The dynamical integrity, a new concept proposed by J.M.T. Thompson, and developed by the authors, is used to interpret experimental results. After reviewing the main issues involved in this analysis, including the proposal of a new integrity measure able to capture in an easy way the safe part of basins, attention is dedicated to two experiments, a rotating pendulum and a micro-electro-mechanical system, where the theoretical predictions are not fulfilled. These mechanical systems, the former at the macro-scale and the latter at the micro-scale, permit a comparative analysis of different mechanical and dynamical behaviours. The fact that in both cases the dynamical integrity permits one to justify the difference between experimental and theoretical results, which is the main achievement of this paper, shows the effectiveness of this new approach and suggests its use in practical situations.
The men of experiment are like the ant, they only collect and use; the reasoners resemble spiders, who make cobwebs out of their own substance. But the bee takes the middle course: it gathers its material from the flowers of the garden and field, but transforms and digests it by a power of its own. Not unlike this is the true business of philosophy (science); for it neither relies solely or chiefly on the powers of the mind, nor does it take the matter which it gathers from natural history and mechanical experiments and lay up in the memory whole, as it finds it, but lays it up in the understanding altered and digested. Therefore, from a closer and purer league between these two faculties, the experimental and the rational (such as has never been made), much may be hoped.
(Francis Bacon 1561–1626)
But are we sure of our observational facts? Scientific men are rather fond of saying pontifically that one ought to be quite sure of one's observational facts before embarking on theory. Fortunately those who give this advice do not practice what they preach. Observation and theory get on best when they are mixed together, both helping one another in the pursuit of truth. It is a good rule not to put overmuch confidence in a theory until it has been confirmed by observation. I hope I shall not shock the experimental physicists too much if I add that it is also a good rule not to put overmuch confidence in the observational results that are put forward until they have been confirmed by theory.
(Arthur Stanley Eddington 1882–1944)
The goal of this paper is to develop theoretical concepts and tools for understanding experimental results, following seminal works of Thompson [1–3], to whom this paper is dedicated. The main idea is that classical, or Lyapunov , stability in certain circumstances is not enough for the existence of the associated attractor in the real world. In fact, classical stability refers to infinitesimal changes in initial conditions, whereas in the real world, the perturbations are always finite, although small, so that owing to actual perturbations a theoretically stable solution might not be seen in experiments.
This idea is somehow reminiscent of the Koiter  major contributions on the effects of imperfections, but it is different, indeed. In fact, while Koiter refers to perturbations of the structure/model (geometry, materials, connections, etc.), and thus operates in the realm of structural stability (with major effects even in the static case), Thompson, like Lyapunov, refers to perturbations in initial conditions, and contributes to the theory of (nonlinear) dynamical systems.
Thompson's idea can be reformulated by saying that 1.1
Thompson and co-workers developed this idea through a series of papers. Initially [1,6], attention was paid to the escape from the potential well, showing how it is intimately associated with homoclinic tangles, fractal basins and a variety of chaotic bifurcations. The basin boundary metamorphoses are discussed in , where an engineering integrity diagram is proposed, seemingly for the first time together with , to quantify the rapid erosion of the area under increasing excitation amplitude. A more detailed study of the integrity measures quantifying basin erosion is reported in , where it clearly highlighted the problem of eliminating the fractal parts from the measure of integrity, the latter being unsafe from a practical point of view.
The study of the effects of fractality is continued in , where the erosion by incursive fractals is shown to be a robust phenomenon facilitating the optimal escape from a potential well, in , where two distinct bifurcation scenarios are investigated and in , where the role of the underlying global bifurcations and of manifold organization is highlighted.
A noise excitation superimposed to a basic deterministic (harmonic) excitation is investigated in , where the effects are quantified in terms of a stochastic integrity measure, and correlated with the geometric changes experienced by the deterministic basin of attraction. In , the engineering significance of the control cross section in the four-dimensional phase-control space is discussed. Absolute and transient basins are examined too.
In the Thompson group, erosion profiles, i.e. the integrity measure versus the magnitude of a driving parameter, have been determined for increasing excitation amplitudes. In , instead, erosion profiles for time-dependent variations of the excitation frequency have been determined.
The relevance of the proposed investigation for the capsize of a ship is discussed in a series of papers [9,13,14], and in a review , where the relevance of the dynamical integrity to assess the actual level of safety is pointed out for the specific problem at hand. Moving along these lines, in  the concept of safe basin erosion has been applied to study ship capsizing, focusing on the effects of wind and waves.
Although the basic idea (1.1) is simple, as all major advances in science, its practical implementation is far from trivial, and gives rise to the research field today known as ‘dynamical integrity’ , which attempts at practicing the Thompson statement ‘…it is argued that in engineering design…a study of safe basins should augment, and perhaps entirely replace, conventional analysis of the steady-state attracting solutions’ , an affirmation that is totally shared by the authors.
The dynamical integrity concepts have been initially used by the authors to check the performances of a method for controlling chaos by eliminating, or, better, shifting in the parameter space, the global bifurcations triggering basin erosion [18,19]. Then, various theoretical issues including the correct definition of the safe basin and an adequate definition of integrity measure have been discussed in [17,20]. In particular, attention was paid to looking for a measure balancing the opposite requirements of ruling out the unsafe fractal parts from the integrity definition/evaluation and of being simple to implement. This leads to the so-called integrity factor, which is the normalized radius of the largest sphere belonging to the safe basin. Also the effects of hardening versus softening, symmetry versus asymmetry, smoothness versus non-smoothness, homoclinic versus heteroclinic bifurcations, etc., have been discussed with reference to four different mechanical systems.
Then, these concepts and tools have been applied to different mechanical systems with the aim (i) of highlighting their relevance and (ii) of further investigating other mechanical/integrity issues that arise in practical cases. In , the behaviour of a single-degree-of-freedom (d.f.) piecewise-linear model of a suspended bridge is analysed by the combined use of safe basin pictures and erosion profiles and the underlying dynamical phenomena are discussed.
The planar pendulum parametrically excited by the vertical motion of the pivot is considered in . Here, attention was paid to studying the competition between oscillating (in-well) and rotating (out-of-well) solutions. A detailed discussion of the interaction and mutual erosion of the various attractors-basins was presented. A dynamical integrity analysis of the rotating solutions of an experimental pendulum is performed in , which is the first starting point for the developments in this paper.
In [24,25], a two-dimensional reduced-order model of a cylindrical shell is considered, and, apparently for the first time, basins of attraction of dimension larger than two have been used in a preliminary way to evaluate the dynamical integrity. The dynamical integrity of Augusti's model is preliminarily considered in , and then reconsidered in depth in  together with the dynamical integrity of a similar archetypal model of a guyed tower. For this type of systems, the load carrying capacity is the main mechanical performance, and thus in [28,29] the relationships between load carrying capacity and dynamical integrity have been investigated in depth. In , it is shown that the requirement of a large basin also applies in the absence of dynamical excitation, because close to the static bifurcation point the basin of attraction shrinks to zero and thus the associated equilibrium point is no longer robust, the situation being unsafe. In , on the other hand, the dynamical excitation is added, and the interactions between the excitation amplitude and the axial load in reducing the global safety are investigated, analysing in-depth both the basin extent and its compactness.
The dynamical integrity of a micro-electro-mechanical system (MEMS) was initially studied in  at a theoretical/numerical level. More recently, the theoretical results have been compared with experimental results [31–33], showing the usefulness of the dynamical integrity concept in justifying apparently strange outcomes of the experiments. This constitutes the second starting point of the present paper.
The previous works highlight that the dynamical integrity becomes particularly important at least in the following two circumstances.
(1) Close to the local and to some global bifurcations determining the disappearance of static or dynamic solutions. If fact, as the driving parameter approaches the bifurcation value, the safe basin of the solution shrinks to zero , so that in the neighbourhood of the bifurcation threshold the basin becomes unsafely small, although the solution is still stable in the Lyapunov sense. Here, the basins are simply small, not necessarily eroded, and we call this a ‘non-robust’ situation with respect to finite dynamic perturbations, and ‘robustness’ profiles the curves reporting the integrity measure (that in this case can also be the magnitude of the basin).
(2) In the actually dynamic case, when the basins of attraction are very fractal. The extended fractality is due to various homo/heteroclinic tangles that cause penetration of eroding tongues inside the safe basin, and lead to the well-known ‘butterfly effect’ , namely to sensitivity to initial conditions. Even small changes in initial conditions, i.e. small perturbations, give rise to time histories that approach different attractors. When anyone of these is unwanted, we are in an unsafe situation. In this case, we use the name ‘eroded’ basins, and refer to the process leading to this situation as ‘erosion’. It is worth noting that it is particularly in this case that the strict magnitude of the safe basin is an inadequate measure of integrity.
Of course, the combination of cases (1) and (2) represents the worst situation, as we have a double effect running in reducing the integrity.
A further unsafe and potentially dangerous situation is related to some other global bifurcations, like, for example, the boundary crises of an in-well chaotic attractor. Here, in fact, the basin of attraction might remain uncorrupted up to the bifurcation point, where the attractor suddenly disappears and its former basin is captured by another coexisting attractor (e.g. cross-well chaotic attractor). This situation, however, does not fall within the realm of dynamical integrity, because the dangerous situation occurs for (Koiter-like) perturbations of the structure, by means of small changes of the relevant parameters, and not by (Lyapunov-like) perturbations in the initial conditions. In fact, below the bifurcation threshold, if one is guaranteed that the parameters do not change, the state of the system is safe.
In both situations (1) and (2) dynamical integrity helps in understanding experimental results; in particular, it helps to illustrate why we are not able to experimentally detect a solution which is theoretically stable. In fact, when the basin of attraction is too small or too eroded, just small perturbations, always present in the real world, are sufficient to get the response out of the safe basin towards a different, more robust, attractor.
To illustrate this fact, we consider two cases in which dynamical integrity becomes very useful to interpret experimental results. To stress that this conclusion is ‘scale independent’, we consider both a macro-experiment, a rotating pendulum [23,35], and a micro-experiment, a MEMS device [31–33]. In the first case, we pay attention to the robustness of a special class of solutions, the rotating ones, against competing oscillations; in the latter case, we pay attention to the overall escape from the system potential well—irrespective of the associated solutions—leading to the so-called dynamic pull-in. In both cases, we found that the theoretical limit of existence of the considered system state cannot be reached in practice. The practical limit, on the other hand, is associated to an acceptable level of the integrity measure, different from case to case but fixed for a specific mechanical system, thus providing evidence of the role played by dynamical integrity.
The two considered cases differ not only for being one macro- and one micro-system, but also for their dynamical characteristics and the underlying technical interest. In fact, in the MEMS there is a unique potential well surrounded by a homoclinic orbit (in the Hamiltonian case), and the main loss of integrity is due to its overall erosion from the out-of-well attractor, which is possibly unwanted. By contrast, in the case of the pendulum, we may be interested in obtaining rotations, i.e. a class of solutions outside the well bounded by a heteroclinic orbit which, in general, is not robust enough against the erosion from in-well attractors, namely oscillations and/or rest position. Thus, also from these points of view, we are considering complementary examples, thus implicitly confirming the general value of the dynamic integrity analysis.
Within the former general framework, this paper has also a second goal, namely introducing and using a refined measure of basin magnitude that rules the fractal parts out of the integrity evaluation and is simple to implement. For the considered mechanical systems, this new measure is calculated and compared with other ones previously used in the literature, in order to establish its performances, advantages and disadvantages.
The paper is organized as follow. In §2, the basic tools of a dynamical integrity analysis are reviewed, and a new integrity measure is proposed. These arguments are applied to the rotations of a pendulum in §3, and to the escape (pull-in) of a MEMS in §4. The paper ends with some conclusions (§5).
2. How to measure the dynamical integrity
The safe basin is the union, in phase space, of all initial conditions sharing a given property. Often this is the convergence towards an attractor, in this case coinciding with the classical basin of attraction, but it can also be the non-escape from a given potential well, or whatever else. When considering periodically excited systems, one often considers stroboscopic cross sections of the safe basin. In this case, the phase at which this section is taken can play a role. To take this aspect into account, the ‘true’ safe basin is defined as the intersection of the previously defined safe basin when the excitation phase ranges over the excitation period. The ‘true’ safe basin is then the smallest phase-independent set of initial conditions sharing the desired dynamical property .
In this paper, we deal with periodically excited one mechanical d.f. systems, and consider the two-dimensional safe basin of the stroboscopic Poincaré section of the flow at t=0.
After having defined the safe basin, the second key point in any dynamical integrity analysis is to properly measure it. In the past, three measures have been used.
The global integrity measure (GIM), defined by Thompson , is the normalized hyper-volume (area in two-dimensional cases) of the safe basin. It is the most intuitive and easy integrity measure, but it is not satisfactory in all cases in which the safe basin is strongly fractal, even if, possibly, only in a certain part of it where, indeed, the dynamics is not safe. Normalized means that the hyper-volume of the safe basin corresponding to the actual value of a varying control parameter is divided by the one corresponding to a reference value, so that GIM is a dimensionless number.
The local integrity measure (LIM)  is the normalized minimum distance between the attractor and its basin boundary. As this measure accounts for the attractor remoteness from the boundary, it is a good measure of the attitude of the system to support perturbations in initial conditions without escaping from the basin itself. It also has the advantage, contrary to GIM, of ruling out the fractal parts from the integrity evaluation. However, it has the disadvantage that it is difficult to be computed in practice, especially when the attractor is chaotic, and that it is meaningless when the safe basin is not a basin of attraction.
To overcome the previous points, the authors introduced the integrity factor (IF), which is the normalized radius of the largest hyper-sphere  (circle in two-dimensional cases) entirely belonging to the safe basin. IF is as computationally easy as GIM, and succeeds in eliminating the unsafe fractal tongues from the integrity evaluation such as LIM. In fact, it is a measure of the compact part of the safe basin, which is the largest convex set entirely belonging to the basin and the sole region that guarantees the system dynamical integrity. Contrary to LIM, IF is not a property of the attractor, but just of the safe basin. LIM and IF differ significantly from each other when there is a spread chaotic attractor, or when the periodic attractor is not ‘centred’ in its basin.
As already mentioned, both LIM and IF have been introduced mainly to rule out the unsafe fractal parts from the integrity evaluation. Here, a new refined measure of integrity aimed at obtaining this goal is proposed, along with the relevant computational procedure. It moves from the observation that, in practice, safe basins are always obtained in a discrete way, i.e. by approximating the continuous phase space with a finite number of cells, which are pixels in any graphical representation. Each cell contains a number (the colour in the picture) making reference to the considered dynamical property (the kind of attractor, for basins of attraction). By contrast with the previously defined ‘nominal’ safe basin, the ‘actual’ one is defined as that obtained by eliminating from the former all cells that are not surrounded by cells of initial conditions having the same number. Clearly, by definition, the fractal parts are no longer present in the actual safe basin, which has a substantially compact shape. Note that the actual safe basin can be considered the equivalent domain, in the presence of dynamic excitations, to the safe basin also characterizing a static stability problem . An example of nominal safe basin and of the associated actual one is reported in figure 1, which clearly shows the elimination of the fractal parts.
It is also worth noting that actual safe basins can be disconnected (figure 1b). Furthermore, in addition to fractal parts, the procedure for evaluating the new measure also eliminates a strip, of one cell width, along the entire boundary of the compact part of the nominal safe basin. However, if the basin discretization accuracy (i.e. the number of cells) is large enough, this does not entail meaningful underestimations.
Having preliminarily eliminated the fractal parts, we no longer need to pay attention to the definition of the measure, and so we can take the easiest one, i.e. the magnitude of the actual safe basin. We name its normalized version the actual global integrity measure (AGIM).
Clearly, by definition, if the normalization is done with respect to the same reference hyper-volume and distance, respectively, it is AGIM<GIM and LIM<IF, and the larger is the fractal part, the larger is the difference.
To have a preliminary idea of the performances of the new measure AGIM, we note that in the case of figure 1a, the magnitude of the safe basin is 43 519 pixels, whereas in the case of figure 1b, the magnitude of the reduced basin is 5786 pixels. This permits us to conclude that in this fractal case the AGIM is 13.3 per cent of the GIM, i.e. the basin safety is 752 per cent more reliable.
The above defined four integrity measures are considered and compared with each other in the following sections.
3. The rotating pendulum
(a) Mechanical system description
To start with the macro-scale mechanical systems, we consider the experimental rotating solutions of the pendulum illustrated in figure 2. From a dynamical integrity point of view, rotating solutions are more interesting than oscillations because they are less robust and thus more affected by perturbations. The experiments, and the green energy production motivation that is behind them, are described in [23,35]. From the experiments, we have identified the damping coefficient as h=0.015. Note that a purely numerical dynamical integrity analysis of the pendulum is reported in , where the integrity of the competing oscillations is also comparatively considered.
The dimensionless equation of motion, which is used to provide the theoretical results, is 3.1where θ is the position angle measured from the downward vertical (rest) position, p and ω are the excitation amplitude and (circular) frequency, respectively.
By numerically integrating (3.1), it is possible to find the theoretical range of existence of period 1 rotating solutions. A typical bifurcation diagram for increasing excitation amplitude is reported in figure 3, where we see that the rotations are born through a saddle-node (SN) bifurcation, lose stability by a period-doubling (PD) bifurcation, undergo a PD cascade (barely visible in figure 3), and finally disappear via a boundary crisis (BC). By determining several bifurcation diagrams for different values of ω, it is possible to obtain the region of existence of rotations in parameter space, which is reported in figure 4.
In the experiments, we investigated the effects of the excitation frequency and amplitude on the rotational behaviour of the pendulum. More precisely, we aimed at detecting in the (ω,p) parameter space the region where rotations can be found experimentally. With this objective, we spanned the frequency range ω∈(1.15;2.30), and for each considered frequency, we applied different increasing amplitudes, paying special attention to determine the lowest and highest amplitudes where rotations can be found experimentally.
The detected rotations are reported in figure 4, where the region of existence of experimental rotations is clearly seen to be a strip of finite magnitude, shrinking for low frequencies and belonging to the central part of the region of theoretical existence.
It is worth remarking that not only were we unable to find experimental rotations for large excitation amplitudes, this being somehow expected because for large amplitudes the accuracy of the experiments decreases, but also for low excitation amplitudes, where instead we would have expected to detect them.
The difference between the experimental and theoretical regions of existence of rotations in figure 4 is too large to be attributed only to experimental uncertainties. In the following section, we justify it by means of a dynamical integrity analysis.
(b) Dynamical integrity and integrity profiles
The key tool for understanding why rotating solutions can be practically observed only in a subset of the stability domain is the integrity profile, i.e. a curve that reports the integrity measure for increasing excitation amplitude p. In this section, the safe basin is the basin of attraction of the clockwise rotating solution. We consider all four integrity measures defined in §2.
The integrity profiles for ω=1.3 are reported in figure 5, where they have been normalized with respect to their own maximum values (accordingly, the maximum of each curve is 1). The associated bifurcation diagram is that of figure 3.
From figure 5, we see that, just after being born via a SN bifurcation at pSN≅0.0476, the basin of attraction enlarges and the integrity suddenly increases. The subsequent indented region is due to the appearing and sudden disappearing of secondary rotations (in particular, at p≅0.080–0.090 a secondary rotation of period 6, R6, and at p≅0.100–0.110 a secondary rotation of period 5, R5). They appear, by SN bifurcations, inside the basin of attraction of the main rotation, thus entailing an instantaneous decrement of the compact part of its safe basin and a fall down of the integrity profiles. When the secondary solution disappears, the whole former safe basin is recaptured by the main rotation, which regains (at least partially) its integrity.
At p≅0.190, a period 3 rotation (R3) appears by a SN bifurcation inside the basin of attraction of the main rotation R1. It is more robust than the previous secondary rotations, and accordingly there is a large decrement of integrity. The R3 has an interval of existence and stability that is larger than those of R6 and R5, but it is in any case small. Thus, it suddenly disappears and leaves the main rotation R1, which has, however, a merely residual integrity, because the disappeared attractor had previously tangled with the surrounding fractal part, so that integrity is definitely lost.
After p≅0.200, the dynamical integrity is residual. The basin of attraction is definitely small and almost completely eroded. At pBC≅0.369, the bifurcation path ensuing from the rotation solution, which previously underwent a PD cascade, disappears by a BC. This is the last point of the integrity profiles.
For the purpose of interpreting figure 4, the main property of figure 5 is that all integrity measures of the period 1 rotation are high only in the central part, where the attractor is relatively robust and its basin is substantially uneroded. This is where we found the experimental rotations, which provides the theoretical justification for experimentally observing rotations only in the central strip of the excitation amplitude range.
This justification is not limited to the case ω=1.3, because integrity profiles built for different values of the excitation frequency share the same qualitative properties .
It is worth remarking that, looking at the dynamical integrity, not only do we understand why we have not observed rotations for large excitation amplitudes, which is somehow expected owing to experimental limitations, but also why we have not observed rotations for excitation amplitudes just above the SN bifurcation, which is much less intuitive.
To have a better comparison, in figure 6, we report the measures normalized with respect to the same value. More precisely, in figure 6a, we report together GIM and AGIM, which are both measures of a hyper-volume, normalized with respect to the maximum of GIM. The inequality AGIM<GIM is clearly satisfied. In figure 6b, on the other hand, we report together LIM and IF, which are both measures of a distance, normalized with respect to the maximum of IF. Again, the theoretical inequality LIM<IF is satisfied.
Figure 6a is particularly expressive in showing how large is the difference between GIM and AGIM, the former being not reliable because it still accounts for the fractal parts and thus overestimates dynamical integrity. On the contrary, the differences between IF and LIM are minor; in the central part of larger integrity, where we have observed rotations experimentally, LIM≅0.5IF, whereas in the subsequent plateau, the differences are much more minor. A meaningful difference between LIM and IF is a consequence of the fact that the attractor is lateral and not in the centre of its safe basin.
To have a better comparison between numerical and experimental results, and to further justify the latter in terms of dynamical integrity, we continue the analysis for different values of the excitation frequency ω. We start by noting that the basin of attraction of rotations for low excitation frequencies is smaller than for large frequencies. An example is reported in figure 7, where the basins and the associated circles used in the IF evaluation are reported, and where we see how they increase for increasing values of ω and p. The difference between figure 7a and figure 7b is remarkable, both in absolute value (the IF-circle of figure 7b is 4.4 times larger than that of figure 7a) and with respect to the competing attractors. Figure 7 also highlights the out-of-well character of the rotating solutions with respect to the domain of the oscillating solutions and/or rest position, bounded by heteroclinic orbits in the Hamiltonian case.
We have then built several integrity profiles, like those in figure 5, for different values of the excitation frequency ω. They have been normalized with respect to a unique reference case in order to have comparable results. By means of these integrity profiles, we were able to obtain contour plots of the various integrity measures, which summarize in an effective way the results of the extended numerical simulations.
We initially note that the integrity increases by jointly increasing ω and p, a fact that confirms and extends the results of figure 7.
For ‘low’ excitation frequencies, say ω<1.6, the experimental points are above the main ridge of IF, the lower experimental points being observed approximately just along the ridge. This means that the largest possible safe basin, i.e. the largest dynamical integrity, is necessary to trigger the experimental observation of rotations.
For ‘large’ values of excitation frequencies, say ω>1.6, the experimental points are clearly around the main ridge of IF, which is a definitive confirmation that only rotations with large dynamical integrity can be practically observed. The fact that for ‘very large’ frequencies, say ω>2.2, the points no longer follow the ridge is a consequence of the fact that in the present experiment the amplitudes have a technical upper bound , which cannot be overcome.
The bottom curve of existence of rotations now approximately stands on a contour level of IF, showing the minimal dynamical integrity necessary for the onset of experimental rotations.
4. The micro-electro-mechanical system
(a) Mechanical system description
Another mechanical system that can be fruitfully analysed by the dynamical integrity concept is the capacitive accelerometer reported in figure 9. It consists of a proof mass between two cantilever beams. The upper electrode is formed by the proof mass, while the lower electrode is placed underneath it on a silicon substrate; it is of the same length as the proof mass and of slightly smaller width. When electrically excited, the proof mass oscillates in the out-of-plane direction x. The device has been experimentally tested in [32,36], which are referred to for the data and for further details.
While in the pendulum problem attention was paid to a special class of solutions, here attention is dedicated to the overall escape phenomenon, i.e. the dynamic pull-in which, from one side, constitutes the most dangerous practical situation for the accelerometer (in fact, in-well jumps between resonant and non-resonant attractors, although undesirable in some applications, do not destroy the structure) and, from the other side, is the most interesting phenomenon. Indeed, it is related to subsequent bifurcations and to strong erosion, thus having a double shrinking effect on the safe basin, namely its reduction in extent owing to approaching the bifurcation, and its erosion owing to fractal tongues. To be more precise, we focus on the so-called inevitable escape, i.e. on those situations where the pull-in is the unique attractor in the phase space. Actually, depending on the initial conditions, we can have escape also in the presence of bounded attractors.
The dynamic pull-in data come from a frequency-sweeping process, where the electrodynamic voltage V AC is kept fixed and the frequency Ω is increased or decreased quasi-statically. They refer to the case of primary resonance, with electrostatic voltage V DC=40.1 V, and pressure very close to an ultra-high-vacuum environment, 153 mTorr. In this range, the identified single-d.f. equation of motion is  4.1
As an example, an experimental frequency sweeping is reported in figure 10 together with the corresponding frequency response curve obtained numerically. Both experimental and numerical curves have a softening behaviour. For increasing frequency, the non-resonant oscillation loses stability through a SN bifurcation, which in the experimental case leads to pull-in, whereas in the theoretical case leads to the resonant attractor. In fact, in figure 10, we can preliminarily appreciate the differences between experimental and numerical/theoretical results. In the former case there is an interval (approx. between 180 and 188 Hz) without bounded solutions, i.e. an interval where, because there are no other attractors, the escape is inevitable , as previously evidenced also for MEMS [32,38]. In the latter case, on the other hand, there is always a bounded solution and the escape, while being possible depending on the type of perturbation, is not inevitable.
For larger excitation amplitudes, the inevitable escape region appears also in the theoretical curve, as shown in figure 11. So, this phenomenon is not a peculiarity of the experiments. However, it practically occurs in a region where it is not theoretically predicted, as shown in figure 10.
By building several frequency response curves like those of figure 10 for different values of the excitation amplitude, it is possible to identify the experimental and the numerical/theoretical escape regions in the parameter space, which are reported in figure 12.
The theoretical escape region is similar to those observed with other softening mechanical systems [30,39], and has the classical V-shape, with vertex at Ω≅178.3 Hz and V AC≅22.2 V, bounded on the left by the SN where the non-resonant solution disappears for increasing frequency (like in figure 11), and on the right by the BC at the end of the existence region of the resonant oscillation for decreasing frequency (figure 11). The Δ-shaped region where the resonant and non-resonant oscillations coexist, with the degenerate cusp bifurcation at Ω≅187.5 Hz, also occurs. However, in this section, we are interested in the erosion of the overall in-well dynamics, so we do not care which one is the bounded attractor.
Like in the case of the pendulum, the difference between experimental and theoretical regions of escape in figure 12 is too large to be attributed only to experimental uncertainties. In the following section, we justify it by means of a dynamical integrity analysis.
(b) Dynamical integrity analysis
We start by illustrating the erosion mechanism by means of two examples in which we vary the excitation frequency (figure 13), one for V AC=8 V and the other for V AC=15 V. For both cases, we have no theoretical escape but we have practical escape (figure 12).
The fractal tongues of escape (white) enter the potential well through the resonant basin (light grey). The non-resonant one (dark grey), instead, is not involved because it is protected by the stable manifolds of its saddle.
For V AC<8 V, the erosion concerns only a small part of the resonant basin boundary (not reported in the figures). At about V AC=8 V (figure 13a–c), instead, the fractal escape area starts developing more rapidly and, remarkably, it enters the potential well and separates the two basins. A part of the resonant basin yet surrounds the non-resonant one (figure 13b), even if it is embedded with fractal tongues. This extended in-well fractality, along with the disconnection of the two basins (contrary to what occurs in figure 13c), is likely the cause of practical escape in experiments.
For increasing values of the voltage, figure 13d–f, the situation is drastically changed. The erosion is much more evident. The resonant attractor is marginal for low values of Ω; it is theoretically stable, but of course there is no hope to observe it in practice. For increasing Ω, the non-resonant attractor disappears through the SN bifurcation, while the resonant one recovers robustness (figure 13f) and becomes the sole in-well attractor.
The erosion profiles are obtained from many basins of attractions like those of figure 13. Some examples are reported in figure 14. Here, the safe basin is the basin of attraction of the dominant bounded attractor, i.e. the non-resonant attractor for low values of the excitation frequencies and the resonant attractor for high values of the excitation frequency. The normalization has been done with respect to the basins at V AC=0.1 V and Ω=180 Hz. More precisely, like in figure 6, GIM and AGIM, which are both measures of a hyper-volume, are normalized with respect to the value of GIM at V AC=0.1 V and Ω=180 Hz; LIM and IF, which are both measures of a distance, are normalized with respect to the value of IF at V AC=0.1 V and Ω=180 Hz.
From figure 14, we see that for V AC=15 V, where there is no theoretical escape region (see figure 11a), there is a region where the two in-well attractors coexist. However, in the coexistence region the integrity of the resonant attractor is so small that there is no hope to observe it in experiments. Where the profiles of the non-resonant attractor approach zero, we expect that also this latter attractor cannot be observed. We conclude that in this intermediate frequency range, although we have two stable attractors, we may expect inevitable escape in experiments.
For V AC=30 V, on the other hand, there is a theoretical region of escape (figure 11b). Here, the integrity measures ‘rapidly’ approach the zero values of the escape interval. Thus, just outside the theoretical escape region, the dynamical integrity is so low that there is no hope to observe in practice the associated attractor. It can be observed when the integrity is large enough, i.e. for lower and higher excitation frequencies, suggesting that the practical escape region is larger than the theoretical one: this is exactly what has been observed in experiments (figure 11). Note that the practical escape region in figure 12 is meaningfully larger on the right-hand side, consistent with the slower increase of dynamic integrity of the resonant attractor in figure 14. Thus, once more, dynamical integrity provides the justification for non-coincidence between theoretical and experimental results.
Figure 14, like figure 5, permits one to compare the performances of the four integrity measures. Contrary to the pendulum (figure 6a), here, there are practically no differences between GIM and AGIM, showing that the fractality is not so extended in the phase space as in the pendulum. This is likely associated with the fact that we are now dealing with the erosion of in-well attractors, instead of out-of-well ones, as in that case. Also the differences between IF and LIM are less than those of the pendulum (compare with figure 6b). Here, however, for large excitation frequencies the difference is no longer negligible, denoting the non-central position of the (resonant) attractor with respect to its basin.
Building several erosion profiles, like those of figure 14, for different values of the AC voltage, it is possible to build the contour plot of the integrity measures, which provides an overall understanding of the dynamical integrity of the system. That of the IF is reported in figure 15, whereas those corresponding to the other measures are similar and are not reported. Figure 15 is the counterpart for the MEMS of figure 8 for the pendulum.
The main conclusion that can be drawn from figure 15 is that the experimental threshold of escape approximately stands along a level curve of the integrity measure (the fact that this value is about 30% is irrelevant for the present purposes). The curves of constant percentage of IF then succeed in interpreting—and possibly predicting—the effects of disturbances in the experiments; the theoretical escape curves represent the limit case when disturbances are absent.
This is a further straightforward justification of the experimental results in terms of dynamical integrity, and confirms that also in this case, as for the pendulum, to observe experimentally an attractor it is necessary that its basin of attraction be large and compact enough.
Of course, irrespective of experimental results, the meaningful lowering of theoretical escape thresholds (corresponding to zero dynamical integrity), which occurs when a minimal residual integrity is required for system safety, poses important problems of properly accounting for robustness and erosion phenomena in the design stage.
The problem of interpreting experimental results that do not fulfil the theoretical predictions has been addressed by means of the dynamical integrity concept proposed by Thompson and subsequently developed by the authors.
With the aim of showing how the main ideas work well in different situations, two mechanical systems have been considered, one at the macro-scale (the pendulum) and the other at the micro-scale (a MEMS). There are differences also of mechanical (e.g. for the considered aspects of the pendulum resonance is not important while it is very important for the MEMS) and dynamical nature (e.g. out-of-well versus in-well behaviour), so that we claim that cases considered in this paper are representative of much more general situations, and the conclusions are not limited to the two specific mechanical systems.
We were able to show how the dynamical integrity is able to justify, or even possibly predict, the experimental results. Indeed, we have shown that the solutions of interest exist only in the region where the basins of attractions are sufficiently robust and/or uneroded; on the contrary, in the regions where the integrity is compromised, we have not observed the attractor in the experiments, even if it is theoretically stable.
The key tool for drawing the previous conclusions are the integrity profiles, and the associated behaviour chart, which permits us to immediately localize the (unsafe) region of erosion, i.e. the regions in parameter space where we may expect that Lyapunov-based theoretical predictions on stability are not fulfilled in real situations.
As a secondary result, we introduced a new measure of integrity, named AGIM, which is conceptually simple enough, easy to implement and which rules out the fractal parts from the magnitude-based measure of integrity. This measure has been compared with other existing measures in the considered cases, and it has been shown that it performs better than the classical measure based on the hyper-volume of the safe basin (GIM), while being comparable with other measures that rule out fractality.
This paper constitutes a further step in showing the relevance of dynamical integrity arguments for practical applications.
The authors wish to thank Prof. Mohammad I. Younis for kindly providing the experimental results upon which §4 is based. The work of S.L. is partially supported by the Italian Ministry of Education, Universities and Research by the PRIN funded programme 2010/11 N. 2010MBJK5B entitled ‘Dynamics, Stability and Control of Flexible Structures’.
One contribution of 17 to a Theme Issue ‘A celebration of mechanics: from nano to macro’.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.