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Reshaping-induced order-chaos routes in a damped driven Helmholtz oscillator
Chaos, Solitons and Fractals 24 (2005) 459–470 www.elsevier.com/locate/chaos
Reshaping-induced order-chaos routes in a damped driven Helmholtz oscillator F. Balibrea
a,*
n b, M.A. Lo pez , R. Chaco
c
a
Departamento de Matematicas, Facultad de Matematicas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain Departamento de Electronica e Ingenierıa Electromecanica, Escuela de Ingenierıas Industriales, Universidad de Extremadura, P.O. Box 382, 06071 Badajoz, Spain Departamento de Matematicas, Escuela Universitaria Politecnica de Cuenca, Universidad de Castilla-La Mancha, 16071 Cuenca, Spain b
c
Accepted 5 May 2004
Abstract We study the structural stability of the Helmholtz oscillator under changes of the shape of periodic nonlinear perturbations. In particular, we determine the order-chaos threshold by means of Melnikov’s method for the case of periodic perturbations given in terms of Jacobian elliptic functions. Computer simulations confirm our theoretical predictions. 2004 Elsevier Ltd. All rights reserved.
1. Introduction In this paper we shall concentrate on simple model for a universal chaotic escape situation, namely the Helmholtz oscillator [1–3]. Escape from a potential well is an ubiquitous phenomenon in science. Examples are known in mechanics [4], chemistry [5], quantum optics [6], astrophysics [7], hydrodynamics [8–10], etc., where complex escape phenomena can often be well described by a low-dimensional system of differential equations. In [11,12] the structural stability has been considered for systems modelled by nonlinear ordinary differential equations which are both damped and forced periodically and where the shape of the wave varies. The simplest periodical functions to fulfil these requirements and where is possible to change the shape of the waves are the Jacobian elliptic functions (JEF) [13–15]. If one considers that polynomials are the simplest nonlinear solutions of linear oscillators, it can be seen ([13,14]) that they can be given in terms of JEF’s. In comparison with harmonic functions, JEF functions add a new variable to the parameter space of the system, the elliptic parameter m which is responsible for the shape of the perturbation, i.e., for the temporal rate at which energy is transferred from the excitation mechanism to the system, when the period is fixed. This fact leads us to expect new aspects on the behavior of the system, nonexisting in the harmonic case, when m is varied and the remaining parameters are constant. Following [13,14] let us denote generically a JEF function by pq and by f ðt; A; T ; mÞ ¼ Apqðxt; mÞ a generic modulation of amplitude A and period T and take x xðmÞ ¼ lKðmÞ=T where KðmÞ denotes the complete elliptic integral of the first kind [13,14] and lKðmÞ the period of pq (i.e., 4K or 2K). When A and T are fixed, if we change the elliptic parameter m 2 ½0; 1 then we obtain a change in the shape of the perturbation.
*
Corresponding author. E-mail address: [email protected] (F. Balibrea).
0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.05.016
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F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
In this paper, we show how by altering uniquely the shape of a weak external nonlinear forcing, a system can change from a regular to a chaotic state, and vice-versa. The inhibition or reduction of an initial chaotic state depends on a fully nonlinear situation on the three parameters of the periodic perturbation causing the phenomenon: period, amplitude, and shape (besides, of course, the initial conditions and remaining parameters of the system). To apply the former ideas to particular models, we have selected the Helmholtz oscillator, partly because its chaotic transitions can be predicted, albeit approximately by Melnikov analysis (MM), and partly because is the simple model for a universal chaotic escape situation [16]. In the first place, we consider the Helmholtz oscillator, damped and periodically forced, €x ¼ x bx2 d_x þ c snðxt; mÞ;
ð1Þ
where x denotes the displacement, ðc; d 1Þ, x, d, and c are the normalized parameters of frequency, damping coefficient, and forcing term amplitude, respectively, and snðxt; mÞ is the JEF sine-amplitude (senam) of elliptic parameter m and period T (for a picture of this function, see for example [17]). A nice introduction of this function via a system of differential equations can be seen in [15]. When m ! 0, is snðxt; mÞ ! sinðxtÞ; i.e., one recovers the limit case of the harmonic forcing. This is fundamental when we compare the structural stability of the system when only the forcing shape is varied. In this situation, the parameter space of Eq. (1) is four-dimensional due to the addition of the parameter m to the three-dimensional parameter space ðb; d; cÞ of the corresponding harmonic counterpart. In [11] it is shown that lim snð4Kt=T ; mÞ ¼
m!1
1 4X 1 sin½ð2n þ 1Þ2pt=T : p n¼0 2n þ 1
Note that this is the Fourier expression of the square wave function of period T whose shape is not exactly a square wave, due to the well-known Gibb’s phenomenon [18]. Now if we fixed the parameters b, d, c and T , and vary m from 0 to 1 we are able to appreciate the variations in the shape of the perturbation. Since we are interested in the stability of the chaotic behavior under these changes, a good procedure is to apply Melnikov’s method (MM) to Eq. (1). In the second place, we will consider the equation, €x ¼ x bx2 d_x þ c cnðxt; mÞ;
ð2Þ
where cnðxt; mÞ is the JEF cosine-amplitude (cosam) of parameter m (see again [15,17]). When m ! 0 is cnðxt; mÞ ! cosðxtÞ, i.e., we recover the case of harmonic forcing. When m ! 1 the function tends to a sequence of symmetric pulses of decreasing width. Similarly to the previous case, we study the structural stability of the system when only the forcing shape is varied. Now, we fix the parameters b, d, c, and T , and vary m from 0 to 1 to study the pure effect of variations in the shape of the perturbation. Additionally we also consider what happens when we perturbate only the shape of a weak periodic parametric perturbation of the quadratic term (C), that is, €x ¼ x b½1 þ g snðxt; mÞx2 d_x;
ð3Þ
taking 0 < gb 1; and also the same problem for perturbation of the linear term (L), €x ¼ x½1 þ g0 snðxt; mÞ bx2 d_x;
ð4Þ
for 0 < g0 1. Finally we compare the results.
2. Melnikov analysis Melnikov’s method (MM) is an analytical technique currently available to provide a criterion for the occurrence of homoclinic (and heteroclinic) chaos in continuous systems (see [25]), and is considered as a standard procedure. As it is well known, MM predictions are both approximate (the MM is a first-order perturbative method) and limited (only valid for orbits starting at points sufficiently near to a separatrix). Many applications of the method can be found in the literature, see for examples references [19–26] for further details.
F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
461
3. Helmholtz oscillator forced with the JEF sn 3.1. Melnikov function The application of MM to Eq. (1) involves calculating the Melnikov function, Z 1 Mðt0 Þ ¼ fdu20 ðtÞ þ cu0 ðtÞ sn½xðt þ t0 Þ; mgdt;
ð5Þ
1
where x0 ¼
t 3 sec h2 ; 2b 2
x_ 0 ðtÞ u0 ðtÞ ¼
t t 3 sec h2 tan h 2b 2 2
ð6Þ
is the parametric representation of the separatrix of the underlying conservative system ðd ¼ c ¼ 0Þ: Using the Fourier expansion of sn [13], plugging Eq. (6) into Eq. (5), and after some simple algebraic manipulations, Eq. (5) is obtained in the form 0 Z 1 1 9d 3pc X pK 2n þ 1 pxt0 1 4 t 2 t p ffiffiffi ffi nþ sec h csch Mðt0 Þ ¼ 2 tan h dt cos K K 2 2 2 2 2bK m 1 4b 1 Z 1 t t pxt 1 tan h sin nþ dt; ð7Þ sec h2 2 2 K 2 1 where K 0 is the complementary complete integral of the first kind. The resulting integrals can be evaluated using standard integral tables [27]. Finally, with xðmÞ 4KðmÞ ¼ T , we obtain 1 6d 96p4 c X ð2n þ 1Þ2pt0 ; ð8Þ an ðmÞbn ðT Þ cos Mðt0 Þ ¼ 2 b n¼0 T 5b where an ðmÞ ¼
1 pffiffiffiffi csch K m
bn ðT Þ ¼
1 T2
1 pK 0 ; nþ 2 K
ð8aÞ
and
2n þ 1 2
2
csch
ð2n þ 1Þ2p2 : T
ð8bÞ
3.2. Threshold function analysis From Eq. (8), it is straightforward to prove that the existence of a homoclinic bifurcation and then chaos is guaranteed for trajectories whose initial conditions are sufficiently near the unperturbed separatrix (6) whenever d < Uf ;sn ðb; m; T Þ; c
ð9Þ
where Uf ;sn ðb; m; T Þ ¼ 80bp4
1 X
an ðmÞbn ðT Þ:
ð10Þ
n¼0
Uf ;sn ðb; m; T Þ will be called the sn-chaotic threshold or simply sn-threshold of the oscillator and has been found calculating the simple zeros of the Melnikov function (8), i.e., studying the values t0 where changes the sign of the Melnikov function. With T and b constant, we represent the sn-threshold as a function of the forcing shape parameter m. A plot of Uf ;sn ðmÞ is shown in Fig. 1. The qualitative form of this function remains the same when b and T are varied. When the coefficient b is varied we can have different situations. If b ! 0 is recovered the linear limit since Uf ;sn is being decreasing, and Uf ;sn ðb ¼ 0; m; T Þ ¼ 0; i.e., in the limit is not possible the chaotic behavior. For the limits cases of harmonic function and square wave we obtain, respectively,
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F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
Fig. 1. sn-Threshold function Uf ;sn ðmÞ vs. m [Eq. (10)] for b ¼ 1=2, and T ¼ 2p.
2 20bp3 2p ; Uf ;sn ðb; m ¼ 0; T Þ ¼ csch T2 T Uf ;sn ðb; m ¼ 1; T Þ ¼
1 80bp2 X ð2n þ 1Þ2p2 ; ð2n þ 1Þcsch T 2 n¼0 T
ð11Þ ð12Þ
with lim Uf ;sn ðb; m ¼ 0; T Þ ¼ 0;
ð13Þ
lim Uf ;sn ðb; m ¼ 1; T Þ ¼ 0;
ð14Þ
lim Uf ;sn ðb; m ¼ 0; T Þ ¼ 0;
ð15Þ
lim Uf ;sn ðb; m ¼ 1; T Þ ¼ 0:
ð16Þ
T !1
T !1
T !0
and T !0
In sum, the sn-threshold functions Uf ;sn ðb; m; T Þ for fixed b, seen as functions of T only, have the same qualitative shape for every m 2 ½0; 1; moreover the case m ! 1 (square wave) is not qualitatively different of the harmonic case ðm ¼ 0Þ. Let us suppose that the system (1) is in a periodic motion state denoted by A (see Fig. 1). Then, increasing m from m1 to m2 , and keeping constant the remaining parameters b, T , d, and c, the system may reach a state (point B) able of being chaotic. On the contrary, if B represents a chaotic escape state, the route B ! A regularizes such a state. Now, fixing b, T , and m we can increase the ratio d=c (increasing d, decreasing c, or both) reaching a periodic state C. This is a known procedure for taming chaos [23,25]. Note that the pathways of types B ! A and B ! C are only particular routes to not escape states. The most common situation consists on a simultaneous variation of d=c and m, as, for example., in the paths B ! D, and B ! E.
4. Helmholtz oscillator forced with the JEF cn 4.1. Melnikov function The application of MM to Eq. (2) involves calculating the Melnikov function, Z 1 Mðt0 Þ ¼ fdu20 ðtÞ þ cu0 ðtÞcn½xðt þ t0 Þ; mgdt:
ð17Þ
1
Using the Fourier expansion of cn [13], plugging Eq. (6) into Eq. (17), and after some simple algebraic manipulation, Eq. (17) can be obtained in the form
F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
0 1 1 t t 9d 3c p X pK 2n þ 1 pxt0 1 nþ Mðt0 Þ ¼ 2 sec h4 sec h tan h2 dt þ pffiffiffiffi sin K K 2 2 2b K m n¼0 2 2 4b 1 Z 1 t t pxt 1 sec h2 tan h sin nþ dt: 2 2 K 2 1
463
Z
ð18Þ
Finally, we obtain Mðt0 Þ ¼
1 6d 96p4 c X ð2n þ 1Þ2pt0 ; c ðmÞb ðT Þ sin n n b T 5b2 n¼0
ð19Þ
where cn ðmÞ ¼
1 1 pK 0 pffiffiffiffi sec h n þ ; K m 2 K
bn ðT Þ ¼
1 T2
ð19aÞ
and
2n þ 1 2
2
csch
ð2n þ 1Þ2p2 : T
ð19bÞ
4.2. Threshold function analysis From Eq. (19), it is straightforward to prove that a homoclinic bifurcation and then chaos is guaranteed for trajectories whose initial conditions are sufficiently near the unperturbed separatrix (6) if d < Uf ;cn ðb; m; T Þ; c
ð20Þ
where the cn-chaotic threshold or simply cn-threshold function is Uf ;cn ðb; m; T Þ ¼ 80bp4
1 X
cn ðmÞbn ðT Þ:
ð21Þ
n¼0
With T and b constant, we study the cn-threshold as a function of the forcing shape parameter m. A plot of Uf ;cn ðmÞ is shown in Fig. 2. The qualitative form of this function remains the same when b and T are varied. For the limits cases of harmonic function and symmetric pulses of low width we obtain, respectively, 2 20bp3 2p Uf ;cn ðb; m ¼ 0; T Þ ¼ ; ð22Þ csch T2 T Uf ;cn ðb; m ¼ 1; T Þ ¼ 0: The limits for these functions when T ! 1, and T ! 0 are 0.
Fig. 2. cn-Threshold function Uf ;cn ðmÞ vs. m [Eq. (21)] for b ¼ 1=2, and T ¼ 2p.
ð23Þ
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F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
Let us suppose that the system (2) is in a chaotic escape state denoted by A (see Fig. 2). Then, increasing m from m1 to m2 , and keeping constant the remaining parameters b, T , d, and c, the system may reach a state (point B) of periodic motion. On the contrary, if B represents a regular state, the route B ! A rouses such a escape. Now, fixing b; T , and m we can increase the ratio d=c (increasing d, decreasing c ; or both) starting from F reaching a periodic state at point C. Note that the pathways of types B ! A, B ! D, and B ! E lead to chaotic escape states.
5. Case of perturbation of the quadratic term 5.1. Melnikov function For Eq. (3) the Melnikov function is, 1 6d 192p4 g X ð2n þ 1Þ2pt0 ; Mðt0 Þ ¼ 2 an ðmÞdn ðT Þ cos T 5b 5b2 n¼0
ð24Þ
where an ðmÞ ¼
1 2n þ 1 pK 0 pffiffiffiffi csch ; 2 K K m
ð24aÞ
and 1 dn ðT Þ ¼ 2 T
(
2n þ 1 2
2 "
4p T
2
2n þ 1 2
2
#) (" # 2 ) 2 2p 2n þ 1 ð2n þ 1Þ2p2 : þ1 þ 1 csch T 2 T
ð24bÞ
5.2. Threshold function analysis From Eq. (24), it is immediate to prove that a homoclinic bifurcation is guaranteed for trajectories whose initial conditions are sufficiently near the unperturbed separatrix (6) if d < Uc;sn ðm; T Þ; g
ð25Þ
where the sn-threshold function is Uc;sn ðm; T Þ ¼ 32p4
1 X
an ðmÞdn ðT Þ:
ð26Þ
n¼0
With T constant, we represent the chaotic threshold as a function of only the forcing shape parameter m. A plot of Uc;sn ðmÞ is shown in Fig. 3. The qualitative form of this function remains the same when T is varied. For the limit cases 2 2 2 p 4p2 4p 4p 2p Uc;sn ðm ¼ 0; T Þ ¼ ; ð27Þ þ1 þ 4 csch 2 T2 T2 T2 T Uc;sn ðm ¼ 1; T Þ
" # " # 2 2 2 2 1 32p2 X 4p 2n þ 1 2p 2n þ 1 ð2n þ 1Þ2p2 ¼ 2 ð2n þ 1Þ ; þ1 þ 1 csch T 2 T 2 T n¼0 T
ð28Þ
The limits for these functions when T ! 1 and T ! 0 are 0. The comments which can be stated are similar to those previously done, we observe that Uc;sn ðmÞ increases very quickly which indicates that the change on the state of the system varying the parameter m from m1 to m2 , and keeping constant the other parameters T , g, and d, will be reached before than in the sn case, i.e., that the interval [m1 ; m2 ] for the change of the state will be smaller than a forcing given for sn. Also, fixing T , and b; limm!0 Uc;sn ðmÞ is greater than limm!0 Uf ;sn ðmÞ, which indicates that the change of the state for the forcing starts for a ratio d=c smaller than the ratio d=g in a parametric perturbation of the quadratic term.
F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
465
Fig. 3. sn-Threshold function Uc;sn ðmÞ vs. m [Eq. (26)] for T ¼ 2p.
6. Case of a perturbation of the linear term 6.1. Melnikov function For Eq. (4) the Melnikov function is, Mðt0 Þ ¼
1 6d 48p4 g0 X ð2n þ 1Þ2pt0 ; a ðmÞe ðT Þ cos n n T b2 n¼0 5b2
ð29Þ
where 1 1 pK 0 pffiffiffiffi csch n þ ; 2 K K m ( # 2 " 2 2 ) 1 2n þ 1 4p 2n þ 1 ð2n þ 1Þ2p2 en ðT Þ ¼ 2 : þ 1 csch T 2 T 2 T
an ðmÞ ¼
ð29aÞ
ð29bÞ
6.2. Threshold function analysis From Eq. (29), it is straightforward to prove that a homoclinic bifurcation is guaranteed for trajectories whose initial conditions are sufficiently near the unperturbed separatrix (6) if d < Ul;sn ðm; T Þ; g0
ð30Þ
where the sn-threshold function is Ul;sn ðm; T Þ ¼ 40p4
1 X
an ðmÞen ðT Þ:
ð31Þ
n¼0
For the limit cases of harmonic function and square wave we obtain, respectively, 2 2 5p 4p2 4p 2p ; þ 1 csch Ul;sn ðm ¼ 0; T Þ ¼ T2 T2 T 2 " # 2 2 1 40p2 X 4p 2n þ 1 ð2n þ 1Þ2p2 Ul;sn ðm ¼ 1; T Þ ¼ 2 ; ð2n þ 1Þ þ 1 csch T n¼0 T T 2 The limits for these functions when T ! 1, and T ! 0 are 0. The comments and analysis that can be done are similar to those made in 5.2.
ð32Þ
ð33Þ
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F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
6.3. Numerical analysis and comparison of results To compare the effect in the chaotic escape behavior of the system of using the functions senam or cosam in the periodical forcing, we introduce the map Cf ;sn;cn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Uf ;cn ðb; m; T Þ; for 8m 2 ½0; 1; T constant, and b constant. In Fig. 4 we can see that Cf ;sn;cn P 1 and if we fix d=c in Uf ;sn ðb; m; T Þ and Uf ;cn ðb; m; T Þ then for each m we have Uf ;sn ðb; m; T Þ P Uf ;cn ðb; m; T Þ; and since if d=c < Uf ;sn ðb; m; T Þ we have chaotic escape. Therefore taking the rest of the parameters constant, chaotic escape exists easier when the forcing is given by the function sn than if it is given by the function cn. Cf ;sn;cn ðmÞ is plotted in Fig. 4 for b ¼ 1=2 and T ¼ 2p. The calculated points have been joined by a dotted curve. Now in order to compare these theoretical points with the real situation, we have computed the probability of escaping towards infinite vs. m of the points of the system. To this end, let us denote by P ðmÞ the number of such points. We normalize them dividing into the harmonic case Psn ðm ¼ 0ÞðPcn ðm ¼ 0ÞÞ and obtain Psn ðmÞ Pcn ðmÞ Pbsn ðmÞ ¼ Pbcn ðmÞ ¼ Psn ðm ¼ 0Þ Pcn ðm ¼ 0Þ where it is easy to see that Psn ðm ¼ 0Þ ¼ Psin ðm ¼ 0Þ ¼ Pcos ðm ¼ 0Þ ¼ Pcn ðm ¼ 0Þ: Finally we compute Pbsn ðmÞ Pbcn ðmÞ which we have plotted also in Fig. 4 with black points. We appreciate that real calculations are well fitted to the theoretical ones. We introduce a similar representation for Cc;l;sn ðmÞ ¼ Uc;sn ðm; T Þ=Ul;sn ðm; T Þ vs: m for T ¼ 2p; which is plotted in Fig. 5, where we appreciate that the chaotic escape is easier using the function senam as a perturbation in the quadratic term than using the same function in the linear one.
Fig. 4. Function Cf ;sn;cn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Uf ;cn ðb; m; T Þ vs. m for b ¼ 1=2 and T ¼ 2p. Function Pbsn ðmÞ= Pbcn ðmÞ vs. m (black points).
F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
467
Fig. 5. Function Cc;l;sn ðmÞ ¼ Uc;sn ðm; T Þ=Ul;sn ðm; T Þ vs. m for T ¼ 2p.
With Cf ;c;sn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Uc;sn ðm; T Þ vs: m for b ¼ 1=2 and T ¼ 2p we can compare the effect of using the function senam in the forced and in the quadratic terms. In this case we have a decreasing representation (see Fig. 6) which means that when m is increased we have chaotic escape easier using senam as a perturbation of the quadratic term than of the forced term. Similarly, for b ¼ 1=2, and T ¼ 2p we obtain the functions Cf ;l;sn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Ul;sn ðm; T Þ; Cf ;cn;sn ðmÞ ¼ Uf ;cn ðb; m; T Þ=Ul;sn ðm; T Þ; and Cf ;cn;c;sn ðmÞ ¼ Uf ;cn ðb; m; T Þ=Uc;sn ðm; T Þ:
Fig. 6. Function Cf ;c;sn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Uc;sn ðm; T Þ vs. m for b ¼ 1=2, and T ¼ 2p.
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F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
Fig. 7. Basin of attraction of the system €x ¼ x bx2 d_x þ c cnð4KðmÞ t; mÞ, for b ¼ 1, d ¼ 0:1, c ¼ 0:075, and T ¼ 7:391982714 in the T window 0:8 < x < 1:9, 0:8 < x_ < 0:9: (a) m ¼ 0, (b) m ¼ 0:5, (c) m ¼ 0:8, (d) m ¼ 1–1015 .
Fig. 8. Basin of attraction of the system €x ¼ x bx2 d_x þ c sn 4KðmÞ t; m , for b ¼ 1, d ¼ 0:1, c ¼ 0:075, and T ¼ 7:391982714 in the T window 0:8 < x < 1:9, 0:8 < x_ < 0:9: (a) m ¼ 0, (b) m ¼ 0:5, (c) m ¼ 0:8, (d) m ¼ 1–1015 .
The comparisons and remarks we can do are similar to the previous cases. A detailed discussion of all situations can be seen in [17]. It is interesting also plotting for some different initial conditions of the system, the behaviour of xðtÞ when t ! 1 This has been done for the case of periodical forcing with senam and cosam functions where we can state the differences. In the case of perturbation of the quadratic or linear case we can obtain similar representations. With white color is represented the nonescaping basin of attraction zone and with black the escaping one. To generate numerically the basin sets of attraction we have considered a grid of 400 · 400 starting points uniformly distributed in the phase space fxð0Þ 2 ½0:8; 1:9; x_ ð0Þ 2 ½0:8; 0:9g. From the points of this grid each integration of the system is continued until x exceeds at which the system is considered to have escaped (attracted to infinity) or until the maximum number allowable of cycles, here 20 has been reached. When we have an harmonic perturbation ðm ¼ 0Þ, the system presents a stratification of the basin of attraction into zones of escaping and others of nonescaping. If m is increased (in the case of cosam function) the strata of escaping inside the basin are becoming rare and practically there is no of them when m ¼ 1 (see Fig. 7). We have the opposite situation with senam function. Starting in m ¼ 0 (harmonic situation) we have a particular stratification of the basin of attraction (see Fig. 8a). When m is increased, the black strata grow and when m ¼ 1, practically all points escape to infinite (see Fig. 8d).
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469
6.4. Conclusions Taking the Helmholtz oscillator as a model for a situation of universal chaotic escape from a potential well, we have introduced changes in the shape of a nonlinear periodical perturbation in the linear, quadratic and forced terms of the equation. The changes in the shape of the perturbation have been observed comparing the effects of using the elliptic functions senam and cosam. Numerically we have compared those effects varying the modulus parameter m in the case of the periodical forced perturbation. We conclude that the escape from the potential well is easier when we use the function senam than if we use the function cosam. Also is shown the effect that in the basin of the nonescaping set has the variation in the parameter m in the different situations of senam or cosam. We have also shown a way to reduce or suppress chaotic escape from a potential well, by the method of altering the geometrical shape of weak periodic perturbations, (see Figs. 1 and 2), by changing the parameter m. In resume, changes in the form of the wave of a periodical perturbation depends on the FEJ chosen and on the type of perturbation applied to the system. It is an interesting conclusion that the existence of chaotic escape is easier when the forcing is perturbed by senam function than in the case of cosam function. For a similar situation, see [28].
Acknowledgements The authors acknowledge the MCYT (Ministerio de Ciencia y Tecnologia, Spain) Projects BFM2002-03512 and BFM2002-00010, the JCCM (Junta de Comunidades de Castilla-La Mancha, Spain) Project PAC-02-002, and Fundaci on Seneca (Comunidad Aut onoma de la Regi on de Murcia), Project PI-8/00807/FS/01, for financial support.
References [1] Hemlholtz HLF. On the sensation of tone as a physiological basis for the theory of music. New York: Dover; 1954. [2] El Naschie MS, Kapitaniak T. Helmholtz and Duffing oscillators, Chaos, Solitons and Fractals Basin. In: Second International Conference on Industrial and Applied Math. ICIAM 91, Washington, proceedings, 1991. p. 64. [3] Hussein AA, Al Athel S. A note on the Helmholtz oscillator. Chaos, Solitons & Fractals 2001;12:1835–8. [4] Poston T, Stewart IN. Catastrophe theory and its applications. London: Pitman; 1978. [5] Goggin ME, Milonni PW. Driven Morse oscillator: classical chaos, quantum theory, and photodissociation. Phys Rev A 1988;37:796–806. [6] Chac on R, Cirac JI. Chaotic and regular behavior of a trapped ion interacting with a laser field. Phys Rev A 1995;51:4900–6. [7] Buchler JB, Einchhorn H, editors. Chaotic phenomena in astrophysics. Annals of the New York Academy of Sciences, vol. 497. New York: New York Academy of Sciences; 1987. [8] Thompson JMT. Chaotic phenomena triggering the escape from a potential well. Proc R Soc Lond A 1989;421:195–225. [9] Thompson JMT, Soliman MS. Fractal control boundaries of driven oscillators and their relevance to safe engineering design. In: Engineering applications of dynamics of chaos. CISM courses and lectures, vol. 319. Vienna: Springer; 1991. p. 311–25. [10] Soliman MS. Basin boundaries with fractal and smooth accumulation properties in systems with single potential well. Chaos, Solitons & Fractals 1998;9(6):989–94. [11] Chac on R, Dı´az Bejarano J. Routes to suppressing chaos by weak periodic perturbations. Phys Rev Lett 1993;71:3103–6. [12] Chac on R. Geometrical resonance in a driven symmetric bistable system subjected to strong or to weak damping. Phys Rev E 1996;54:6153–9. [13] Byrd PF, Friedman MD. Handbook of elliptic integrals for engineers and scientists. New York: Springer-Verlag; 1971. [14] Milne-Thompson LM. In: Abramovitz M, Stegun IA, editors. Handbook of mathematical functions. New York: Dover; 1972 and references therein. [15] Meyer KR. Jacobi elliptic functions from a dynamical systems point of view. Amer Math Month 2001;108:729–37. [16] Kapitaniak T, editor. Chaotic oscillators. Singapore: World Scientific; 1992. [17] L opez MA. Umbral orden-caos y Bifurcaciones en Sistemas no Aut onomos bajo Perturbaciones Peri odicas Generalizadas. Ediciones de la Universidad de Castilla- La Mancha, Cuenca, Spain, 1999. [18] Zygmund A. Trigonometric series. Cambridge: Cambridge University Press; 1990. [19] Arnold VI. Instability of dynamical systems with several degrees of freedom. Sov Math Dokl 1964;5:581–5. [20] Balibrea F, Chac on R, L opez MA. Inhibition of chaotic escape by an additional driven term. Int J Bifurc Chaos 1998;8(8):1719– 23.
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[21] Chac on R, Balibrea F, L opez MA. Inhibition of chaotic escape from a potential well using small parametric modulations. J Math Phys 1996;37:5518–23. [22] Chac on R, Balibrea F, L opez MA. Role of parametric resonance in the inhibition of chaotic escape from a potential well. Phys Lett A 1997;235:153–8. [23] Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Applied mathematical sciences, vol. 42. New York: Springer-Verlag; 1983. [24] Melnikov VK. On the stability of the center for time periodic perturbations. Trans Moscow Math Soc 1963;12:1–57. [25] Wiggins S. Global bifurcations and chaos, analytical methods. Applied mathematical sciences, vol. 73. New York: SpringerVerlag; 1988. [26] Wiggins S. Introduction to applied nonlinear dynamical systems and chaos. Applied mathematics, vol. 2. New York: SpringerVerlag; 1990. [27] Gradshteyn I, Ryzhik I. Table of integrals, series and products. New York: Academic; 1981. [28] Chac on R, Martınez Garcıa-Hoz A. Modifying fractal basin boundaries by reshaping periodic terms. J Math Phys 2002;43(7):3586–91.
Reshaping-induced order-chaos routes in a damped driven Helmholtz oscillator F. Balibrea
a,*
n b, M.A. Lo pez , R. Chaco
c
a
Departamento de Matematicas, Facultad de Matematicas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain Departamento de Electronica e Ingenierıa Electromecanica, Escuela de Ingenierıas Industriales, Universidad de Extremadura, P.O. Box 382, 06071 Badajoz, Spain Departamento de Matematicas, Escuela Universitaria Politecnica de Cuenca, Universidad de Castilla-La Mancha, 16071 Cuenca, Spain b
c
Accepted 5 May 2004
Abstract We study the structural stability of the Helmholtz oscillator under changes of the shape of periodic nonlinear perturbations. In particular, we determine the order-chaos threshold by means of Melnikov’s method for the case of periodic perturbations given in terms of Jacobian elliptic functions. Computer simulations confirm our theoretical predictions. 2004 Elsevier Ltd. All rights reserved.
1. Introduction In this paper we shall concentrate on simple model for a universal chaotic escape situation, namely the Helmholtz oscillator [1–3]. Escape from a potential well is an ubiquitous phenomenon in science. Examples are known in mechanics [4], chemistry [5], quantum optics [6], astrophysics [7], hydrodynamics [8–10], etc., where complex escape phenomena can often be well described by a low-dimensional system of differential equations. In [11,12] the structural stability has been considered for systems modelled by nonlinear ordinary differential equations which are both damped and forced periodically and where the shape of the wave varies. The simplest periodical functions to fulfil these requirements and where is possible to change the shape of the waves are the Jacobian elliptic functions (JEF) [13–15]. If one considers that polynomials are the simplest nonlinear solutions of linear oscillators, it can be seen ([13,14]) that they can be given in terms of JEF’s. In comparison with harmonic functions, JEF functions add a new variable to the parameter space of the system, the elliptic parameter m which is responsible for the shape of the perturbation, i.e., for the temporal rate at which energy is transferred from the excitation mechanism to the system, when the period is fixed. This fact leads us to expect new aspects on the behavior of the system, nonexisting in the harmonic case, when m is varied and the remaining parameters are constant. Following [13,14] let us denote generically a JEF function by pq and by f ðt; A; T ; mÞ ¼ Apqðxt; mÞ a generic modulation of amplitude A and period T and take x xðmÞ ¼ lKðmÞ=T where KðmÞ denotes the complete elliptic integral of the first kind [13,14] and lKðmÞ the period of pq (i.e., 4K or 2K). When A and T are fixed, if we change the elliptic parameter m 2 ½0; 1 then we obtain a change in the shape of the perturbation.
*
Corresponding author. E-mail address: [email protected] (F. Balibrea).
0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.05.016
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In this paper, we show how by altering uniquely the shape of a weak external nonlinear forcing, a system can change from a regular to a chaotic state, and vice-versa. The inhibition or reduction of an initial chaotic state depends on a fully nonlinear situation on the three parameters of the periodic perturbation causing the phenomenon: period, amplitude, and shape (besides, of course, the initial conditions and remaining parameters of the system). To apply the former ideas to particular models, we have selected the Helmholtz oscillator, partly because its chaotic transitions can be predicted, albeit approximately by Melnikov analysis (MM), and partly because is the simple model for a universal chaotic escape situation [16]. In the first place, we consider the Helmholtz oscillator, damped and periodically forced, €x ¼ x bx2 d_x þ c snðxt; mÞ;
ð1Þ
where x denotes the displacement, ðc; d 1Þ, x, d, and c are the normalized parameters of frequency, damping coefficient, and forcing term amplitude, respectively, and snðxt; mÞ is the JEF sine-amplitude (senam) of elliptic parameter m and period T (for a picture of this function, see for example [17]). A nice introduction of this function via a system of differential equations can be seen in [15]. When m ! 0, is snðxt; mÞ ! sinðxtÞ; i.e., one recovers the limit case of the harmonic forcing. This is fundamental when we compare the structural stability of the system when only the forcing shape is varied. In this situation, the parameter space of Eq. (1) is four-dimensional due to the addition of the parameter m to the three-dimensional parameter space ðb; d; cÞ of the corresponding harmonic counterpart. In [11] it is shown that lim snð4Kt=T ; mÞ ¼
m!1
1 4X 1 sin½ð2n þ 1Þ2pt=T : p n¼0 2n þ 1
Note that this is the Fourier expression of the square wave function of period T whose shape is not exactly a square wave, due to the well-known Gibb’s phenomenon [18]. Now if we fixed the parameters b, d, c and T , and vary m from 0 to 1 we are able to appreciate the variations in the shape of the perturbation. Since we are interested in the stability of the chaotic behavior under these changes, a good procedure is to apply Melnikov’s method (MM) to Eq. (1). In the second place, we will consider the equation, €x ¼ x bx2 d_x þ c cnðxt; mÞ;
ð2Þ
where cnðxt; mÞ is the JEF cosine-amplitude (cosam) of parameter m (see again [15,17]). When m ! 0 is cnðxt; mÞ ! cosðxtÞ, i.e., we recover the case of harmonic forcing. When m ! 1 the function tends to a sequence of symmetric pulses of decreasing width. Similarly to the previous case, we study the structural stability of the system when only the forcing shape is varied. Now, we fix the parameters b, d, c, and T , and vary m from 0 to 1 to study the pure effect of variations in the shape of the perturbation. Additionally we also consider what happens when we perturbate only the shape of a weak periodic parametric perturbation of the quadratic term (C), that is, €x ¼ x b½1 þ g snðxt; mÞx2 d_x;
ð3Þ
taking 0 < gb 1; and also the same problem for perturbation of the linear term (L), €x ¼ x½1 þ g0 snðxt; mÞ bx2 d_x;
ð4Þ
for 0 < g0 1. Finally we compare the results.
2. Melnikov analysis Melnikov’s method (MM) is an analytical technique currently available to provide a criterion for the occurrence of homoclinic (and heteroclinic) chaos in continuous systems (see [25]), and is considered as a standard procedure. As it is well known, MM predictions are both approximate (the MM is a first-order perturbative method) and limited (only valid for orbits starting at points sufficiently near to a separatrix). Many applications of the method can be found in the literature, see for examples references [19–26] for further details.
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461
3. Helmholtz oscillator forced with the JEF sn 3.1. Melnikov function The application of MM to Eq. (1) involves calculating the Melnikov function, Z 1 Mðt0 Þ ¼ fdu20 ðtÞ þ cu0 ðtÞ sn½xðt þ t0 Þ; mgdt;
ð5Þ
1
where x0 ¼
t 3 sec h2 ; 2b 2
x_ 0 ðtÞ u0 ðtÞ ¼
t t 3 sec h2 tan h 2b 2 2
ð6Þ
is the parametric representation of the separatrix of the underlying conservative system ðd ¼ c ¼ 0Þ: Using the Fourier expansion of sn [13], plugging Eq. (6) into Eq. (5), and after some simple algebraic manipulations, Eq. (5) is obtained in the form 0 Z 1 1 9d 3pc X pK 2n þ 1 pxt0 1 4 t 2 t p ffiffiffi ffi nþ sec h csch Mðt0 Þ ¼ 2 tan h dt cos K K 2 2 2 2 2bK m 1 4b 1 Z 1 t t pxt 1 tan h sin nþ dt; ð7Þ sec h2 2 2 K 2 1 where K 0 is the complementary complete integral of the first kind. The resulting integrals can be evaluated using standard integral tables [27]. Finally, with xðmÞ 4KðmÞ ¼ T , we obtain 1 6d 96p4 c X ð2n þ 1Þ2pt0 ; ð8Þ an ðmÞbn ðT Þ cos Mðt0 Þ ¼ 2 b n¼0 T 5b where an ðmÞ ¼
1 pffiffiffiffi csch K m
bn ðT Þ ¼
1 T2
1 pK 0 ; nþ 2 K
ð8aÞ
and
2n þ 1 2
2
csch
ð2n þ 1Þ2p2 : T
ð8bÞ
3.2. Threshold function analysis From Eq. (8), it is straightforward to prove that the existence of a homoclinic bifurcation and then chaos is guaranteed for trajectories whose initial conditions are sufficiently near the unperturbed separatrix (6) whenever d < Uf ;sn ðb; m; T Þ; c
ð9Þ
where Uf ;sn ðb; m; T Þ ¼ 80bp4
1 X
an ðmÞbn ðT Þ:
ð10Þ
n¼0
Uf ;sn ðb; m; T Þ will be called the sn-chaotic threshold or simply sn-threshold of the oscillator and has been found calculating the simple zeros of the Melnikov function (8), i.e., studying the values t0 where changes the sign of the Melnikov function. With T and b constant, we represent the sn-threshold as a function of the forcing shape parameter m. A plot of Uf ;sn ðmÞ is shown in Fig. 1. The qualitative form of this function remains the same when b and T are varied. When the coefficient b is varied we can have different situations. If b ! 0 is recovered the linear limit since Uf ;sn is being decreasing, and Uf ;sn ðb ¼ 0; m; T Þ ¼ 0; i.e., in the limit is not possible the chaotic behavior. For the limits cases of harmonic function and square wave we obtain, respectively,
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F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
Fig. 1. sn-Threshold function Uf ;sn ðmÞ vs. m [Eq. (10)] for b ¼ 1=2, and T ¼ 2p.
2 20bp3 2p ; Uf ;sn ðb; m ¼ 0; T Þ ¼ csch T2 T Uf ;sn ðb; m ¼ 1; T Þ ¼
1 80bp2 X ð2n þ 1Þ2p2 ; ð2n þ 1Þcsch T 2 n¼0 T
ð11Þ ð12Þ
with lim Uf ;sn ðb; m ¼ 0; T Þ ¼ 0;
ð13Þ
lim Uf ;sn ðb; m ¼ 1; T Þ ¼ 0;
ð14Þ
lim Uf ;sn ðb; m ¼ 0; T Þ ¼ 0;
ð15Þ
lim Uf ;sn ðb; m ¼ 1; T Þ ¼ 0:
ð16Þ
T !1
T !1
T !0
and T !0
In sum, the sn-threshold functions Uf ;sn ðb; m; T Þ for fixed b, seen as functions of T only, have the same qualitative shape for every m 2 ½0; 1; moreover the case m ! 1 (square wave) is not qualitatively different of the harmonic case ðm ¼ 0Þ. Let us suppose that the system (1) is in a periodic motion state denoted by A (see Fig. 1). Then, increasing m from m1 to m2 , and keeping constant the remaining parameters b, T , d, and c, the system may reach a state (point B) able of being chaotic. On the contrary, if B represents a chaotic escape state, the route B ! A regularizes such a state. Now, fixing b, T , and m we can increase the ratio d=c (increasing d, decreasing c, or both) reaching a periodic state C. This is a known procedure for taming chaos [23,25]. Note that the pathways of types B ! A and B ! C are only particular routes to not escape states. The most common situation consists on a simultaneous variation of d=c and m, as, for example., in the paths B ! D, and B ! E.
4. Helmholtz oscillator forced with the JEF cn 4.1. Melnikov function The application of MM to Eq. (2) involves calculating the Melnikov function, Z 1 Mðt0 Þ ¼ fdu20 ðtÞ þ cu0 ðtÞcn½xðt þ t0 Þ; mgdt:
ð17Þ
1
Using the Fourier expansion of cn [13], plugging Eq. (6) into Eq. (17), and after some simple algebraic manipulation, Eq. (17) can be obtained in the form
F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
0 1 1 t t 9d 3c p X pK 2n þ 1 pxt0 1 nþ Mðt0 Þ ¼ 2 sec h4 sec h tan h2 dt þ pffiffiffiffi sin K K 2 2 2b K m n¼0 2 2 4b 1 Z 1 t t pxt 1 sec h2 tan h sin nþ dt: 2 2 K 2 1
463
Z
ð18Þ
Finally, we obtain Mðt0 Þ ¼
1 6d 96p4 c X ð2n þ 1Þ2pt0 ; c ðmÞb ðT Þ sin n n b T 5b2 n¼0
ð19Þ
where cn ðmÞ ¼
1 1 pK 0 pffiffiffiffi sec h n þ ; K m 2 K
bn ðT Þ ¼
1 T2
ð19aÞ
and
2n þ 1 2
2
csch
ð2n þ 1Þ2p2 : T
ð19bÞ
4.2. Threshold function analysis From Eq. (19), it is straightforward to prove that a homoclinic bifurcation and then chaos is guaranteed for trajectories whose initial conditions are sufficiently near the unperturbed separatrix (6) if d < Uf ;cn ðb; m; T Þ; c
ð20Þ
where the cn-chaotic threshold or simply cn-threshold function is Uf ;cn ðb; m; T Þ ¼ 80bp4
1 X
cn ðmÞbn ðT Þ:
ð21Þ
n¼0
With T and b constant, we study the cn-threshold as a function of the forcing shape parameter m. A plot of Uf ;cn ðmÞ is shown in Fig. 2. The qualitative form of this function remains the same when b and T are varied. For the limits cases of harmonic function and symmetric pulses of low width we obtain, respectively, 2 20bp3 2p Uf ;cn ðb; m ¼ 0; T Þ ¼ ; ð22Þ csch T2 T Uf ;cn ðb; m ¼ 1; T Þ ¼ 0: The limits for these functions when T ! 1, and T ! 0 are 0.
Fig. 2. cn-Threshold function Uf ;cn ðmÞ vs. m [Eq. (21)] for b ¼ 1=2, and T ¼ 2p.
ð23Þ
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F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
Let us suppose that the system (2) is in a chaotic escape state denoted by A (see Fig. 2). Then, increasing m from m1 to m2 , and keeping constant the remaining parameters b, T , d, and c, the system may reach a state (point B) of periodic motion. On the contrary, if B represents a regular state, the route B ! A rouses such a escape. Now, fixing b; T , and m we can increase the ratio d=c (increasing d, decreasing c ; or both) starting from F reaching a periodic state at point C. Note that the pathways of types B ! A, B ! D, and B ! E lead to chaotic escape states.
5. Case of perturbation of the quadratic term 5.1. Melnikov function For Eq. (3) the Melnikov function is, 1 6d 192p4 g X ð2n þ 1Þ2pt0 ; Mðt0 Þ ¼ 2 an ðmÞdn ðT Þ cos T 5b 5b2 n¼0
ð24Þ
where an ðmÞ ¼
1 2n þ 1 pK 0 pffiffiffiffi csch ; 2 K K m
ð24aÞ
and 1 dn ðT Þ ¼ 2 T
(
2n þ 1 2
2 "
4p T
2
2n þ 1 2
2
#) (" # 2 ) 2 2p 2n þ 1 ð2n þ 1Þ2p2 : þ1 þ 1 csch T 2 T
ð24bÞ
5.2. Threshold function analysis From Eq. (24), it is immediate to prove that a homoclinic bifurcation is guaranteed for trajectories whose initial conditions are sufficiently near the unperturbed separatrix (6) if d < Uc;sn ðm; T Þ; g
ð25Þ
where the sn-threshold function is Uc;sn ðm; T Þ ¼ 32p4
1 X
an ðmÞdn ðT Þ:
ð26Þ
n¼0
With T constant, we represent the chaotic threshold as a function of only the forcing shape parameter m. A plot of Uc;sn ðmÞ is shown in Fig. 3. The qualitative form of this function remains the same when T is varied. For the limit cases 2 2 2 p 4p2 4p 4p 2p Uc;sn ðm ¼ 0; T Þ ¼ ; ð27Þ þ1 þ 4 csch 2 T2 T2 T2 T Uc;sn ðm ¼ 1; T Þ
" # " # 2 2 2 2 1 32p2 X 4p 2n þ 1 2p 2n þ 1 ð2n þ 1Þ2p2 ¼ 2 ð2n þ 1Þ ; þ1 þ 1 csch T 2 T 2 T n¼0 T
ð28Þ
The limits for these functions when T ! 1 and T ! 0 are 0. The comments which can be stated are similar to those previously done, we observe that Uc;sn ðmÞ increases very quickly which indicates that the change on the state of the system varying the parameter m from m1 to m2 , and keeping constant the other parameters T , g, and d, will be reached before than in the sn case, i.e., that the interval [m1 ; m2 ] for the change of the state will be smaller than a forcing given for sn. Also, fixing T , and b; limm!0 Uc;sn ðmÞ is greater than limm!0 Uf ;sn ðmÞ, which indicates that the change of the state for the forcing starts for a ratio d=c smaller than the ratio d=g in a parametric perturbation of the quadratic term.
F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
465
Fig. 3. sn-Threshold function Uc;sn ðmÞ vs. m [Eq. (26)] for T ¼ 2p.
6. Case of a perturbation of the linear term 6.1. Melnikov function For Eq. (4) the Melnikov function is, Mðt0 Þ ¼
1 6d 48p4 g0 X ð2n þ 1Þ2pt0 ; a ðmÞe ðT Þ cos n n T b2 n¼0 5b2
ð29Þ
where 1 1 pK 0 pffiffiffiffi csch n þ ; 2 K K m ( # 2 " 2 2 ) 1 2n þ 1 4p 2n þ 1 ð2n þ 1Þ2p2 en ðT Þ ¼ 2 : þ 1 csch T 2 T 2 T
an ðmÞ ¼
ð29aÞ
ð29bÞ
6.2. Threshold function analysis From Eq. (29), it is straightforward to prove that a homoclinic bifurcation is guaranteed for trajectories whose initial conditions are sufficiently near the unperturbed separatrix (6) if d < Ul;sn ðm; T Þ; g0
ð30Þ
where the sn-threshold function is Ul;sn ðm; T Þ ¼ 40p4
1 X
an ðmÞen ðT Þ:
ð31Þ
n¼0
For the limit cases of harmonic function and square wave we obtain, respectively, 2 2 5p 4p2 4p 2p ; þ 1 csch Ul;sn ðm ¼ 0; T Þ ¼ T2 T2 T 2 " # 2 2 1 40p2 X 4p 2n þ 1 ð2n þ 1Þ2p2 Ul;sn ðm ¼ 1; T Þ ¼ 2 ; ð2n þ 1Þ þ 1 csch T n¼0 T T 2 The limits for these functions when T ! 1, and T ! 0 are 0. The comments and analysis that can be done are similar to those made in 5.2.
ð32Þ
ð33Þ
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F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
6.3. Numerical analysis and comparison of results To compare the effect in the chaotic escape behavior of the system of using the functions senam or cosam in the periodical forcing, we introduce the map Cf ;sn;cn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Uf ;cn ðb; m; T Þ; for 8m 2 ½0; 1; T constant, and b constant. In Fig. 4 we can see that Cf ;sn;cn P 1 and if we fix d=c in Uf ;sn ðb; m; T Þ and Uf ;cn ðb; m; T Þ then for each m we have Uf ;sn ðb; m; T Þ P Uf ;cn ðb; m; T Þ; and since if d=c < Uf ;sn ðb; m; T Þ we have chaotic escape. Therefore taking the rest of the parameters constant, chaotic escape exists easier when the forcing is given by the function sn than if it is given by the function cn. Cf ;sn;cn ðmÞ is plotted in Fig. 4 for b ¼ 1=2 and T ¼ 2p. The calculated points have been joined by a dotted curve. Now in order to compare these theoretical points with the real situation, we have computed the probability of escaping towards infinite vs. m of the points of the system. To this end, let us denote by P ðmÞ the number of such points. We normalize them dividing into the harmonic case Psn ðm ¼ 0ÞðPcn ðm ¼ 0ÞÞ and obtain Psn ðmÞ Pcn ðmÞ Pbsn ðmÞ ¼ Pbcn ðmÞ ¼ Psn ðm ¼ 0Þ Pcn ðm ¼ 0Þ where it is easy to see that Psn ðm ¼ 0Þ ¼ Psin ðm ¼ 0Þ ¼ Pcos ðm ¼ 0Þ ¼ Pcn ðm ¼ 0Þ: Finally we compute Pbsn ðmÞ Pbcn ðmÞ which we have plotted also in Fig. 4 with black points. We appreciate that real calculations are well fitted to the theoretical ones. We introduce a similar representation for Cc;l;sn ðmÞ ¼ Uc;sn ðm; T Þ=Ul;sn ðm; T Þ vs: m for T ¼ 2p; which is plotted in Fig. 5, where we appreciate that the chaotic escape is easier using the function senam as a perturbation in the quadratic term than using the same function in the linear one.
Fig. 4. Function Cf ;sn;cn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Uf ;cn ðb; m; T Þ vs. m for b ¼ 1=2 and T ¼ 2p. Function Pbsn ðmÞ= Pbcn ðmÞ vs. m (black points).
F. Balibrea et al. / Chaos, Solitons and Fractals 24 (2005) 459–470
467
Fig. 5. Function Cc;l;sn ðmÞ ¼ Uc;sn ðm; T Þ=Ul;sn ðm; T Þ vs. m for T ¼ 2p.
With Cf ;c;sn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Uc;sn ðm; T Þ vs: m for b ¼ 1=2 and T ¼ 2p we can compare the effect of using the function senam in the forced and in the quadratic terms. In this case we have a decreasing representation (see Fig. 6) which means that when m is increased we have chaotic escape easier using senam as a perturbation of the quadratic term than of the forced term. Similarly, for b ¼ 1=2, and T ¼ 2p we obtain the functions Cf ;l;sn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Ul;sn ðm; T Þ; Cf ;cn;sn ðmÞ ¼ Uf ;cn ðb; m; T Þ=Ul;sn ðm; T Þ; and Cf ;cn;c;sn ðmÞ ¼ Uf ;cn ðb; m; T Þ=Uc;sn ðm; T Þ:
Fig. 6. Function Cf ;c;sn ðmÞ ¼ Uf ;sn ðb; m; T Þ=Uc;sn ðm; T Þ vs. m for b ¼ 1=2, and T ¼ 2p.
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Fig. 7. Basin of attraction of the system €x ¼ x bx2 d_x þ c cnð4KðmÞ t; mÞ, for b ¼ 1, d ¼ 0:1, c ¼ 0:075, and T ¼ 7:391982714 in the T window 0:8 < x < 1:9, 0:8 < x_ < 0:9: (a) m ¼ 0, (b) m ¼ 0:5, (c) m ¼ 0:8, (d) m ¼ 1–1015 .
Fig. 8. Basin of attraction of the system €x ¼ x bx2 d_x þ c sn 4KðmÞ t; m , for b ¼ 1, d ¼ 0:1, c ¼ 0:075, and T ¼ 7:391982714 in the T window 0:8 < x < 1:9, 0:8 < x_ < 0:9: (a) m ¼ 0, (b) m ¼ 0:5, (c) m ¼ 0:8, (d) m ¼ 1–1015 .
The comparisons and remarks we can do are similar to the previous cases. A detailed discussion of all situations can be seen in [17]. It is interesting also plotting for some different initial conditions of the system, the behaviour of xðtÞ when t ! 1 This has been done for the case of periodical forcing with senam and cosam functions where we can state the differences. In the case of perturbation of the quadratic or linear case we can obtain similar representations. With white color is represented the nonescaping basin of attraction zone and with black the escaping one. To generate numerically the basin sets of attraction we have considered a grid of 400 · 400 starting points uniformly distributed in the phase space fxð0Þ 2 ½0:8; 1:9; x_ ð0Þ 2 ½0:8; 0:9g. From the points of this grid each integration of the system is continued until x exceeds at which the system is considered to have escaped (attracted to infinity) or until the maximum number allowable of cycles, here 20 has been reached. When we have an harmonic perturbation ðm ¼ 0Þ, the system presents a stratification of the basin of attraction into zones of escaping and others of nonescaping. If m is increased (in the case of cosam function) the strata of escaping inside the basin are becoming rare and practically there is no of them when m ¼ 1 (see Fig. 7). We have the opposite situation with senam function. Starting in m ¼ 0 (harmonic situation) we have a particular stratification of the basin of attraction (see Fig. 8a). When m is increased, the black strata grow and when m ¼ 1, practically all points escape to infinite (see Fig. 8d).
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6.4. Conclusions Taking the Helmholtz oscillator as a model for a situation of universal chaotic escape from a potential well, we have introduced changes in the shape of a nonlinear periodical perturbation in the linear, quadratic and forced terms of the equation. The changes in the shape of the perturbation have been observed comparing the effects of using the elliptic functions senam and cosam. Numerically we have compared those effects varying the modulus parameter m in the case of the periodical forced perturbation. We conclude that the escape from the potential well is easier when we use the function senam than if we use the function cosam. Also is shown the effect that in the basin of the nonescaping set has the variation in the parameter m in the different situations of senam or cosam. We have also shown a way to reduce or suppress chaotic escape from a potential well, by the method of altering the geometrical shape of weak periodic perturbations, (see Figs. 1 and 2), by changing the parameter m. In resume, changes in the form of the wave of a periodical perturbation depends on the FEJ chosen and on the type of perturbation applied to the system. It is an interesting conclusion that the existence of chaotic escape is easier when the forcing is perturbed by senam function than in the case of cosam function. For a similar situation, see [28].
Acknowledgements The authors acknowledge the MCYT (Ministerio de Ciencia y Tecnologia, Spain) Projects BFM2002-03512 and BFM2002-00010, the JCCM (Junta de Comunidades de Castilla-La Mancha, Spain) Project PAC-02-002, and Fundaci on Seneca (Comunidad Aut onoma de la Regi on de Murcia), Project PI-8/00807/FS/01, for financial support.
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