Multistability and synchronization: The co-existence of synchronous patterns in coupled pendula
Introduction
The problem of multistability in complex dynamical systems has been known for decades [1], [2], [3] and has been extensively studied and developed in different areas of nonlinear sciences. The co-existence of various attractors, which occurrence strictly depends on the initial/border conditions of the system and its parameters has to be taken into account when the research is focused on the comprehensive description of the dynamics and possible practical applications of the observed behaviours. The problem becomes even more crucial, when particular solutions lead to catastrophic events, just to mention the crash of aircraft YF-22 Boeing in April 1992 [4].
Multistability has been reported in various systems, including the classical ones like Lorenz model [5] or Chua’s circuit [6], chaotic flows [7], [8], [9], chemical oscillators [10], [11] or delayed networks [12], [13], [14]. The phenomenon is strongly related to the evolution of biological systems [15], [16] and its theory has been applied to describe the phenomenon of the lactose utilization process [17], the activity of the brain [18], [19] or the dynamics of neural networks [13], [20], [21], just to mention a few. The concept of multistability allows to understand better the complex phenomena occurring in practical installations and devices used in modern technique and engineering. In [22] one can find the study on the multistability of highly turbulent flows, while in [1], [23] and [24] the Authors investigate the models of lasers and semiconductor superlattices, respectively. The research on the network-induced multistability in power grids can be found in [25].
One of the main concepts related to multistability is the phenomenon of extreme multistability [26], when the number of co-existing attractors tends to infinity and the variety of possible solutions becomes practically unlimited. This type of property has been described in chaotic flows [8], [9], Chua’s circuit [27] or chemical nodes [10]. In [28] the study on the extensive multistability in oscillator simplexes has been shown, while in [29] Hens et al. propose the methods allowing to obtain the phenomenon in coupled dynamical systems. With the increase of the number of possible co-existing solutions (especially the undesired ones), the problem of their control arises. The results of the control of multistability have been discussed in [30], [31], [32], including the possibility of hidden attractors [32]. The latter phenomenon, i.e. hidden oscillations [33], [34] along with the high multistability of the system can make its analysis even more complex, leading to the scenario, when not only the number of co-existing states is unknown but also the methods of their detection become not straightforward.
In this paper we investigate the dynamics and possible multistability of 3 DOF system of classical pendula (two nodes) suspended on the oscillating beam (the coupling platform). The model is schematically shown in Fig. 1.
The system consists of the beam of mass , which is connected with the support by the spring of stiffness and the damper with the damping coefficient . The variable denotes the displacement of the beam. The angle between the direction of the platform’s oscillations and the horizontal axis is given by parameter (fixed during the oscillations). The beam interacts with two suspended planar pendula (red and blue in Fig. 1), which dynamics can be described using variables . The masses of the pendula and their lengths are denoted by parameters and , respectively.
The dynamics of the model shown in Fig. 1 can be determined using Lagrange equations, which in this case take the following form: where .
The pendula of model (1) are supplied with van der Pol type drives, which allows them to self-excite. The parameters of the drives are given by (the damping coefficient) and , as denoted in Eqs. (1).
During the research we have fixed the following parameters of the considered system: , , , , and (the standard gravity). The damping coefficient has been varied to preserve fixed logarithmic decrement of the nodes (equals to ) and its influence on the observed behaviours has not been studied.
In the uncoupled case (the unmovable beam), the pendula with above parameters (when ) oscillate with amplitude (20) and period .
It should be noted, that for the values of the parameters considered in our study (see Sections 2 The identical pendula, 3 The non-identical pendula for details), the beam is always over the resonance and below the critical damping.
The results discussed in this paper continue and extend the research performed previously, when the direction of the beam’s oscillations has not been considered as one of the model’s parameters. Namely, when (see Fig. 1), the platform is oscillating horizontally, which scenario has been studied e.g. in [35], [36], [37], [38]. The results obtained in this case include both oscillating and rotating nodes [35], [36], the study of the influence of the coupling structure’s parameters on global dynamics [37] or the basin stability problems [38]. On the other hand, when , the beam is moving vertically, which has been investigated e.g. in [39], [40], [41]. The results discussed in these papers exhibit, that when the direction of the beam’s oscillations is changed (from horizontal to vertical), the dynamics also changes and one can observe new types of solutions (e.g., the quarter-phase synchronization) and various scenarios of their co-existence.
The difference between the cases of the horizontal [35], [36], [37], [38] and the vertical [39], [40], [41] oscillations is significant, since depending on the direction of the beam’s movements, the suspension points of the pendula are excited differently.
In this paper we extend the research presented in the previous works, considering the case, when the beam is oscillating in various directions. Parameter determines the direction of the oscillations (see Fig. 1 for details) and apart from the previously studied scenarios ( – horizontal ; – vertical), it can be set at any value in range . We investigate the influence of the direction of the beam’s oscillations (parameter ) on the behaviour of the suspended pendula, which has not been considered previously. Our results include the occurrence of different synchronization patterns (e.g., the phase-locked state, which is possible only in particular range of ), the study of the co-existence (multistability) and uniqueness, as well as the use of typical tools of dynamical systems analysis (e.g., bifurcation and energy diagrams or basin stability).
The synchronization problems can be found widely in different types of mechanical systems [42] and involve such phenomena as stability [43] and control [44], chaotic behaviours [45] or dynamics of complex networks of oscillators [46], just to mention a few. The results discussed in this paper fill the research gap between the horizontal [35], [36], [37], [38] and the vertical [39], [40], [41] scenarios of the suspension’s oscillations and allow to describe the behaviour of pendula systems with additional coupling structure (the beam — an additional degree of freedom) in more general way.
The results of this paper are ordered as follows. In Section 2 we investigate the model with identical pendula (), studying the influence of beam’s mass and angle on the dynamics and synchronization of the nodes. Then, in Section 3 we vary mass to determine the changes in possible dynamical patterns, when the pendula are non-identical. The final conclusions and generalizations of the results are given in Section 4.
Section snippets
The identical pendula
In this Section we consider model (1) with the identical pendula, i.e. . Depending on the values of the beam’s mass and the angular direction of its oscillations , one can observe different types of synchronous configurations, which have been discussed in Fig. 2, Fig. 3.
The two most universal types of synchronization, which can be observed in classical, mechanical oscillators, are the in-phase and the anti-phase patterns [36], [39]. The study of model (1) exhibits, that the
The non-identical pendula
In this Section we investigate model (1) with non-identical pendula, fixing the mass of the 1st node and varying the mass of the 2nd one: (the natural periods of the pendula in the uncoupled case remain the same). Parameter describes the ratio between the pendula masses and has been used to present possible dynamics along with the variation of angle . To obtain the results, we have fixed beam’s mass , focusing on the influence of varied and
Conclusions
In this paper we have investigated the dynamics of two self-excited pendula suspended on the beam oscillating in various directions. Our results generalize the studies performed for previous models (with only the horizontal and the vertical movements of the support) and exhibit, that the direction of the support’s oscillations can influence on the existence of possible synchronization configurations. As we have shown, depending on the angular direction , the pendula can synchronize in various
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work has been supported by the National Science Centre, Poland, SONATA Programme (Project No 2019/35/D/ST8/00412) and OPUS Programme (Project No 2018/29/B/ST8/00457).
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