Milena Petrini, Lucio Demeio
Dipartimento di Ingegneria Industriale e Scienze Matematiche Universit`a Politecnica delle Marche, Ancona, Italy
[email protected], [email protected] Stefano Lenci
Dipartimento di Architettura, Costruzioni e Strutture Universit`a Politecnica delle Marche, Ancona, Italy
[email protected] July 7, 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 [3]. 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- harmonic excitation has been considered in [7] for the gearbox dynam- ics, 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 microbeam resonator [9]. At the macro-scale, multi-harmonic excitations in plates and shells have been considered in [10, 11], while multi-harmonic exci- tations in beams have been investigated in [12, 13], and in [14] for the case of a moving beam. A two-frequency parametric excitation 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
Mf(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/θmax, 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) = −x22, |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
The time history and the phase portrait of this periodic attractor are outlined in Figs. 3a and 3b, respectively.
In Figures 4-7 we show the bifurcation diagrams of this attractor with respect to γ for σ = 0, 0.1 (Figures 4a and 4b), 0.15, 0.28 (Figures 5a and 5b), 0.3, 0.32 (Figures 6a and 6b), 0.4 and 0.5 (Figures 7a and 7b) 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. 4a).
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. 4b). 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
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 t/T
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
x
-1 -0.5 0 0.5 1
x -2.5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
a) b)
Figure 3: The scattered periodic attractor 1-2-s for ω = 2, σ = 0. a) Time history; b) phase portrait.
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
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 4: Bifurcation diagrams of attractor 1-2-s, ω = 2 : a) σ = 0; b) σ = 0.1.
period doubling (PD) cascade and disappears after the boundary cri- sis (barely visible in Fig. 4b). As γ increases, the bifurcation diagram displays the 1− 2 − s periodic attractor of the single-frequency system (shown in Fig. 4a).
By further increasing σ (Fig. 5a) 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 5: Bifurcation diagrams of attractor 1-2-s, ω = 2 : a) σ = 0.15; b) σ = 0.28.
The new bifurcational event around γ = 1 becomes more important for larger σ (Fig. 5b). 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. 6a): 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. 6b). The former 1− 2 − s no longer exists.
The bifurcation scenario becomes much richer for larger values of σ (see Fig. 7): the scattered chaotic attractors occupy larger and larger intervals in γ (Fig. 7a) and they become the most persistent type of
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
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
a) b)
Figure 6: 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 7: Bifurcation diagrams of attractor 1-2-s, ω = 2 : a) σ = 0.4 ; b) σ = 0.5.
attractor (Fig. 7b).
In Figures 8 and 9 we show a sequence of bifurcation diagrams with respect to σ for two of the three 1− 2 − s attractors shown in Fig. 5a for selected values of γ. For γ = 0.4 (first attractor) a PD bifurcation starts at about σ ≈ 0.62 (see Figure 8a), while for γ = 1.7 (second
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 8: Bifurcation diagrams of attractor 1-2-s, ω = 2 : a) γ = 0.4; b) γ = 1.7.
attractor) a PD bifurcation starts at about σ ≈ 0.3, followed by a chaotic attractor for σ ≥ 0.4 (see Figure 8b) .
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 9: Bifurcation diagrams of attractor 1-2-s, ω = 2, γ = 1.9.
In Fig. 9 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. 6. 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
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
a) b)
Figure 10: Stability domains of (a) the first attarctor and of (b) the second attractor 1-2-s (shown in Figs. 8(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. 10. 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
The time history and the phase portrait of this periodic attractor are shaped in Figs. 11a and 11b, respectively.
For ω = 2, in the single-frequency system this periodic attractor has an existence domain in the range γ ∈ [0.511, 0.6985].
In Figures 12-17 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 appears (see Figures 12-13, where σ = 0, 0.35, 0.37 and 0.38): the left attractor starts a period
0 0.5 1 1.5 2 2.5 3 3.5 t/T
-1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7
x
-1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7
x -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
a) b)
Figure 11: The confined periodic attractor 1-2-c for ω = 2, σ = 0. a) Time history; b) phase portrait.
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 12: Bifurcation diagrams of attractor 1-2-c, ω = 2 : a) σ = 0; b) σ = 0.35.
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. 14a), while the main 1− 2 − c attractor
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
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
x
a) b)
Figure 13: Bifurcation diagrams of attractor 1-2-c, ω = 2 : a) σ = 0.37;
b) σ = 0.38.
shows a reverse (left) period doubling cascade before recovering its stability (see Fig. 14b).
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. 16, the left one undergoing a reverse period doubling cascade.
In Figures 17-18 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. 19. As in Figures 10a and 10b, 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 14: 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 15: 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
x
Figure 16: 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 17: 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 18: 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 19: 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. 20a 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 case, among many others with many attractors, because we believe that it is the most interesting.
In Figures 20-24 we report the basins of attraction for ω = 4.4, γ = 0.61 and for increasing values of σ.
In Fig. 20a the nine attractors which appear at σ = 0 are portrayed.
In Fig. 20b, corresponding to σ = 0.1, there are five attractors: the two left and right rest positions (green blue and orange yellow colors, respectively), two 1−1−c attractors (green and light blue colors) and a left 1− 2 − c attractor (blue color).
In Fig. 21a, corresponding to σ = 0.2, the attractors are three: the two left and right rest positions (orange yellow and green colors, re- spectively), a left 1− 1 − c attractor which has just started the period doubling (green blue color). In Fig. 21b, corresponding to σ = 0.3, the attractors are the same of Fig. 21a, the confined attractor is now 4− 4 − c (light blue color) and the chattering in the rest positions has slightly modified.
a) b)
Figure 20: Basins of attraction for w = 4.4, γ = 0.61 : a) σ = 0; b) σ = 0.1.
In Fig. 22a, corresponding to σ = 0.4, we find the 3− 2 − s attractor that here is 6−4−s (green color) with the left and right rest positions (orange yellow and green blue colors).
In Fig. 22b, corresponding to σ = 0.5, we find only the lateral posi- tions that present chattering.
In Fig. 23a, corresponding to σ = 0.6, we have the left and right posi- tions (light blue and orange yellow colors) and three chaotic confined right attractors (with different ranges in x).
In Fig. 23b, corresponding to σ = 0.7, we find the left rest position with a modified chattering (orange yellow color), a chaotic right at- tractor that from confined becomes scattered (light blue color) and two chaotic confined right attractors (green and green blue colors).
In Fig. 24a, corresponding to σ = 0.8, we see seven chaotic attractors that from confined become scattered, except the one with blue color that remains confined.
Eventually in Fig. 24b, corresponding to σ = 0.9, we find nine chaotic attractors among which the one with orange yellow color remains con- fined, whereas the others are confined in the beginning and then scat-
tered.
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 21 - 24). 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. 22b (σ = 0.5) where we notice the presence of a confined chaotic attractor, and in Figs. 24 (σ = 0.8 and σ = 0.9) where only chaotic attractors are present. From a practical point of view, the conclusion is that the
”imperfections” (due to the superharmonic terms), as a consequence of the symmetry breaking, tend to regularize somewhat the dynam- ical outcome, according to the fact that a symmetrical situation is structurally unstable.
a) b)
Figure 21: basin of attraction for w = 4.4, γ = 0.61 : a) σ = 0.2; b) σ = 0.3.
a) b)
Figure 22: basin of attraction for w = 4.4, γ = 0.61 : a) σ = 0.4; b) σ = 0.5.
a) b)
Figure 23: basin of attraction for w = 4.4, γ = 0.61 : a) σ = 0.6; b) σ = 0.7.
a) b)
Figure 24: basin of attraction for w = 4.4, γ = 0.61, : a) σ = 0.8; b) σ = 0.9.
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 = 1.95
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 < 1.656 and scattered for γ ≥ 1.656, consis- tently with the multistability map presented as Fig. 14 of [1], where it is clearly shown that in the range of large frequencies (ω > ωcr ' 10.8) confined attractors are possible and are the first to appear as γ in- creases.
In this section we investigate the effect of the second harmonic term on this attractor for γ = 1.95 and ω = 18 and show the results in Figures 25 - 28 in phase space. In Figures 25a and 25b the attractor is shown for σ = 0 and σ = 0.1, respectively; here it remains a scattered attractor. In Figures 26a (σ = 0.19) and 26b (σ = 0.2) we see that the attractor becomes confined. It remains confined (see Figs 27a and 27b) until σ = 3.3, when it is barely visible; then from σ = 3.4 on it is again scattered as shown in Figures 28a and 28b when σ = 3.8 and σ = 4.1.
It appears from these results that the attractor becomes confined at σ ∼= 0.2. Since the transition from confined to scattered chaotic attractor is governed, in this case, by the homoclinic bifurcation of the hilltop saddle,we recall the definition of the critical amplitude for the homoclinic bifurcation, γcrh in the case of harmonic forcing and γcr in the present case, respectively (see [1], [2]):
a) b)
Figure 25: scattered chaotic attractor, γ = 1.95, ω = 18 : a) σ = 0; b) σ = 0.1.
a) b)
Figure 26: scattered and confined chaotic attractor, γ = 1.95, ω = 18 : a) σ = 0.19; b) σ = 0.2.
γcrh = [(1− r)√
1 + δ2+ (1 + r)δ]p(1 + ω2)2+ (2δω)2 q
[(1− r)√
1 + δ2+ (1 + r)δ]2+ [ω(1 + r)]2
, (4)
γcr = γcrh 1
M, M = maxτ ∈[0,2π]{h(τ)} (5)
a) b)
Figure 27: confined chaotic attractor, γ = 1.95, ω = 18 : a) σ = 1.0; b) σ = 3.0.
a) b)
Figure 28: scattered chaotic attractor, γ = 1.95, ω = 18 : a) σ = 3.8; b) σ = 4.1.
where
h(τ ) = sin(τ + Φ1) + h2sin(2τ + Φ2), h2 = σC2
γC1
, (6)
Cj = Λj
rΓj
> 0, Λj = q
(1 + r)2(jω)2+ [(1− r)√
1 + δ2+ (1 + r)δ]2,
Γj = p(1 + (jω)2)2 + (2δjω)2, Φj = φj + νj, tan(φj) = 2δjω 1 + (jω)2, π ≤ φj ≤ 3π/2, tan(νj) = (1 + r)jω
(1− r)√
1 + δ2+ (1 + r)δ, 0 ≤ νj ≤ 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 + Φ1) and there- fore M = 1; the associated critical curve γcrh is considered as a refer- ence for measuring the relative reduction of the chaotic zone in the parameter plane (see [2], [6]).
In our case, when the second harmonic term is present in the ex- citation we have M < 1: a good approximation of M is given by M1 = 1− σ
γ · 0.5009. Therefore the critical curve γ = γcr will be higher than the critical curve γ = γcrh of the single-frequency system.
As a consequence, the zone of possibly chaotic response of the system is reduced (see [2]).
For γ = γcrh one has M1 = 1− σ · 0.0040 and the value γcr in (5) differs from γcrh in the third decimal place from σ = 0.14 to σ = 0.28.
As considered in [1], for γ only slightly larger than γcrh the onset of the scattered chaotic attractor requires an extremely long transient and with a shorter transient we see the phenomenon at a larger value of γ. This is the reason of the choice γ = 1.95, in which the scattered chaotic attractor appears more clearly.
We observed that when γ < 1.656, for σ > 0 the confined attractor reduces its size and it disappears before σ = 1; then another confined attractor appears nearby σ = 2.5 and it becomes scattered for growing σ. When 1.656≤ γ ≤ 1.81 we noticed a reduction of the gap between the value of σ in which the confined attractor disappear and the new attractor appears. At last, when γ ≥ 1.82 we have not find a gap: the
confined attractor, when it is barely visible, becomes more pronounced for growing σ, enlarging until it transforms into a scattered chaotic attractor.
5.2 ω = 3, γ = 1.04 and γ = 1.2
Another robust scattered chaotic attractor was observed at ω = 3, a frequency value belonging to the range of small frequencies ω <
ωcr (ωcr ' 10.8) where only scattered chaotic attractors have been observed in the single-frequency case [1]. This attractor lives when γ ≥ 1.025, for γ ∼= 1.222 it touches its basin boundary and disappears.
Successively, the periodic 1− 2 − s attractor (with the orange colored basin of attraction) that coexists with the chaotic attractor becomes globally attracting.
In Figures 29 - 34 we show the attractor for several values of γ and σ.
When γ = 1.04, Figures 29 - 31 show how the coexistence between the
a) b)
Figure 29: scattered chaotic attractor, γ = 1.04, ω = 3 : a) σ = 0; b) σ = 0.2.
periodic and the chaotic attractors change when σ > 0. The scattered chaotic attractor lives inside its basin of attraction (see Fig. 29a and
a) b)
Figure 30: confined chaotic attractor, γ = 1.04, ω = 3 : a) σ = 0.45; b) σ = 0.7.
29b) and it is still present after touched the basin boundary (see Fig.
30a when σ = 0.45). When σ ∼= 0.72 it disappears (Figs. 30b and 31a), whereas the size of the 1 − 2 − s basin of attraction reduces as σ grows (Figs. 31 and 32). The basin of attraction of the chaotic attractor is still the place of a chaotic scattered attractor (not robust), visible for some value of σ as in Figs. 30b and 31b.
The situation looks somewhat different at γ = 1.2 (see Figs. 33 - 34). For σ = 0 the chaotic attractor is close to its basin boundary (see Fig. 33a); we observe that it still exists until σ = 0.09 when it touches its basin boundary (as in Fig. 33b) and then disappears (Fig.
34a where σ = 0.1). Then in its basin we see more and more points of the 1− 2 − s basin of attraction (Fig. 34b where σ = 0.2).
At last we observed that, when γ > 1.222, for σ > 0 the coexisting 1− 2 − s attractor becomes globally attracting even if it remains the
’ghost’ of the basin of attraction of the chaotic attractor (not showed).
The conclusion is that the added superharmonic terms not always
a) b)
Figure 31: scattered chaotic attractor, γ = 1.04, ω = 3 : a) σ = 0.72; b) σ = 0.73.
a) b)
Figure 32: scattered chaotic attractor, γ = 1.04, ω = 3 : a) σ = 0.8; b) σ = 0.9.
destroy the scattered chaotic attractor, which in some cases can sur- vive this ”imperfection”.
a) b)
Figure 33: scattered chaotic attractor, γ = 1.2, ω = 3 : a) σ = 0; b) σ = 0.09.
a) b)
Figure 34: scattered chaotic attractor, γ = 1.2, ω = 3 : a) σ = 0.1; b) σ = 0.2.
6 Conclusions
We continued a systematic investigation of the nonlinear dynamics of an inverted pendulum between lateral walls started in [1] and report the behaviour of two periodic attractors, some basins of attraction and
two chaotic attractors.
In the evolution of the periodic attractors in the parameters plane (σ, γ) for a fixed frequency ω, we remark the ways in which they lose stability, as they may connect with a periodic attractor that originates at γ = 0 (in the case of confined attractors) and the formation of isolas of periodic orbits.
A common feature observed in the behaviour charts of all periodic attractors is that the region of existence collapses for increasing σ and almost all of them disappear for some σ < 1 : this is confirmed also by the basins of attraction in Section 4.
For the chaotic attractors, we distinguish what happens in a case occurring for a large frequency from a low frequency one.
In the former case we have that, with an excitation no longer harmonic, the critical curve γ = γcr is higher than the critical curve γ = γcrh corresponding to the harmonic forcing. As the mechanism of transition from confined to scattered chaotic attractors occurring in symmetric systems and named merging-type symmetric boundary crisis, verifies on the critical curve γ = γcr, when ω = 18 we can observe a confined attractor for σ ≥ 0.2.
For a low frequency, where in case of harmonic forcing only scat- tered chaotic attractors arereported, we observe how much the chaotic attractor is still present for increasing σ and the evolution of its basin of attraction.
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