Milena Petrini, Lucio Demeio
Dipartimento di Ingegneria Industriale e Scienze Matematiche Universit` a Politecnica delle Marche, Ancona, Italy
m.petrini@univpm.it, l.demeio@univpm.it Stefano Lenci
Dipartimento di Architettura, Costruzioni e Strutture Universit` a Politecnica delle Marche, Ancona, Italy
s.lenci@univpm.it May 31, 2020
Abstract
In this work we examine the nonlinear dynamics of an in- verted pendulum between lateral rebounding barriers. We con- tinue the numerical investigation started in [1] by adding the contribution of the second harmonic in the external forcing term. We investigate the behaviour of the periodic attractors by bifurcation diagrams with respect to each amplitude and by behaviour charts of single attractors in the amplitude pa- rameters plane for fixed frequency. We study the effects of the activation of the second harmonic term on the existence domain of each attractor, on local bifurcations and on the changes in the basins of attraction. The behaviour of some robust chaotic attractor is also considered.
Keywords. Inverted pendulum, non-smooth dynamics, chaos, im-
pact damper.
1 Introduction
The impacting inverted pendulum is the basic and the simplest sys- tem in the class of impacting oscillators; a literature survey on some applications of the inverted pendulum and the dynamics of impacting oscillators is given in [1, 2, 3]. In a previous work [1] we performed a detailed analysis of the nonlinear dynamics of an inverted pendu- lum bouncing between lateral walls. An external harmonic excitation was considered, with the presence of a single frequency; systematic numerical simulations on a wide range of frequency and amplitude parameters were performed.
In this paper we extend the study presented in [1] by taking into ac- count a second harmonic term in the external excitation, (see equation (1)).
The motivation for studying the effect of an additional external harmonic term on the dynamics of the impacting inverted pendulum is both theoretical and practical. Theoretically, it is important to un- derstand how the bifurcation diagrams and the basins of attraction are modified as the amplitude of the first harmonic rises from zero and, also, whether new nonlinear phenomena are observed. On the prac- tical side, this study represents another step towards determining the optimal excitation form for chaos control purposes ??. Moreover, the added superharmonic excitation term may be thought of as the rep- resentative of an imperfection or a defect, whose effects are important to study.
Actually, systems with multiharmonic excitation have recently been
studied in the literature. In addition to [2] and [3], where the multi-
harmonic excitations of the impacting inverted pendulum are prop-
erly chosen for the optimal control of the chaotic behaviour, a multi-
harmonmic excitation has been considered in [7] for the gearbox dy-
namics, also in the realm of nonsmooth systems . At the micro- and
nano-scale, multi-harmonic excitations have been proposed to improve the performances of atomic force microscopy [8] and to study a mi- crobeam resonator [9]. At the macro-scale, multi-harmonic excitations in plates and shells have been considered in [10, 11], while multi- harmonic excitations in beams have been investigated in [12, 13], and in [14] for the case of a moving beam. A two-frequency parametric ex- citation of a simple 1 DOF systems has been studied in [15] by means of asymptotic methods.
In order to investigate the behaviour of the main classes of attrac- tors, periodic and robust chaotic attractors, we performed systematic numerical simulations by varying the parameters which characterize the system, namely the frequency and the amplitudes of the two har- monic terms of the external excitation. In particular, we are inter- ested in studying what happens the to the rich dynamical behaviour observed in [1] when a second harmonic term is added, in particular if this latter simplifies or complexifies the dynamical response.
We shall refer to the system studied in [1] as “single-frequency” ex- cited system, while we use the term “double-frequency” excited system for the system studied in this work.
The paper is organized as follows. After the mathematical model
of the pendulum described in Section 2, two main classes of system
responses are investigated separately: periodic attractors (Section 3)
and robust chaotic attractors (Section 5). To complement the previous
analyses, in Section 4 we show the behaviour of some relevant basins
of attraction.
θ θ M
f(t)
V
x
x
( )
−
−
b a ( )
( )
Figure 1: (a) The impacting inverted pendulum; (b) the potential of the unforced conservative system.
2 The model
The dynamics of an excited inverted pendulum impacting between lateral walls is governed by the equations [2]
¨
x + 2δ ˙x − x = f(t), |x| < 1,
˙x(t + ) = −r ˙x(t − ), |x| = 1, (1) where x = θ/θ max , θ is the deflection angle measured from the vertical, δ is the damping coefficient of free motion (0 < δ < 1), r ∈]0, 1] is the restitution coefficient (r = 1 for elastic impacts), t − and t + are the instants of time immediately before and after an impact, and f (t) is the excitation force on the pendulum representing the horizontal acceleration of the base (see Fig. 1 (a)).
In this work we assume an excitation of the form
f (t) = γ sin(ωt) + σ sin(2ωt), (2)
thus adding the second harmonic term to the excitation considered in the single-frequency system [1].
In the absence of the forcing term (corresponding to f (t) = 0) the system (1) represents the unforced impacting pendulum, which has three equilibrium positions: x = 0, which corresponds to a saddle, and x = ±1, which represent stable centers. In the absence of damping and with elastic impacts (δ = 0 and r = 1), the unforced system becomes conservative and possesses the potential energy given by
V (x) = − x 2
2, |x| < 1,
V (x) = ∞, |x| = 1, (3)
which is a double well potential energy, where each well is on a side of the hilltop saddle at x = 0 (see Fig. 1 (b)).
x x
-2 -1 2
1 0
0
Figure 2: The phase space of the conservative system.
In the conservative unforced system two classes of periodic orbits
are observed in phase space [1]: a class of scattered orbits and a class of
confined orbits. The scattered orbits live in both wells of the potential,
the confined orbits live on one well only (see Fig. 2); they are divided
by the homoclinic loops of the hilltop saddle, highlighted with a thick
line in Fig. 2, and maintain their existence when the excitation force is added. From the symmetry of the system, we observe that, when σ = 0, at each confined attractor x(t) there corresponds a specular one, confined in the other potential well. The scattered orbits, on the other hand, can be self-symmetric or, alternatively, they also have their symmetric counterpart. This property is not guaranteed to hold when the second harmonic term in the excitation is turned on (σ > 0), confirming that by the presence of a second harmonic we can describe the effect of the imperfections, which usually break the symmetry of the system.
As the damping is turned on, the equilibrium point at x = 0 disap- pears, while the two attractors at x = ±1 survive, as long as the exci- tation amplitude is small. In the single-frequency system [1], we iden- tified three classes of attractors: periodic attractors, robust chaotic attractors and chattering oscillations.
In this paper, we carry out a systematic numerical analysis of the evolution and the properties of the periodic and chaotic attractors by varying the parameters σ and γ for some fixed values of the frequency ω.
For δ and r we choose δ = 0.05 and r = 0.92 in all our numerical solutions.
3 Two periodic attractors at ω = 2
It is customary to denote periodic attractors by the label n − m − y,
where n represents the period (in units of the period T = 2π/ω of the
excitation force), m the number of impacts with the walls per period
and y = c for confined attractors, and y = s for scattered attractors
[1, 2, 3, 5]. In the numerical investigation on the single-frequency
excited system [1] we observed the coexistence and competition of
several attractors. More precisely, we detected five confined attractors, 1 −1−c, 1−2−c, 1−3−c, 2−1−c and 3−3−c, each living for γ < 1 in the frequency range ω ∈ [0.5, 5] (see Fig. 8 of [1]). In addition, we met one scattered attractor for each of the types 1 − 2 − s, 1 − 4 − s, 3 − 2 − s, 3 − 6 − s, respectively, three of type 1 − 3 − s (with their symmetric counterpart) and two attractors of type 2 − 6 − s.
In this Section we focus our numerical simulations on two of these attractors, the 1 −2−s and 2−1−c, and construct bifurcation diagrams with respect to γ and σ for selected values of the frequency ω. The bifurcation diagrams are accompanied by behaviour charts where the loci of saddle-node (SN) bifurcation and final crisis are reported in the parameter space (σ,γ) with reference to two 1 − 2 − s attractors.
3.1 The 1 − 2 − s attractor for ω = 2
In figures 3-6 we show the bifurcation diagrams of this attractor with respect to γ for σ = 0, 0.1 (figures 3(a) and (b)), 0.15, 0.28 (figures 4(a) and (b)), 0.3, 0.32 (figures 5(a) and (b)), 0.4 and 0.5 (figures 6(a) and (b)) and ω = 2. In the single-frequency system with ω ∈ [1, 5] this attractor lives in a wide range of the amplitude parameter γ: for ω = 2, for example, it is born by a saddle-node bifurcation at γ = 0.372, it undergoes a pitchfork bifurcation at γ = 2.5 and it disappears after a boundary crisis at γ = 3.249 (see Fig. 3(a)).
By increasing σ, we note that the pitchfork bifurcation disappears,
due to the fact that with σ 6= 0 the symmetry is broken by the ex-
citation (Fig. 3b). The range of existence of the scattered chaotic
attractor after the boundary crisis is strongly reduced, and it is re-
placed by a 1 − 2 − s periodic attractor, which undergoes a standard
period doubling (PD) cascade and disappears after the boundary cri-
sis (barely visible in Fig. 3b). As γ increases, the bifurcation diagram
displays the 1 − 2 − s periodic attractor of the single-frequency system
0 0.5 1 1.5 2 2.5 3 3.5 -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
x
P F
0 0.5 1 1.5 2 2.5 3 3.5
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0
x
a) b)
Figure 3: Bifurcation diagrams of attractor 1-2-s, ω = 2 : a) σ = 0.0; b) σ = 0.1.
(shown in Fig. 3a).
By further increasing σ (Fig. 4a) we note that a period doubling bifurcation appears at about γ = 1. Here, the original 1 − 2 − s attractor loses stability in a small interval of γ; the stability is however restored at γ ≈ 1.18 when the period doubling cascade ends.
0 0.5 1 1.5 2 2.5 3 3.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
0 0.5 1 1.5 2 2.5 3 3.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
a) b)
Figure 4: Bifurcation diagrams of attractor 1-2-s, ω = 2 : a) σ = 0.15; b)
σ = 0.28.
The new bifurcation event around γ = 1 becomes more important for larger σ (Fig. 4b). Here the (left) period doubling cascade is more visible; a new, robust, scattered chaotic attractor is born at γ = 1.2 and exists up to γ = 1.495, where a reverse (right) period doubling cascade is observed before the main 1 − 2 − s attractor recovers its stability.
0 0.5 1 1.5 2 2.5 3 3.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
0 0.5 1 1.5 2 2.5 3 3.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
a) b)
Figure 5: Bifurcation diagrams of attractor 1-2-s, ω = 2 : a) σ = 0.3; b) σ = 0.32.
For σ ' 0.3 a new interesting phenomenon occurs (Fig. 5a): the branches of reverse PD cascade to the right of the chaotic attractor at γ ≈ 1.5 and of the PD cascade to the left of the attractor at γ ≈ 2.6 coalesce and give rise to two separate periodic solutions of type 2 −4−s (Fig. 5b). The former 1 − 2 − s no longer exists.
The bifurcation scenario becomes much richer for larger values of σ (see Fig. 6): new scattered chaotic attractors appear (Fig. 6a), they occupy larger and larger intervals in γ and they become the most persistent type of attractor (Fig. 6b).
In figures 7 and 8 we show a sequence of bifurcation diagrams with
respect to σ for two of the three 1 − 2 − s attractors shown in Fig. 4a
0 0.5 1 1.5 2 2.5 3 3.5 -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
0 0.5 1 1.5 2 2.5 3 3.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
a) b)
Figure 6: Bifurcation diagrams of attractor 1-2-s, ω = 2 : a) σ = 0.4 ; b) σ = 0.5.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5
x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
a) b)
Figure 7: Bifurcation diagrams of attractor 1-2-s, ω = 2 : a) γ = 0.4; b) γ = 1.7.
for selected values of γ. For γ = 0.4 (first attractor) a PD bifurcation starts at about σ ≈ 0.62 (see figure 7a), while for γ = 1.7 (second attractor) a PD bifurcation starts at about σ ≈ 0.3, followed by a chaotic attractor for σ ≥ 0.4 (see figure 7b) .
In Fig. 8 we observe an isola transition where a right period dou-
0 0.1 0.2 0.3 0.4 0.5 0.6 -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
Figure 8: Bifurcation diagrams of attractor 1-2-s, ω = 2, γ = 1.9.
bling cascade at a certain point coalesces with a left period doubling cascade of the 2 − 4 − s attractor which appears in Fig. 5. A similar phenomenon has been observed, for example, in the nonlinear dynam- ics of cables [16].
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.5 1 1.5 2 2.5 3 3.5
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
a) b)
Figure 9: Stability domains of (a) the first attarctor and of (b) the second attractor 1-2-s (shown in Figs. 7(a) and (b) respectively), ω = 2.
The stability domains of the first and second attractor in the pa-
rameter space (σ, γ) are shown in Fig. 9. The red curve (lower curve)
represents the loci of SN bifurcation, the blue curve (upper curve) the
loci of the final crisis.
3.2 The 1 − 2 − c attractor for ω = 2
For ω = 2, in the single-frequency system this periodic attractor has an existence domain in the range γ ∈ [0.511, 0.6985].
0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 -1
-0.95 -0.9 -0.85 -0.8 -0.75
x
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
-1 -0.95 -0.9 -0.85
x
a) b)
Figure 10: Bifurcation diagrams of attractor 1-2-c, ω = 2 : a) σ = 0; b) σ = 0.35.
In figures 10-15 we show the bifurcation diagrams with respect to γ for selected values of σ.
In the interval 0.24 ≤ σ ≤ 0.37 we observe that the attractor coalesces with a 1-2-c attractor originating at γ = 0 (whose orbits are slightly different); as a result, an isola bifurcation appears (see Figures 10-11, where σ = 0, 0.35, 0.37 and 0.38): the left attractor starts a period doubling while the right attractor begins a reverse period doubling and they merge creating an isola of a 2 − 4 − c periodic attractor.
At σ = 0.39 the left 1 − 2 − c attractor loses stability by a pe-
riod doubling cascade at the end of which a chaotic attractor appears
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 -1
-0.98 -0.96 -0.94 -0.92 -0.9 -0.88 -0.86 -0.84
x
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
-1 -0.98 -0.96 -0.94 -0.92 -0.9 -0.88 -0.86 -0.84
x
a) b)
Figure 11: Bifurcation diagrams of attractor 1-2-c, ω = 2 : a) σ = 0.37;
b) σ = 0.38.
around γ = 0.128 (see Fig. 12a), while the main 1 − 2 − c attractor shows a reverse (left) period doubling cascade before recovering its stability (see Fig. 12b).
At σ = 0.8, the main 1 − 2 − c attractor is no longer present and we have instead two unconnected 1 − 2 − c attractors (see Fig. 14, the left one undergoing a reverse period doubling cascade.
Servirebbe qualche figura intermedia tra σ = 0.39 e σ = 0.7. I due attrattori sconnessi sono gli stessi delle Fig. 10b e 11a ?
In Figures 15-16 we show the bifurcation diagrams with respect to the parameter σ. We observe the wide σ-range in which the two unconnected attractors live.
The stability domain of the 1-2-c for ω = 2 in the (σ, γ) parameter
plane is shown in Fig. 17. As in figures 9(a) and (b), the red curve
(lower curve) represents the loci of SN bifurcation, the blue curve
(upper curve) the loci of the final crisis. The separation between the
two 1-2-c attractors noted at σ ≈ 0.8 corresponds to the jump in the
red curve of the SN bifurcation.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 -1
-0.98 -0.96 -0.94 -0.92 -0.9 -0.88 -0.86 -0.84
x
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
-1 -0.98 -0.96 -0.94 -0.92 -0.9 -0.88 -0.86 -0.84
x
a) b)
Figure 12: Bifurcation diagram of attractors 1-2-c, ω = 2 and σ = 0.39 : a) left 1 − 2 − c attractor; b) main 1 − 2 − c attractor.
0.24 0.26 0.28 0.3 0.32 0.34 0.36
-1 -0.99 -0.98 -0.97 -0.96 -0.95 -0.94
x
0.24 0.26 0.28 0.3 0.32 0.34 0.36
-1 -0.99 -0.98 -0.97 -0.96 -0.95 -0.94 -0.93
x
a) b)
Figure 13: Bifurcation diagram of the main 1 − 2 − c attractor, ω = 2 : a)
σ = 0.7; b) σ = 0.75.
0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 -1
-0.98 -0.96 -0.94 -0.92 -0.9 -0.88 -0.86 -0.84 -0.82
x
Figure 14: Bifurcation diagram of the main 1 − 2 − c attractor, ω = 2, σ = 0.8.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7
x
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-1 -0.99 -0.98 -0.97 -0.96 -0.95 -0.94 -0.93 -0.92 -0.91
x
a) b)
Figure 15: Bifurcation diagrams of attractor 1-2-c, ω = 2 : a) γ = 0.0; b)
γ = 0.3.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -1
-0.99 -0.98 -0.97 -0.96 -0.95 -0.94 -0.93 -0.92 -0.91
x
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-1 -0.99 -0.98 -0.97 -0.96 -0.95 -0.94 -0.93 -0.92
x
a) b)
Figure 16: Bifurcation diagrams of attractor 1-2-c, ω = 2 : a) γ = 0.31;
b) γ = 0.32.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
Figure 17: Stability domain of attractor 1-2-c, ω = 2.
4 Basins of attraction for ω = 4.4, γ = 0.61
The impacting inverted pendulum exhibits the interesting and impor- tant dynamical feature of multistability, that is coexistence of several attractors, both periodic and chaotic. In [1] we constructed a map of coexisting periodic attractors for several values of ω and γ; in partic- ular, we observed that multistability is relevant at large frequencies and small amplitudes (see Fig. 8 in [1]). We also gave an example of multistability through the portrait of the basins of attraction of nine coexisting attractors for ω = 4.4 and γ = 0.61, the nine attractors being the two left and right rest positions, two 1 − 1 − c, 1 − 2 − c, 3 − 3 − c attractors and one 3 − 2 − s attractor (see Fig. 9 in [1] and Fig. 18a here below). In this section we are interested in studying how this very rich attractor scenario is modified by the addition of the superharmonic term in the external excitation. We select this partic- ular attractor, among many others, because we believe that it is the most interesting.
In Figs. 18-21 we report the basins of attraction for ω = 4.4, γ = 0.61 and for increasing values of σ.
In Fig. 18a the nine attractors which appear at σ = 0 are portrayed
(green = , blue = ... etc.). In Fig. 18b, corresponding to σ = 0.1,
quanti attrattori? descrivi con i colori - lo stesso per le figure successive
These figures show that the number of periodic attractors decreases
as σ increases and that the fractal dimension of the basins becomes
more pronounced (see Figures 19 - 20). From the bifurcation diagrams
with respect to σ (not shown), we observed that all periodic attractors
disappear for σ ≥ 0.4, leaving only the rest positions and the chaotic
attractors in place. This is also illustrated in Fig. 20b (σ = 0.5) where
we notice the presence of a confined chaotic attractor, and in Fig. 22
(σ = 0.8) where only a chaotic attractor is present. From a practical
point of view, the conclusion is that the ”imperfections” (due to the
a) b)
Figure 18: Basins of attraction for w = 4.4, γ = 0.61 : a) σ = 0; b) σ = 0.1.
superharmonic terms), as a consequence of the symmetry breaking, tend to regularize somewhat the dynamical outcome, according to the fact that a symmetrical situation is structurally unstable.
a) b)
Figure 19: basin of attraction for w = 4.4, γ = 0.61 : a) σ = 0.2; b)
σ = 0.3.
a) b)
Figure 20: basin of attraction for w = 4.4, γ = 0.61 : a) σ = 0.4; b) σ = 0.5.
a) b)
Figure 21: basin of attraction for w = 4.4, γ = 0.61 : a) σ = 0.6; b)
σ = 0.7.
Figure 22: basin of attraction for w = 4.4, γ = 0.61, σ = 0.8
5 Two robust chaotic attractors
With the aim of exploring different regions of the stability diagram shown in [1], we investigate in this section the behaviour of two robust chaotic attractors of the single-frequency system at ω = 18 and at ω = 3.
5.1 ω = 18, γ = 5.6935
The numerical simulations of the single-frequency system reported in [1] showed a robust chaotic attractor at the frequency ω = 18. This attractor is confined for γ < 5.6935 and scattered for γ ≥ 5.6935, consistently with the multistability map presented as Fig. 14 of [1], where it is clearly shown that in the range of large frequencies (ω >
ω cr ' 10.8) confined attractors are possible and are the first to appear as γ increases.
In this section we investigate the effect of the second harmonic term on this attractor for γ = 5.6935 and ω = 18 and show the results in Figures 23 - 25 in phase space. In figures 23(a) and (b) the attractor is shown for σ = 0 and σ = 0.05, respectively; here it remains a scattered attractor. In figures 24(a) and (b) we have σ = 0.1 and σ = 0.3 and in figures 25(a) and (b) σ = 1 and σ = 1.1. It appears from these results that the attractor becomes confined at σ = 0.1 and remains confined for σ > 0.1.
Nella figura 21a si vede un secondo attrattore sbiadito. Cos`e ? Va menzionato e descritto
Since the transition from confined to scattered chaotic attractor is
governed, in this case, by the homoclinic bifurcation of the hilltop sad-
dle, we recall the definition of the critical amplitude for the homoclinic
bifurcation, γ h cr in the case of harmonic forcing and γ cr in the present
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2
-1.5 -1 -0.5 0 0.5 1 1.5
2 gamma = 5.6935, sigma = 0.0, w = 18.0, x0 = -0.2566, p0 = 0.5691
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5
2 gamma = 5.6935, sigma = 0.05, w = 18.0, x0 = -0.2566, p0 = 0.5691
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a) b)
Figure 23: scattered chaotic attractor, γ = 5.6935, ω = 18 : a) σ = 0; b) σ = 0.05.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5
2 gamma = 5.6935, sigma = 0.1, w = 18.0, x0 = -0.2566, p0 = 0.5691
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5
2 gamma = 5.6935, sigma = 0.3, w = 18.0, x0 = -0.2566, p0 = 0.5691
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a) b)
Figure 24: confined chaotic attractor, γ = 5.6935, ω = 18 : a) σ = 0.1; b) σ = 0.3.
case, respectively (see [1], [2]):
γ cr h = [(1 − r) √
1 + δ 2 + (1 + r)δ]p(1 + ω 2 ) 2 + (2δω) 2 q
[(1 − r) √
1 + δ 2 + (1 + r)δ] 2 + [ω(1 + r)] 2
, (4)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2
-1.5 -1 -0.5 0 0.5 1 1.5
2 gamma = 5.6935, sigma = 1.0, w = 18.0, x0 = -0.2566, p0 = 0.5691
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5
2 gamma = 5.6935, sigma = 1.1, w = 18.0, x0 = -0.2566, p0 = 0.5691
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a) b)
Figure 25: confined chaotic attractor, γ = 5.6935, ω = 18 : a) σ = 1.0; b) σ = 1.1.
γ cr = γ cr h 1
M , M = max τ ∈[0,2π] {h(τ)} (5) where
h(τ ) = sin(τ + Φ 1 ) + h 2 sin(2τ + Φ 2 ), h 2 = σC 2
γC 1
, (6)
C j = Λ j
rΓ j
> 0, Λ j = q
(1 + r) 2 (jω) 2 + [(1 − r) √
1 + δ 2 + (1 + r)δ] 2 , Γ j = p(1 + (jω) 2 ) 2 + (2δjω) 2 , Φ j = φ j + ν j , tan(φ j ) = 2δjω
1 + (jω) 2 , π ≤ φ j ≤ 3π/2, tan(ν j ) = (1 + r)jω
(1 − r) √
1 + δ 2 + (1 + r)δ , 0 ≤ ν j ≤ π/2, j = 1, 2.
The 2π-periodic function h(τ ) has zero mean value in the period;
as a consequence, the maximum value M in (5) is non negative.
In the single-frequency case we have h(τ ) = sin(τ + Φ 1 ) and there- fore M = 1; the associated critical curve γ cr h is considered as a refer- ence for measuring the relative reduction of the chaotic zone in the parameter plane (see [2], [6]).
When the second harmonic term is present in the excitation, we
have M < 1 (see [2]), therefore the critical curve γ = γ cr will be higher than the critical curve γ = γ cr h of the single-frequency system.
As a consequence, the zone of possibly chaotic response of the system is reduced (see [2]).
The mechanism of transition from confined to scattered chaotic attractors occurring in symmetric systems and named merging-type symmetric boundary crisis, happens on the critical curve γ = γ cr ; since γ cr is higher than the critical curve γ cr h , the transition from the scattered to the confined chaotic attractor for σ ≥ 0.1 occurs at γ >
γ cr h .
5.2 ω = 3, γ = 1.3 and γ = 2.26
Another robust scattered chaotic attractor was observed at ω = 3, a frequency value belonging to the range of small frequencies ω <
ω cr (ω cr ' 10.8) where only scattered chaotic attractors have been observed in the single-frequency case [1]. This attractor lives in the interval γ ∈ [1.25, 2.2562]; for γ = 2.2562 it touches its basin boundary and disappears (at γ = 2.2563 it is no longer observed).
In figures 26 - 29 we show the attractor for several values of γ and σ. Figures 26 and 27, where γ = 1.3, show that the attractor remains scattered for σ < 0.48, it turns into a confined attractor for σ ≥ 0.48, it shrinks as σ increases and it disappears at σ = 1.275.
From these results we conclude that this kind of attractor is possible for the double-frequency system (1) at small frequencies.
The situation looks somewhat different at γ = 2.26 (see Fig. 28
- 29). As the attractor moves close to the boundary crisis (dov’`e ),
we observe that the scattered attractor still exists when σ = 0.066
and it does not disappear as σ increases. The conclusion is that the
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2
-1.5 -1 -0.5 0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a) b)
Figure 26: scattered chaotic attractor, γ = 1.3, ω = 3 : a) σ = 0; b) σ = 0.45.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a) b)
Figure 27: confined chaotic attractor, γ = 1.3, ω = 3 : a) σ = 0.48; b) σ = 1.2.
added superharmonic terms not always destroy the scattered chaotic
attractor, which in some cases can survive this ”imperfection”.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2
-1.5 -1 -0.5 0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a) b)
Figure 28: scattered chaotic attractor, γ = 2.26, ω = 3 : a) σ = 0.066; b) σ = 0.4.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
