (t), A (t), h (t), h (t)] dt
(3.6)
(with [2] denoting transpose). We split the space D of di!erentiable functions into two subspaces DM and DI where DM contains both constant and periodic functions with the smallest frequencies, and DI the rest of the functions. To system (3.6), we associate the following reduced system: dl "eHM (l, t), dt
(3.7)
by using the transformation u"l#eGI (l, t). (Note that u"l at e"0.)
(3.8)
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Inserting Eq. (3.8) into Eq. (3.6) using Eq. (3.7), we get HM #eHM
*GI *GI # "hM (l#eGI , t)#hI (l#eGI , t), *l *t
(3.9)
where h(u, t)"hM (u, t)#hI (u, t).
(3.10)
Separating the terms in DM and DI we get HM "hM (l#eGI , t),
(3.11)
*GI *GI "hI (l#eGI , t)!ehM (l#eGI , t) . *l *t
(3.12)
After inserting the expansions: HM (l, t)"HM #eHM #2#eL\HM #2, L
(3.13a)
GI (l, t)"GI #eGI #2#eL\GI #2, L
(3.13b)
into Eqs. (3.11), (3.12) and comparing coe$cients of e we obtain to lowest order:
*hI HM (l, t)"hM (l, t), HM (l, t)" hI dt,2, *l
(3.14a)
GI " hI (l, t) dt,2.
(3.14b)
Then the solution of system (3.6) reads u"l#eGI #O(e).
(3.15)
The details of this method will be demonstrated, here on the following complex damped system: zK #uz#e z "z" sin 2t"0, 0(e ; 1.
(3.16)
This example has been considered in the "rst part for the purpose of comparison with numerical results. Using our approximate method we can derive "rst- and second-order (in e) analytical formulas for the periodic attractors of Eq. (3.16). Eqs. (3.5a), (3.5b), (3.5c) and (3.5d) for this example become
A (t) d A(t) e "! dt h (t) 4 h (t)
2A sin 2t#(AA !A)S !A A¹ 2A sin 2t#(A A!A)S !AA ¹ , 1 [(AA #A)S #A A¹ ] A 1 [(A#A A)S #AA ¹ ] A
(3.17a)
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where S "sin(2(u#1)t#h #h )!sin(2(u!1)t#h #h ), S "cos(2(u!1)t#h #h )!cos(2(u#1)t#h #h ), ¹ "sin(2(2u#1)t#2(h #h ))!sin(2(2u!1)#2(h #h )), (3.17b) ¹ "cos(2(2u!1)t#2(h #h ))!cos(2(2u#1)#2(h #h )). As is seen from Eq. (3.17b), small frequency components occur at u 1, . This means that system (3.16) has periodic attractors with periods 2n, 4n by the "rst approximation. To "nd periodic attractors with other periods we need to go to second or higher approximations. We consider the case u which has been studied in the "rst part. For this case the reduced system (3.7) using Eqs. (3.13a) and (3.14a) becomes dl "eHM (l, t)#eHM (l, t)#O(e), dt
(3.18a)
where l"[A , A , h , h ], A A sin(2(2u!1)t#2(h #h )) A A sin(2(2u!1)t#2(h #h )) , HM (l, t)" !A cos(2(2u!1)t#2(h #h )) !A cos(2(2u!1)t#2(h #h ))
0
0
1 1 A #2A A HM (l, t)"! 2(A #A ) # # 2(u#1) 2(u!1) 2u#1
(3.18b)
1 1 A #2A A 2(A #A ) # # 2(u#1) 2(u!1) 2u#1
and A , h are the initial values of A (t) and h (t), respectively (i"1, 2). G G G G To satisfy the conditions dA /dt"0, we make the corresponding contributions of HM to vanish by setting G 2(2u!1)t#2(h #h )"kn, k"0, 1, 2,2. (3.19) From Eq. (3.19) one obtains dh dh 2u!1"! ! . dt dt
(3.20)
By substiting from Eqs. (3.18a) and (3.18b) using Eq. (3.19) into Eq. (3.20) we obtain the following expressions for u to second order in e: 1 e e u" # (!1)I(A #A )# [5A #5A #16A A ]#O(e). 4 8 384
(3.21)
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For k"0 (or even) and k"1 (or odd) we get from Eq. (3.21) two di!erent values of u, corresponding to two di!erent periodic attractors. To compare these results with those of the indicatrix method (see Table 1) we use the same initial conditions which are x "!0.1, x "0, y "0, and y "0.3 (i.e A "0.25, A "0.35, h "0, h "n). Then Eq. (3.21) becomes u"!0.023125e#0.000565267e#O(e),
(3.22a)
u"#0.023125e#0.000565267e#O(e).
(3.22b)
Expressions (3.22a) and (3.22b) agree very well with numerical results of Table 1 using the indicatrix method up to e)0.15 (e.g. if e"0.15, Eqs. (3.22a) and (3.22b) yield u"0.246543968, u"0.253481468 while the numerical results of Table 1 are u"0.246706, u"0.2532697). In order to determine the amplitudes A (t) (i"1, 2) and the phases h (t) and hence derive an approximate G G expression for the periodic attractors with 4n-period z( (t)"x( (t)#iy( (t), we need to calculate the function GI . Using Eq. (3.14b) we get GI and then, from Eq. (3.15), substitute A (t) and h (t) in Eq. (3.3) to obtain G G z( (t)"x( (t)#iy( (t),
(3.23a)
where
t t t e x( (t)"A cos #h #A cos #h # 2A A cos !2h !h 2 2 8 2
t 3t 3t #2A A cos !h !h !A cos !h !A cos #h 2 2 2
1 1 1 5t 5t 7t # A cos #h # A cos #h # A A cos #2h #h 3 3 6 2 2 2 1 7t # A A cos #h #2h 6 2 y( (t)"A
sin
(3.23b)
#O(e),
e t t t #h !A sin #h # !2A A sin !2h !h 2 2 8 2
t 3t 3t #2A A sin !h !2h #A sin !h !A sin !h 2 2 2
1 5t 1 5t 1 7t # A sin #h ! A sin #h # A A sin #2h #h 3 2 3 2 6 2 1 7t ! A A cos #h #2h 6 2
#O(e),
(3.23c)
We now use the above results to obtain the boundaries of the instability regions of the periodic attractors of Eq. (3.16). Writing the unperturbed frequency in the form: u"u#ea #ea #2, u "0, , 1,2.
(3.24)
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We study here the case u ", looking for uniformly valid (in t) solutions of Eq. (1.1) (or Eq. (2.2)), linearized about its solution z( (t)"x( (t)#iy( (t); cf. Eqs. (3.23a), (3.23b) and (3.23c) above. Thus, we set x(t)"x( (t)#g(t), y(t)"y( (t)#m(t), g, m `smalla
(3.25)
and linearize Eq. (2.2), keeping the terms up to "rst order in g(t) and f(t) to obtain gK #ug#e sin 2t[2x( (x( g#y( m)#g(x( #y( )]"0,
(3.26a)
m$ #um#e sin 2t[2y( (x( g#y( m)!mQ (x( #y( )]"0.
(3.26b)
Using the standard `multiple-scalinga techniques of perturbation theory [21,22], we write the solutions of Eqs. (3.26a) and (3.26b) as series expansions in e: g(t, e)"g (t, tI )#eg (t, tI )#2,
(3.27a)
m(t, e)"m (t, tI )#em (t, tI )#2,
(3.27b)
where tI ,et
(3.27c)
is the slow time variable. Substituting from Eqs. (3.27a), (3.27b) and (3.27c) into Eqs. (3.26a) and (3.26b) using Eqs. (3.23a), (3.23b) and (3.23c) and equating like powers of e yields a doubly in"nite hierarchy of equations which tend to zero and "rst order read: *g 1 # g "0, *t 4
(3.28a)
*m 1 # m "0, 4 *t
(3.28b)
*g 1 *g *g # g "!2 !a g ! (x( #y( ) #2x( x( g #2y( x( m sin 2t 4 *t*tI *t *t
(3.29a)
*m 1 *m *m # m "!2 !a m ! (x( #y( ) #2y( y( m #2x( y( g sin 2t. *t 4 *t*tI *t
(3.29b)
The general solutions of Eqs. (3.28a) and (3.28b) are: t t g (t, tI )"A (tI ) cos #B (tI ) sin , 2 2
(3.30a)
t t m (t, tI )"C (tI ) cos #D (tI ) sin , 2 2
(3.30b)
where A (tI ), B (tI ), C (tI ), D (tI ) are as yet undetermined functions of the slow vairable tI . Inserting now Eqs. (3.30a) and (3.30b) into Eqs. (3.29a) and (3.29b) making use of "rst-order expressions for x( (t), y( (t), cf.
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Eqs. (3.23a), (3.23b) and (3.23c) we obtain
1 dA t dB t *g !a B sin ! #a A cos # g " dtI dtI *t 4 2 2
1 ! 4
0.33 A !0.031D 4
#
1.11 ! A #0.09D 4
3t 5t 0.37 cos !cos # B #0.03C 4 2 2
t 7 0.33 cos !cos t ! B #0.03C 4 2 2
5 3 sin t#sin t 2 2
7 t sin t#sin 2 2
,
(3.31a)
*m 1 t t dC dD # m " !a D sin ! #a C cos *t 4 dtI 2 dtI 2
! !
1.07 0.03 C ! B 4 16
#
0.03 0.37 C ! B 4 16
3 5 0.09 1.11 cos t!cos t # D ! A 2 2 4 16
t 7 1.07 0.03 cos !cos t ! D # A 2 2 4 16
5 3 sin t#sin t 2 2
7 t sin t#sin 2 2
.
(3.31b)
(Note that we use the initial conditions x "!0.1, x "0, y "0 and y "0.3, i.e. A "0.25, A "0.35, h "0, h "n.) Since we want the solutions to be bounded for 0(t(R, we cannot allow secular terms proportional to sin t/2, cos t/2 on the right-hand side of Eqs. (3.31a) and (3.31b). This requires that A , B , C and D satisfy a system of coupled linear ordinary di!erential equations which in matrix form is written as follows:
A (tI ) d B(tI ) " dtI C (tI ) D(tI )
0.33 a ! 16
0
0.33 a # 16 0.03 ! 4 0
0.03 ! 4
0
0
0
0
0.03 4
1.07 ! a # 16
0 0.03 4 1.07 a ! 16
0
A B . C D
(3.32)
The eigenvalues of the 4;4 matrix in Eq. (3.32) are given by the roots of the characteristic equation j!2bj#c"0,
(3.33a)
where
1 b" 2
1.07 a # 16
0.33 c" a # 16
1.07 0.33 a ! # a # 16 16
0.33 a ! 16
0.33 # a # 16
1.07 a # 16
1.07 a ! 16
#
0.33 a ! 16
,
1.07 0.03 a ! ! 4 16
0.33 a ! 16
0.03 . 4
(Note that the last term in Eq. (3.33b) is very small and can be neglected.)
1.07 a # 16
(3.33b)
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Clearly, if the quantity j"!b$(b!c(0, then Eq. (3.33a) has purely imaginary roots, and the solutions g(t) and m(t) are oscillatory, then x( (t), y( (t) are stable solutions. Otherwise if j'0, g(¹) and m(t) are unbounded (unstable) and the solutions x( (t) and y( (t) are unstable. Therefore, on the boundary of the instability regions 1.07 0.33 , a "$ a "$ 16 16
(3.34)
whence, the boundary curves of the instability regions in the (u, e) plane for x(t) component are given by 1 0.33 u" ! e#O(e), 4 16
(3.35a)
1 0.33 u" # e#O(e), 4 16
(3.35b)
Fig. 3. (a) Instability region [Eqs. (3.35a) and (3.35b)] for the component x(t) of the periodic attractors with period 4n; (b) instability region [Eqs. (3.36a) and (3.36b)] for the component y(t) of the periodic attractors with period 4n.
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while for y(t) component in the (u, e) plane are: 1 1.07 u" ! e#O(e). 4 16
(3.36a)
1 1.07 u" # e#O(e), 4 16
(3.36b)
Expressions (3.35a), (3.35b), (3.36a) and (3.36b) are plotted in Fig. 3a and Fig. 3b, respectively. The predictions of our expressions (3.35a), (3.35b), (3.36a) and (3.36b) are consistent with our numerical checks from fourth order Runge}Kutta integration of Eq. (3.16) or Eq. (2.2). In Fig. 4 we give examples of
Fig. 4. Phase space distance of two nearby trajectories for Eq. (3.16) with e"0.1, x "!0.1, x "0, y "0, y "0.3 and: (a) u"0.2431 (stable case); (b) u"0.2568 (stable case); (c) u"0.24776 (unstable case).
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plots for the phase space distance: d(t)"[(x !x )#(x !x )#(y !y )#(y !y )] . (3.37) In Fig. 4a and Fig. 4b we observe stability over many periods since the values of u"0.2431, 0.2568 at e"0.1 lie outside the instability regions of Fig. 3a and Fig. 3b, while in Fig. 4c, the values of u"0.24776, e"0.1 lie inside the instability region of Fig. 3b (unstable case).
4. Concluding remarks In this investigation the periodic attractors and their stability properties of a class of complex damped non-linear systems described by Eq. (1.1), are presented. Eq. (1.1) appears in many important "elds of physics, mechanics and engineering, for example, in high-energy accelerators, rotor dynamics, robots and shells. These periodic attractors are important in the study of dynamical systems, since they represent stationary or repeatable behavior. The indicatrix method is applied to study the existence of the periodic attractors of Eq. (1.1). The case u , P()t)"sin 2t and f""z" is considered as an example to illustrate these attractors (other examples can be similarly studied). We constructed Poincare& map or (x, x ) surface of section at some of the parameter values u, e of Table 1 which are obtained by the indicatrix method. We observe from Poincare& plots that there exist two periodic attractors with 4n-period, one of them is stable and the other is unstable (see Fig. 2). It brings us to the Poincare& }Birkho! theorem which shows that for eO0 there exist 2k periodic orbits (k*1), half of which are stable and half of which are unstable [22]. Our recent approximate method which is based on the generalized averaging method is used to derive approximate analytical solutions of Eq. (1.1). This method is applied to the above example and approximate expressions for the frequency u are derived up to the second order in e. These expressions are in very good agreement with the results of numerical studies by using the indicatrix method up to e)0.15. The analytical solution of this example is used to get the stability regions of the 4n-periodic attractors. These regions are tested numerically and good agreement is found.
References [1] G.M. Mahmoud, A. Rauh, A.A. Mohamed, On modulated complex non-linear dynamical systems, IL Nuovo Cimento 114 (B.1) (1999) 31}47. [2] G.M. Mahmoud, S.A. Aly, On periodic solutions of parametrically excited complex nonlinear dynamical systems, 4th Int. Conf. on Di!erence Equations and Applications, August 27}31, Poznan, Poland, 1998, accepted for presentation. [3] R.H. Helleman, Self-generated chaotic behavior in nonlinear mechanics, in: G.D. Cohen (Ed.), Fundamental Problems in Statistical Mechanics, Vol. 5, North-Holland, Amsterdam, 1980, pp. 165}233. [4] T. Bountis, G.M. Mahmoud, Synchronized periodic orbits in beam}beam interaction models one and two spatial dimensions, Particle Accelerators 22 (1987) 129}147. [5] A. Muszynska, Whirl and whip-rotor bearing stability problems, J. Sound Vib. 110 (3) (1986) 443}462. [6] Y.H. Kim (Ed.), Dynamics of Rotating Machinery, Hemisphere Publishing Corporation, New York, 1989. [7] A.H. Nayfeh, L.D. Zavodney, Experimental observation of amplitude and phase-modulated responses of two internally coupled oscillators to a harmonic excitation, J. Appl. Mech. 55 (1988) 706}710. [8] C.-C. Chi, R.M. Rosenbery, On damped nonlinear dynamic systems with many degrees of freedom, Int. J. Non-Linear Mech. 20 (5/6) (1985) 371}384. [9] C.A. Ludeke, W.S. Wagner, The generalized Du$ng equation with large damping, Int. J. Non-Linear Mech. 3 (1968) 383}395. [10] T.K. Caughey, M.E. O'Kelly, Classical normal modes in damped linear dynamic systems, Trans. Am. Soc. Mech. Engng, J. Appl. Mech. 32E (1965) 583. [11] J.A. Morrison, A generalized method of averaging with applications to slightly damped nonlinear oscillations, J. Math. Anal. Appl. 15 (1966) 213}227.
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[12] H.R. Baum, On the weakly damped harmonic oscillator, Quart. Appl. Math. (1972) 573}576. [13] G.M. Mahmoud, T. Bountis, On synchronized periodic solutions of a parametrically forced Van der Pol oscillator, Int. Conf. On Singular Behavior and Nonlinear Dynamics, Samos, Greece 1988, pp. 207}221. [14] H.S. Chan, K.W. Chung, Z. Xu, A perturbation incremental method for strongly nonlinear oscillators, Int. J. Non-Linear Mech. 31 (1) (1996) 59}72. [15] J.G. Margallo, J.D. Bejarano, Generalized Fourier series and limit cycles of generalized Van der Pol oscillators, J. Sound Vib. 136 (1990) 453}466. [16] J.D. Bejarano, J.G. Margallo, G.M. Mahmoud, Analytical and numerical studies of periodic solutions outside potential wells, Anales de Fisica 92 (3) (1996) 121}125. [17] Z. Xu, Y.K. Cheung, A non-linear scales methods for strongly nonlinear oscillators, Nonlinear Dyn. 7 (1995) 285}299. [18] G. Villari, Periodic solutions of Lie+nards's equation, J. Math. Anal. Appl. 86 (1982) 379}386. [19] G.M. Mahmoud, T. Bountis, Synchronized periodic solutions of a class of periodically driven nonlinear oscillators, J. Appl. Mech. 55 (1988) 721}728. [20] G.M. Mahmoud, On periodic orbits of nonlinear dynamical systems with many degrees of freedom, Physica A 181 (1992) 385}395. [21] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979. [22] G.M. Mahmoud, Analytical and numerical approaches to periodic orbits of nonlinear dynamical systems, Ph. D. Thesis, Clarkson University, Potsdam, New York, 1987.
Periodic attractors of complex damped non-linear systems Gamal M. Mahmoud *, S.A. Aly Department of Mathematics, Faculty of Science, University of Assiut, Assiut 71516, Egypt Department of Mathematics, Faculty of Science, University of Al-Azhar, Assiut 71511, Egypt
Abstract The aim of the present paper is to investigate the dynamics of a class of complex damped non-linear systems described by the equation zK #uz#ez f(z, z , z , z )P()t)"0,
(*)
where z(t)"x(t)#iy(t), i"(!1, the bar denotes the complex conjugate and e is a small positive parameter. The periodic attractors of Eq. (*) are important in the study of these systems, since they represent stationary or repeatable behavior. This equation appears in several "elds of physics, mechanics and engineering, for example, in high-energy accelerators, rotor dynamics, robots and shells. In the numerical investigation of this work we used the indicatrix method which has been introduced and extended in our previous studies to study the existence of the periodic attractors of our systems. To illustrate these periodic attractors we constructed Poincare& plots at some of the parameter values which are obtained by the indicatrix method for the case u , f""z" and P()t)"sin 2t as an example. Our recent method which is based on the generalized averaging method is used to obtain approximate analytical solutions of Eq. (*) and investigate the stability properties of the solutions. We compared the analytical results of our example with the numerical results and excellent agreement is found. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Periodic attractors; Complex systems; Non-linear; Damped
1. Introduction Recently, approximate analytical and numerical methods have been applied to conservative complex non-linear dynamical systems, and some of their periodic solutions and their stability properties have been studied in detail in Refs. [1,2]. In the present work we wish to turn to the investigation of the dynamics of complex damped systems, which displays many interesting features, di!erent from the conservative cases. Periodic solutions are important in the study of dynamical systems, since they represent stationary or repeatable behavior. Any practical system always contains dissipation.
* Corresponding author. Tel.: 002-088-312564; fax: 002-088-312564. E-mail address: [email protected] (G.M. Mahmoud) 0020-7462/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 9 ) 0 0 0 1 6 - 5
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Herein we study a class of complex damped non-linear systems described by the equation: zK #uz#ez f(z, z , z , z )P()t)"0, 0(e;1,
(1.1)
where z(t)"x(t)#iy(t) is complex with i"(!1 the imaginary unit, z is the complex conjugate function, the frequency u is a constant, f is non-linear analytical function of its arguments, P()t) is a periodic function of frequency ), and dots denote time derivatives. Eq. (1.1) describes a damped model for which the periodic solutions we seek are attractors (see Fig. 2). This equation appears in several "elds of physics, engineering, electronics and statistical mechanics (eg. in high-energy accelerators, rotor dynamics, robots and shells [1}3]). For example, the connection of Eq. (1.1) to the problem of colliding beams has been discussed in Refs. [3,4], while its connection to rotor dynamics is found in Refs. [5,6]. The mathematical models of robots and shells are stated in Ref. [7]. Interesting examples of damped systems have been the subject of many papers (see for example, Refs. [8}19]). Forces that are functions of the velocity are called damping forces. Many researchers have used damping forces such as x f(x) or x g(x, x ), where f(x) and g(x, x ) are analytical functions of their arguments [13}18]. This paper is divided into two parts. The "rst part deals with the numerical investigations of periodic attractors of Eq. (1.1). In this part the indicatrix method which has been introduced and developed in our pervious work [19,20] is applied to study the existence of the periodic attractors of Eq. (1.1). Periodic attractors with period 4n of Eq. (1.1) are studied for the case u , f""z" and P()t)"sin 2t as an example (other cases of u, f and P can be similarly studied). Using this method which is a numerical technique, we "nd two periodic attractors with period 4n for each value of e, one for u(, and the other for u'. These periodic attractors are studied here on the Poincare& surface of section: R"+x(t ), x (t ), t "t #kn, I I I k"0, 1, 2,2, and R"+y(t ), y (t ), t "t #kn, k"0, 1, 2,2,. The parameter values which are suggested I I I by the indicatrix method are used to study the stability properties of these attractors. The second part is concerned with the analytical investigation of these periodic attractors. Our method which is based on the generalized averaging method [1,2] is used to derive approximate analytical solutions of Eq. (1.1). Expressions for the frequency u, valid to second order in e, are obtained for the above example. These expressions are in very good agreement with the results by the indicatrix method of part 1 (up to e)0.15). Approximate formulas are derived for the solution z( (t)"x( (t)#iy( (t) of our example. These formulas are used to derive the stability regions for our periodic attractors with period 4n which turn out to be accurate when compared with the numerical results. Some concluding remarks are presented in Section 4.
2. Numerical investigations In this part we apply the indicatrix method [19,20] for numerically "nding the periodic attractors of a class of complex damped non-linear systems (1.1). For eO0, the sets of points (x(t #¹), x (t #¹)), (y(t #¹), y (t #¹)) with period ¹"2n/u , u u "n/m, (n, m are positive integers) form closed curves parameterized by t , t 3[0, 2n/)] in the spaces (x, x ), (y, y ), respectively, which may or may not pass through the speci"ed initial conditions. These closed curves are called the indicatrices. By varying slightly u and e, we obtain the periodic components x(t), y(t) when the indicatrices in the spaces (x, x ), (y, y ) are observed to pass through the prescribed initial conditions at some values of t , t . This can happen, of course, for more than one set of values of t , t . In this way we obtained the numerical values of the parameters u, e at which periodic components x(t), y(t) of the solutions of Eq. (1.1) with period ¹"2n/u pass through the speci"ed initial conditions (x(t ), x (t ), y(t ), y (t )).
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To explain this method in detail, consider the following damped non-linear system zK #uz#ez "z" sin 2t"0,
(2.1)
where u u " (other values of u can be similarly studied). It is noted that Eq. (2.1) is a special case of Eq. (1.1) with f""z" and P()t)"sin 2t (other cases of f and P()t) can be similarly treated). Eq. (2.1) is a system of two-coupled non-linear second-order di!erential equations: xK #ux#ex (x#y) sin 2t"0,
yK #uy#ey (x#y) sin 2t"0.
(2.2)
Using the indicatrix method we shall determine the values of the parameters u, e at which periodic attractors of Eq. (2.2) with frequency pass by a speci"ed initial conditions e.g. x(t )"!0.1, x (t )"0, y(t )"0, y (t )"0.3. For eO0 the sets of points (x(t #4n), x (t #4n)), (y(t #4n), y (t #4n)) form closed curves parametrized by t , t 3[0, n] in the spaces (x, x ) and (y, y ), respectively, that may or may not pass through our speci"ed initial conditions. These closed curves (indicatrices) are plotted here in Fig. 1 for e"0.1. By varying slightly u and e, we obtain the desired periodic attractors with period 4n when the indicatrices are observed to pass through our initial conditions ((!0.1, 0), (0, 0.3)) at some values of t , t 3[0, n]. This can happen, of course, for more than one set of values of t and t , see Fig. 1a and Fig. 1b for
Fig. 1. Indicatrix plots of Eq. (2.2) for e"0.1, t , t 3[0, n]: (a) u"0.24776: (!0.1, 0) at t "0 (x, x space); (b) u"0.24776: (0, 0.3) at t "n/2 (y, y space); (c) u"0.2521823: (!0.1, 0) at t "n/2 (x, x space); (d) u"0.2521823: (0, 0.3) at t "0 (y, y space).
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Table 1
e
0.001 0.01 0.02 0.04 0.05 0.06 0.08 0.1 0.12 0.14 0.15
u
t ,t
0.249978 0.2500219 0.24978 0.250219 0.249562 0.2504378 0.249124 0.2508746 0.248905 0.251094 0.2486856 0.251312 0.2482467 0.2517465 0.24776 0.2521823 0.247365 0.2526171 0.2469273 0.253054 0.246706 0.2532679
0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0 0, n/2 n/2, 0
t "0, t "n/2, e"0.1 and u"0.24776 and Fig. 1c and Fig. 1d for t "n/2, t "0, e"0.1 and u"0.2521823. We study in this way the existence of the periodic attractors with period 4n. We "nd, with the indicatrix method, two periodic attractors for each value of e, one for u( (t "0 and t "n/2) and the other for u' (t "n/2 and t "0), see Table 1. To illustrate these periodic attractors we use the Poincare& map associated with Eq. (2.2) at the parameter values suggested by the indicatrix method. In fact, using the Poincare& map one can even determine the stability properties of these periodic attractors under small perturbation in the initial condition. Using Poincare& map, or the (x, x ) surface of section R"+[x(t ), x (t )], t "t #kn, k"0, 1, 2,2,, since I I I our system (2.2) is symmetric, we "nd that for several initial conditions the origin is a stable period n orbit and there are two periodic attractors with 4n-period, one of them is stable (marked by 䢇) and the other unstable (marked by ;), see Fig. 2 for e"0.1 and (a) u"0.24776, (b) u"0.2521823. In Fig. 2a we take three initial conditions, (x(0)"0.5, x (0)"0), (x(0)"0.3, x (0)"0), (x(0)"0.25, x (0)"0) and y(t )"y (t )"0, while in Fig. 2b, (x(n/2)"0.5, x (n/2)"0), (x(n/2)"0.35, x (n/2)"0), (x(n/2)"0.25, x (n/2)"0) and y(t )"y (t )"0. The periodic attractors can be viewed here on the Poincare& surface of section as a set of distinct points. We have four points marked by 䢇 for the stable periodic attractor and four points marked by ; for the unstable one since the period of these attractors is 4n. This surface of section shows those attractors in Fig. 2a and Fig 2b. The values of u and e in Fig. 2a correspond to an unstable periodic attractors while in Fig. 2b correspond to the stable one. It is noted that for the initial conditions with y(t )'0, y (t )"0, we obtain a similar "gure as in Fig. 2. In the numerical calculations we used the method of Runge}Kutta of order 4. In the next part we shall present our analytical investigations and a comparison with the results of this part for these periodic attractors.
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Fig. 2. Surface of sections of Eq. (2.2) for e"0.1 and: (a) u"0.24776, y(t )"y (t )"0, t "0 (unstable); (b) u"0.251823, y(t )"y (t )"0, t "n/2 (stable). A stable (marked by 䢇) and an unstable (marked by ;) periodic attractors with
4n-period.
3. Analytical investigations Using the approximate method which is based on the generalized averaging method [1,2], we study in this part the periodic attractors of a class of complex damped non-linear systems whose equation of motion is described by Eq. (1.1). This method starts form the so-called generating solutions z(t)"A exp[i(ut#h )]#A exp[!i(ut#h )], z (t)"iuA exp[i(ut#h )]!iuA exp[!i(ut#h )] (3.1) which satisfy Eq. (1.1) to zero-order (e"0). The constants A and h (i"1, 2) can be determined by the initial G G conditions. In order to solve Eq. (1.1), it is assumed that the constants A and h depend on time, so that in Eq. G G
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(3.1) A PA (t), h Ph (t). Then Eq. (3.1) becomes G G G G z(t)"A (t) exp[i (t)]#A (t) exp[!i (t)],
(3.2a)
z (t)"iuA (t) exp[i (t)]!iuA (t) exp[!i (t)]
(3.2b)
with
(t)"ut#h (t), i"1, 2. G G
(3.3)
By di!erentiating Eq. (3.2a) with respect to t and comparing with Eq. (3.2b) and di!erentiating Eq. (3.2b) and substituting into Eq. (1.1) we get 2[AQ (t)#iA (t)hQ (t)]"!efp()t)+A (t)!A (t) exp[!i( (t)# (t))],, 2[AQ (t)!iA (t)hQ (t)]"efp()t)+A (t) exp[i( (t)# (t))]!A (t),.
(3.4a) (3.4b)
Splitting Eqs. (3.4a) and (3.4b) into real and imaginary parts we get a system of equations which we solve for AQ (t) and hQ (t) to obtain G G e AQ (t)"! Re+ fp()t)(A (t)!A (t) exp[!i( # )]),, 2
(3.5a)
e AQ (t)" Re+ fp()t)(A (t) exp[i( # )]!A (t)),, 2
(3.5b)
e hQ (t)"! Im+ fp()t)(A (t)!A (t) exp[!i( # )]),, 2A (t)
(3.5c)
e hQ (t)"! Im+ fp()t)(A (t) exp[i( # )]!A (t)),. 2A (t)
(3.5d)
Eqs. (3.5a), (3.5b), (3.5c) and (3.5d) can now be solved by the extension of the technique of generalized averaging [20] as follows: Casting the original Eqs. (3.5a), (3.5b), (3.5c) and (3.5d) in the form du "e h(u, t), u"[A (t), A (t), h (t), h (t)] dt
(3.6)
(with [2] denoting transpose). We split the space D of di!erentiable functions into two subspaces DM and DI where DM contains both constant and periodic functions with the smallest frequencies, and DI the rest of the functions. To system (3.6), we associate the following reduced system: dl "eHM (l, t), dt
(3.7)
by using the transformation u"l#eGI (l, t). (Note that u"l at e"0.)
(3.8)
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Inserting Eq. (3.8) into Eq. (3.6) using Eq. (3.7), we get HM #eHM
*GI *GI # "hM (l#eGI , t)#hI (l#eGI , t), *l *t
(3.9)
where h(u, t)"hM (u, t)#hI (u, t).
(3.10)
Separating the terms in DM and DI we get HM "hM (l#eGI , t),
(3.11)
*GI *GI "hI (l#eGI , t)!ehM (l#eGI , t) . *l *t
(3.12)
After inserting the expansions: HM (l, t)"HM #eHM #2#eL\HM #2, L
(3.13a)
GI (l, t)"GI #eGI #2#eL\GI #2, L
(3.13b)
into Eqs. (3.11), (3.12) and comparing coe$cients of e we obtain to lowest order:
*hI HM (l, t)"hM (l, t), HM (l, t)" hI dt,2, *l
(3.14a)
GI " hI (l, t) dt,2.
(3.14b)
Then the solution of system (3.6) reads u"l#eGI #O(e).
(3.15)
The details of this method will be demonstrated, here on the following complex damped system: zK #uz#e z "z" sin 2t"0, 0(e ; 1.
(3.16)
This example has been considered in the "rst part for the purpose of comparison with numerical results. Using our approximate method we can derive "rst- and second-order (in e) analytical formulas for the periodic attractors of Eq. (3.16). Eqs. (3.5a), (3.5b), (3.5c) and (3.5d) for this example become
A (t) d A(t) e "! dt h (t) 4 h (t)
2A sin 2t#(AA !A)S !A A¹ 2A sin 2t#(A A!A)S !AA ¹ , 1 [(AA #A)S #A A¹ ] A 1 [(A#A A)S #AA ¹ ] A
(3.17a)
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where S "sin(2(u#1)t#h #h )!sin(2(u!1)t#h #h ), S "cos(2(u!1)t#h #h )!cos(2(u#1)t#h #h ), ¹ "sin(2(2u#1)t#2(h #h ))!sin(2(2u!1)#2(h #h )), (3.17b) ¹ "cos(2(2u!1)t#2(h #h ))!cos(2(2u#1)#2(h #h )). As is seen from Eq. (3.17b), small frequency components occur at u 1, . This means that system (3.16) has periodic attractors with periods 2n, 4n by the "rst approximation. To "nd periodic attractors with other periods we need to go to second or higher approximations. We consider the case u which has been studied in the "rst part. For this case the reduced system (3.7) using Eqs. (3.13a) and (3.14a) becomes dl "eHM (l, t)#eHM (l, t)#O(e), dt
(3.18a)
where l"[A , A , h , h ], A A sin(2(2u!1)t#2(h #h )) A A sin(2(2u!1)t#2(h #h )) , HM (l, t)" !A cos(2(2u!1)t#2(h #h )) !A cos(2(2u!1)t#2(h #h ))
0
0
1 1 A #2A A HM (l, t)"! 2(A #A ) # # 2(u#1) 2(u!1) 2u#1
(3.18b)
1 1 A #2A A 2(A #A ) # # 2(u#1) 2(u!1) 2u#1
and A , h are the initial values of A (t) and h (t), respectively (i"1, 2). G G G G To satisfy the conditions dA /dt"0, we make the corresponding contributions of HM to vanish by setting G 2(2u!1)t#2(h #h )"kn, k"0, 1, 2,2. (3.19) From Eq. (3.19) one obtains dh dh 2u!1"! ! . dt dt
(3.20)
By substiting from Eqs. (3.18a) and (3.18b) using Eq. (3.19) into Eq. (3.20) we obtain the following expressions for u to second order in e: 1 e e u" # (!1)I(A #A )# [5A #5A #16A A ]#O(e). 4 8 384
(3.21)
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For k"0 (or even) and k"1 (or odd) we get from Eq. (3.21) two di!erent values of u, corresponding to two di!erent periodic attractors. To compare these results with those of the indicatrix method (see Table 1) we use the same initial conditions which are x "!0.1, x "0, y "0, and y "0.3 (i.e A "0.25, A "0.35, h "0, h "n). Then Eq. (3.21) becomes u"!0.023125e#0.000565267e#O(e),
(3.22a)
u"#0.023125e#0.000565267e#O(e).
(3.22b)
Expressions (3.22a) and (3.22b) agree very well with numerical results of Table 1 using the indicatrix method up to e)0.15 (e.g. if e"0.15, Eqs. (3.22a) and (3.22b) yield u"0.246543968, u"0.253481468 while the numerical results of Table 1 are u"0.246706, u"0.2532697). In order to determine the amplitudes A (t) (i"1, 2) and the phases h (t) and hence derive an approximate G G expression for the periodic attractors with 4n-period z( (t)"x( (t)#iy( (t), we need to calculate the function GI . Using Eq. (3.14b) we get GI and then, from Eq. (3.15), substitute A (t) and h (t) in Eq. (3.3) to obtain G G z( (t)"x( (t)#iy( (t),
(3.23a)
where
t t t e x( (t)"A cos #h #A cos #h # 2A A cos !2h !h 2 2 8 2
t 3t 3t #2A A cos !h !h !A cos !h !A cos #h 2 2 2
1 1 1 5t 5t 7t # A cos #h # A cos #h # A A cos #2h #h 3 3 6 2 2 2 1 7t # A A cos #h #2h 6 2 y( (t)"A
sin
(3.23b)
#O(e),
e t t t #h !A sin #h # !2A A sin !2h !h 2 2 8 2
t 3t 3t #2A A sin !h !2h #A sin !h !A sin !h 2 2 2
1 5t 1 5t 1 7t # A sin #h ! A sin #h # A A sin #2h #h 3 2 3 2 6 2 1 7t ! A A cos #h #2h 6 2
#O(e),
(3.23c)
We now use the above results to obtain the boundaries of the instability regions of the periodic attractors of Eq. (3.16). Writing the unperturbed frequency in the form: u"u#ea #ea #2, u "0, , 1,2.
(3.24)
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We study here the case u ", looking for uniformly valid (in t) solutions of Eq. (1.1) (or Eq. (2.2)), linearized about its solution z( (t)"x( (t)#iy( (t); cf. Eqs. (3.23a), (3.23b) and (3.23c) above. Thus, we set x(t)"x( (t)#g(t), y(t)"y( (t)#m(t), g, m `smalla
(3.25)
and linearize Eq. (2.2), keeping the terms up to "rst order in g(t) and f(t) to obtain gK #ug#e sin 2t[2x( (x( g#y( m)#g(x( #y( )]"0,
(3.26a)
m$ #um#e sin 2t[2y( (x( g#y( m)!mQ (x( #y( )]"0.
(3.26b)
Using the standard `multiple-scalinga techniques of perturbation theory [21,22], we write the solutions of Eqs. (3.26a) and (3.26b) as series expansions in e: g(t, e)"g (t, tI )#eg (t, tI )#2,
(3.27a)
m(t, e)"m (t, tI )#em (t, tI )#2,
(3.27b)
where tI ,et
(3.27c)
is the slow time variable. Substituting from Eqs. (3.27a), (3.27b) and (3.27c) into Eqs. (3.26a) and (3.26b) using Eqs. (3.23a), (3.23b) and (3.23c) and equating like powers of e yields a doubly in"nite hierarchy of equations which tend to zero and "rst order read: *g 1 # g "0, *t 4
(3.28a)
*m 1 # m "0, 4 *t
(3.28b)
*g 1 *g *g # g "!2 !a g ! (x( #y( ) #2x( x( g #2y( x( m sin 2t 4 *t*tI *t *t
(3.29a)
*m 1 *m *m # m "!2 !a m ! (x( #y( ) #2y( y( m #2x( y( g sin 2t. *t 4 *t*tI *t
(3.29b)
The general solutions of Eqs. (3.28a) and (3.28b) are: t t g (t, tI )"A (tI ) cos #B (tI ) sin , 2 2
(3.30a)
t t m (t, tI )"C (tI ) cos #D (tI ) sin , 2 2
(3.30b)
where A (tI ), B (tI ), C (tI ), D (tI ) are as yet undetermined functions of the slow vairable tI . Inserting now Eqs. (3.30a) and (3.30b) into Eqs. (3.29a) and (3.29b) making use of "rst-order expressions for x( (t), y( (t), cf.
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319
Eqs. (3.23a), (3.23b) and (3.23c) we obtain
1 dA t dB t *g !a B sin ! #a A cos # g " dtI dtI *t 4 2 2
1 ! 4
0.33 A !0.031D 4
#
1.11 ! A #0.09D 4
3t 5t 0.37 cos !cos # B #0.03C 4 2 2
t 7 0.33 cos !cos t ! B #0.03C 4 2 2
5 3 sin t#sin t 2 2
7 t sin t#sin 2 2
,
(3.31a)
*m 1 t t dC dD # m " !a D sin ! #a C cos *t 4 dtI 2 dtI 2
! !
1.07 0.03 C ! B 4 16
#
0.03 0.37 C ! B 4 16
3 5 0.09 1.11 cos t!cos t # D ! A 2 2 4 16
t 7 1.07 0.03 cos !cos t ! D # A 2 2 4 16
5 3 sin t#sin t 2 2
7 t sin t#sin 2 2
.
(3.31b)
(Note that we use the initial conditions x "!0.1, x "0, y "0 and y "0.3, i.e. A "0.25, A "0.35, h "0, h "n.) Since we want the solutions to be bounded for 0(t(R, we cannot allow secular terms proportional to sin t/2, cos t/2 on the right-hand side of Eqs. (3.31a) and (3.31b). This requires that A , B , C and D satisfy a system of coupled linear ordinary di!erential equations which in matrix form is written as follows:
A (tI ) d B(tI ) " dtI C (tI ) D(tI )
0.33 a ! 16
0
0.33 a # 16 0.03 ! 4 0
0.03 ! 4
0
0
0
0
0.03 4
1.07 ! a # 16
0 0.03 4 1.07 a ! 16
0
A B . C D
(3.32)
The eigenvalues of the 4;4 matrix in Eq. (3.32) are given by the roots of the characteristic equation j!2bj#c"0,
(3.33a)
where
1 b" 2
1.07 a # 16
0.33 c" a # 16
1.07 0.33 a ! # a # 16 16
0.33 a ! 16
0.33 # a # 16
1.07 a # 16
1.07 a ! 16
#
0.33 a ! 16
,
1.07 0.03 a ! ! 4 16
0.33 a ! 16
0.03 . 4
(Note that the last term in Eq. (3.33b) is very small and can be neglected.)
1.07 a # 16
(3.33b)
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Clearly, if the quantity j"!b$(b!c(0, then Eq. (3.33a) has purely imaginary roots, and the solutions g(t) and m(t) are oscillatory, then x( (t), y( (t) are stable solutions. Otherwise if j'0, g(¹) and m(t) are unbounded (unstable) and the solutions x( (t) and y( (t) are unstable. Therefore, on the boundary of the instability regions 1.07 0.33 , a "$ a "$ 16 16
(3.34)
whence, the boundary curves of the instability regions in the (u, e) plane for x(t) component are given by 1 0.33 u" ! e#O(e), 4 16
(3.35a)
1 0.33 u" # e#O(e), 4 16
(3.35b)
Fig. 3. (a) Instability region [Eqs. (3.35a) and (3.35b)] for the component x(t) of the periodic attractors with period 4n; (b) instability region [Eqs. (3.36a) and (3.36b)] for the component y(t) of the periodic attractors with period 4n.
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while for y(t) component in the (u, e) plane are: 1 1.07 u" ! e#O(e). 4 16
(3.36a)
1 1.07 u" # e#O(e), 4 16
(3.36b)
Expressions (3.35a), (3.35b), (3.36a) and (3.36b) are plotted in Fig. 3a and Fig. 3b, respectively. The predictions of our expressions (3.35a), (3.35b), (3.36a) and (3.36b) are consistent with our numerical checks from fourth order Runge}Kutta integration of Eq. (3.16) or Eq. (2.2). In Fig. 4 we give examples of
Fig. 4. Phase space distance of two nearby trajectories for Eq. (3.16) with e"0.1, x "!0.1, x "0, y "0, y "0.3 and: (a) u"0.2431 (stable case); (b) u"0.2568 (stable case); (c) u"0.24776 (unstable case).
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plots for the phase space distance: d(t)"[(x !x )#(x !x )#(y !y )#(y !y )] . (3.37) In Fig. 4a and Fig. 4b we observe stability over many periods since the values of u"0.2431, 0.2568 at e"0.1 lie outside the instability regions of Fig. 3a and Fig. 3b, while in Fig. 4c, the values of u"0.24776, e"0.1 lie inside the instability region of Fig. 3b (unstable case).
4. Concluding remarks In this investigation the periodic attractors and their stability properties of a class of complex damped non-linear systems described by Eq. (1.1), are presented. Eq. (1.1) appears in many important "elds of physics, mechanics and engineering, for example, in high-energy accelerators, rotor dynamics, robots and shells. These periodic attractors are important in the study of dynamical systems, since they represent stationary or repeatable behavior. The indicatrix method is applied to study the existence of the periodic attractors of Eq. (1.1). The case u , P()t)"sin 2t and f""z" is considered as an example to illustrate these attractors (other examples can be similarly studied). We constructed Poincare& map or (x, x ) surface of section at some of the parameter values u, e of Table 1 which are obtained by the indicatrix method. We observe from Poincare& plots that there exist two periodic attractors with 4n-period, one of them is stable and the other is unstable (see Fig. 2). It brings us to the Poincare& }Birkho! theorem which shows that for eO0 there exist 2k periodic orbits (k*1), half of which are stable and half of which are unstable [22]. Our recent approximate method which is based on the generalized averaging method is used to derive approximate analytical solutions of Eq. (1.1). This method is applied to the above example and approximate expressions for the frequency u are derived up to the second order in e. These expressions are in very good agreement with the results of numerical studies by using the indicatrix method up to e)0.15. The analytical solution of this example is used to get the stability regions of the 4n-periodic attractors. These regions are tested numerically and good agreement is found.
References [1] G.M. Mahmoud, A. Rauh, A.A. Mohamed, On modulated complex non-linear dynamical systems, IL Nuovo Cimento 114 (B.1) (1999) 31}47. [2] G.M. Mahmoud, S.A. Aly, On periodic solutions of parametrically excited complex nonlinear dynamical systems, 4th Int. Conf. on Di!erence Equations and Applications, August 27}31, Poznan, Poland, 1998, accepted for presentation. [3] R.H. Helleman, Self-generated chaotic behavior in nonlinear mechanics, in: G.D. Cohen (Ed.), Fundamental Problems in Statistical Mechanics, Vol. 5, North-Holland, Amsterdam, 1980, pp. 165}233. [4] T. Bountis, G.M. Mahmoud, Synchronized periodic orbits in beam}beam interaction models one and two spatial dimensions, Particle Accelerators 22 (1987) 129}147. [5] A. Muszynska, Whirl and whip-rotor bearing stability problems, J. Sound Vib. 110 (3) (1986) 443}462. [6] Y.H. Kim (Ed.), Dynamics of Rotating Machinery, Hemisphere Publishing Corporation, New York, 1989. [7] A.H. Nayfeh, L.D. Zavodney, Experimental observation of amplitude and phase-modulated responses of two internally coupled oscillators to a harmonic excitation, J. Appl. Mech. 55 (1988) 706}710. [8] C.-C. Chi, R.M. Rosenbery, On damped nonlinear dynamic systems with many degrees of freedom, Int. J. Non-Linear Mech. 20 (5/6) (1985) 371}384. [9] C.A. Ludeke, W.S. Wagner, The generalized Du$ng equation with large damping, Int. J. Non-Linear Mech. 3 (1968) 383}395. [10] T.K. Caughey, M.E. O'Kelly, Classical normal modes in damped linear dynamic systems, Trans. Am. Soc. Mech. Engng, J. Appl. Mech. 32E (1965) 583. [11] J.A. Morrison, A generalized method of averaging with applications to slightly damped nonlinear oscillations, J. Math. Anal. Appl. 15 (1966) 213}227.
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323
[12] H.R. Baum, On the weakly damped harmonic oscillator, Quart. Appl. Math. (1972) 573}576. [13] G.M. Mahmoud, T. Bountis, On synchronized periodic solutions of a parametrically forced Van der Pol oscillator, Int. Conf. On Singular Behavior and Nonlinear Dynamics, Samos, Greece 1988, pp. 207}221. [14] H.S. Chan, K.W. Chung, Z. Xu, A perturbation incremental method for strongly nonlinear oscillators, Int. J. Non-Linear Mech. 31 (1) (1996) 59}72. [15] J.G. Margallo, J.D. Bejarano, Generalized Fourier series and limit cycles of generalized Van der Pol oscillators, J. Sound Vib. 136 (1990) 453}466. [16] J.D. Bejarano, J.G. Margallo, G.M. Mahmoud, Analytical and numerical studies of periodic solutions outside potential wells, Anales de Fisica 92 (3) (1996) 121}125. [17] Z. Xu, Y.K. Cheung, A non-linear scales methods for strongly nonlinear oscillators, Nonlinear Dyn. 7 (1995) 285}299. [18] G. Villari, Periodic solutions of Lie+nards's equation, J. Math. Anal. Appl. 86 (1982) 379}386. [19] G.M. Mahmoud, T. Bountis, Synchronized periodic solutions of a class of periodically driven nonlinear oscillators, J. Appl. Mech. 55 (1988) 721}728. [20] G.M. Mahmoud, On periodic orbits of nonlinear dynamical systems with many degrees of freedom, Physica A 181 (1992) 385}395. [21] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979. [22] G.M. Mahmoud, Analytical and numerical approaches to periodic orbits of nonlinear dynamical systems, Ph. D. Thesis, Clarkson University, Potsdam, New York, 1987.