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 22, 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 impact- ing oscillators is given in [1, 2, 3]. In a previously published work [1]
we performed a detailed analysis of the nonlinear dynamics of an in- verted pendulum bouncing between lateral walls (impacting inverted pendulum). 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 account the 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 practical 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.
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.
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). In Section 4 we show the behaviour of some basins of attraction.
2 The model
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.
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)).
In the conservative unforced system two classes of periodic orbits
are observed in phase space in numerical simulations [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). These two classes of orbits maintain their
existence when the excitation force is added and they are divided by
the homoclinic loops of the hilltop saddle. From the symmetry of
the system, we observe that, when σ = 0, at each confined attractor
x x
-2-1 2
1 0
0
Figure 2: The phase space of the conservative system.
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).
As the damping is turned on, the attractor at x = 0 disappears, while the two attractors at x = ±1 survive. In the single-frequency system [1], we observed a very rich dynamical behaviour; in particular, we identified 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 − x, 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 x = c for confined attractors, and x = 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.
Questa frase la toglierei del tutto, non `e collegata a quello che si fa qui: A common feature observed in the behaviour charts of all attractors is that the region of existence collapses for increasing σ and almost all of them disappear at some σ < 1, with the exception of the 1 − 2 − s attractor for ω ≥ 3, and the 1 − 2 − c and 1 − 1 − c attractors which disappear around σ = 1.1 when ω = 5.
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)).
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.
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 (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.
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
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.
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.
respect to σ for two of the three 1 − 2 − s attractors shown in Fig. 4a
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) .
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.
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.
In Fig. 8 we observe an isola transition where a right period dou- bling cascade at a certain point coalesces with a left period doubling cascade of the 2 − 4 − s attractor which appears in Fig. 5.
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.
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 attractors 1-2-s, ω = 2.
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.
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.
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 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 ?
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.
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
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.
(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.
0.1 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 Figs. 18-21 of this Section we report the basins of attraction for ω = 4.4, γ = 0.61 and several values of σ.
a) b)
Figure 18: Basins of attraction for w = 4.4, γ = 0.61 : a) σ = 0; b) σ = 0.1.
In Fig. 18a the nine attractors which appear at σ = 0 are portrayed
(green = , blue = ... etc.). In Fig. 18a, 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.
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
In order to understand this behaviour, we recall the definition of the critical amplitude, γ cr h in the case of harmonic forcing and γ cr in the present 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)
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)
-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.
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.
-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.
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]).
Moreover, we recall that in the single-frequency system the stable and unstable manifolds of the hilltop saddle become tangent when γ reaches the critical value (5) (see [2] and Fig. 14 in [1]). Below this threshold the manifolds stay disjoint, thus preventing the onset of scattered robust chaotic attractors.
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 are possible 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 - ?? we show the attractor for several values of γ and σ. Figures 26 and 27, where γ = 1.3, show that the attractor
-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.
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.
-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.
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 appears again when σ = 0.066
and it does not disappear as σ increases.
-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
