## Abstract

It is well known that by harmonically displacing the point of suspension of a simple pendulum in a vertical direction, it is possible to produce a stable inverted state. Under certain conditions, this inverted state bifurcates into two distinct oscillations that mirror each other about the vertical. For some parameter values these two states, together with a third periodic oscillation that is symmetric about the downward direction, become embedded within a chaotic attractor. With added noise, the system dynamics can consist of endless patterns of escape from a given periodic attractor followed by capture by one of the three. After each escape and before the next capture, the system travels on the chaotic attractor. It is confirmed that the escape process is consistent with a picture of noise induced activation from the effective potential wells associated with each of the attractors.

## 1. Introduction

The curious fact that the act of vibrating the point of suspension of a simple pendulum up and down can convert the inverted position (with the pendulum bob ‘on top’) from one of instability to one of stability has been known for nearly a century (Stephenson 1908). As will be seen, there are two key parameters in the system: the amplitude and frequency of the forced vertical motion. The stability of the inverted state can be understood from the perspective of properties of the Mathieu equation (Baker & Blackburn 2005). Strictly speaking, such an analysis is valid only for small oscillations and negligible damping. The formal analysis can be extended by numerical investigations (Leiber & Risken 1988) where it is found that damping tends to decrease the stability of the inverted state while enhancing the stability of the regular position.

As is commonly found even in such elemental systems, there is a very rich catalogue of possible dynamical modes. Beyond the static up and down configurations, depending on the level of excitation, one may observe: multiperiodic oscillations (Smith & Blackburn 1994), spinning modes, flutter modes (Blackburn *et al*. 1992), multiple-nodding oscillations (Acheson 1995; Clifford & Bishop 1998), and chaos (Leven & Koch 1981; Kim & Hu 1998). A recent contribution to this Journal (Bartuccelli *et al*. 2001) gave a comprehensive overview of these various modes.

Insight into the mechanism of dynamic stabilization of the inverted state is provided by the notion of an *effective potential* (Blackburn *et al*. 1992), an idea that originated with Landau & Lifshitz (1976). The review by Butikov (2001) contains a discussion of the effective potential for pendulums with both vertically and horizontally vibrating axes.

The idea that effective potential wells are the source of the stable states of this system will be extended here to aid in the visualization of new dynamics for this system. In particular, we will show that for certain choices of the excitation parameters, three periodic attractors coexist with and are embedded in a chaotic attractor. Furthermore, it will be demonstrated that when random noise is introduced into this system, the pendulum cannot reside indefinitely in any of the three wells (attractors), but will be activated out of the well, after which it enters a chaotic state. Eventually, the pendulum will re-enter one of the periodic modes, a process that can be visualized as recapture by a potential well. This pattern of noise activation and recapture repeats endlessly, yielding a form of intermittent chaos.

## 2. Periodic orbits

A simple pendulum consisting of a mass *m* attached to a rod of length *l* whose pivot is forced into vertical oscillations of amplitude *A* and frequency *ω* is governed by the equation of motionwhere *θ* is the angle of the rod measured anti-clockwise from the down position, *I* is the total moment of inertia of the rotating components, *b* is a damping coefficient, and *g* has the usual meaning of gravitational acceleration. This is conveniently rescaled in space units of *l* and time units of (*ω*_{0})^{−1}, where is the small-displacement natural frequency of the pendulum, the result being(2.1)where dots denote derivatives in normalized time *t*^{*}=*ω*_{0}*t*; *Ω*=*ω*/*ω*_{0}, *Q*=*Iω*_{0}/*b*, and .

In an earlier work (Blackburn *et al*. 1995), it was shown that with the parameter choice *Q*=22, *ϵ*=2.03, and *Ω*=2.10, there are three coexisting stable periodic states. One of these is a symmetric oscillation around the downward direction. The other two are large amplitude asymmetric oscillations about the up direction; these mirror image orbits have evolved from a symmetry breaking bifurcation of the inverted configuration, as described by Clifford & Bishop (1998). Phase plane representations of these orbits, obtained from numerical solutions of differential equation (2.1) are shown in figure 1. For the two up states, the complementary angle *ϕ*=*π*−*θ*, measured clockwise from the vertical, has been used as the most suitable coordinate.

Poincaré sections will appear in §3; they involve a periodic sampling of the phase space coordinate . The sampling moment within each forcing cycle is arbitrary—the choice adopted for these simulations is marked by small symbols along the orbits in figure 1. These orbits are completed in two forcing cycles, so sampling once per forcing period yields two alternating dots, as indicated.

The motions of the pendulum can be viewed from either of two perspectives: as seen from the external laboratory frame, or as seen by an observer attached to the vertically oscillating point of suspension. Figure 2 illustrates the motions of the tip of the pendulum as they would be registered, for example, on photographic film. In these frames, the dimensionless coordinates of the tip of the pendulum areand for a simple pendulum, .

A time-lapse photograph of the tip of the pendulum recorded by a camera attached to the pivot point would merely show a circular trace. This would hide the details of the rotating and counter-rotating aspects of the motion. To bring out these features, a form of polar plot is adopted in figure 3. Here, the angular position of the tip of the pendulum is combined with a steadily increasing radial distance from the centre. The regular reversals of direction are quite evident in such plots as is the period 2 nature of the modes.

## 3. Noise and mediated hopping

External noise can be introduced to the model by modifying equation (2.1) as follows (Blackburn *et al*. 1995):(3.1)where *σ* is a noise amplitude factor and *N*(0,1) is a number chosen randomly at each moment on the simulation time grid ( was used in the Runge–Kutta routine to give 100 steps per forcing period) from a Gaussian distribution with zero mean. A more detailed discussion of the steps leading to this noise expression and its connection to the fluctuation–dissipation theorem can be found in Blackburn *et al*. (1995).

In (Blackburn *et al*. 1995), it was shown that the three periodic attractors described above coexist with a strange attractor remnant (sometimes referred to as a ‘ghost’) such that if released from an arbitrary initial condition, the system would spend some time on the strange attractor prior to settling into one of the periodic modes. In other words, a chaotic transient preceded the attainment of periodic motion. In the present work, we are interested in the dynamics that arise for slightly higher noise levels than were adopted in the earlier study. What changes is that the enhanced noise is observed to cause the system to hop out of any periodic attractor back on to the strange attractor, over which it will roam for a time before dropping back into one of the three possible periodic states. This is followed again by re-excitation and so on, ad infinitum. Some salient features of this process are now presented.

Given the above summary, it is obvious that with a suitable noise level, just turning the system on from any initial state will lead to a continuous sequence of captures and escapes with interposed chaotic bursts. If Poincaré samples are plotted in the phase plane, each chaotic interval will contribute a number of points that belong to the strange attractor. Allowing this process to continue through many escape/capture cycles will gradually build up a complete picture of the strange attractor. This is the method used to obtain figure 4.

As the figure makes clear, the Poincaré points corresponding to the three periodic states are embedded within the structure of the chaotic strange attractor. Seemingly, the only way for the system to reach a periodic state is by first travelling around on the strange attractor.

An alternate view of the data in figure 4 is presented in figure 5. Here, the location of one of the periodic attractors (U1 was chosen) was employed as a reference point, and the distance in the phase plane of each new Poincaré point from that reference was calculated. Because of the periodicity noted earlier, the distance must be measured to both of the possible U1 locations, yielding two numbers, the smaller of which is utilized for identification purposes according to the following logic. If the system happens at the moment to be in periodic state U1, then the distance would obviously be 0. If, instead, the system is in state U2, the minimum distance to U1 would be 0.6668, based on the exact coordinates of the attractors shown in figure 4. Lastly, if the system falls into state D, an examination of figure 4 reveals that two minimum distances can arise: 5.836 and 6.011. Because noise has been added to the mix, the motion as illustrated in figures 1 and 2 will be blurred, resulting in a smearing around any of these identifying numbers. Therefore, it is necessary to define intervals, labelled Zone #1, Zone #2 and Zone #3, to contain the noise altered characteristic distances.

The sequence plotted in figure 5 clearly shows laminar intervals—when the system is in one of the three (noisy) periodic modes—that are separated by chaotic bursts. We note that this behaviour amounts to *noise-induced intermittent chaos*. If the noise is removed or sufficiently reduced, the intermittency will stop. This scenario for intermittency is quite different from what is usually pictured (Hilborn 1994).

## 4. Capture by periodic attractors

By means of the following heuristic approach, it is possible to extract information about the relative frequency of occurrence of captures by each of the three periodic attractors. A moving window containing 20 data points was stepped along the sequence of 10^{5} (in this example) Poincaré points. At each stage, the following test was applied: do *all* of the most recent ten points lie within any zone AND does *at least one* of the points in the first portion of the window lie outside the zone? When the answer is ‘yes’, the system has been captured by that attractor. Once captured, the system will reside on that attractor for some time, so it is necessary to wait for an escape before retesting for a fresh capture event. This is accomplished with the following test question: do *all* of the first ten points within the window lie within one zone AND does *at least one* of the points in the most recent portion of the window lie outside the zone? By applying these tests as the 20 point window is stepped along the dataset, each capture can be noted and, from the specific distance value, the identity of the periodic attractor can be ascertained. For the simulation run plotted in figure 5, the results were: 81 captures by U1, 80 captures by U2, and 24 captures by D. Hence, the mirror image inverted states have equal likelihood while the non-inverted periodic state is almost four times less common.

The choice of a 20 point window was somewhat arbitrary. Coupled with the tests, it supposes that the system must stay on any attractor for at least 10 successive Poincaré samples. This is equivalent to five complete periodic orbits (two points per orbit) as a minimum condition. The test works very well, but clearly will miss captures of shorter duration. Ultimately, of course, there arises the question of the difference between capture and merely passing through, since a chaotic state will visit the near neighbourhood of U1, U2, or D on occasion.

## 5. Escape from wells

It has already been noted that the stabilization of the inverted state can be interpreted with the aid of the concept of a dynamically induced effective potential well. The depth of such a well is dependent on various system and forcing parameters. A useful phenomenological picture of the escape process is one of noise induced activation from a potential well, a problem first addressed by Kramers (1940). The model consists of a ‘particle’ rolling at the bottom of a well and subject to random perturbations. It attempts to hop out of the well at a rate given by its natural frequency at the bottom of the well, *ω*_{0}. The chance of escaping on any attempt has an exponential dependence on the ratio of the depth of the well *E*_{0} to the mean noise energy *E*_{n}. More explicitly, the probability per unit time that escape is induced is(5.1)where *c* is a constant. This fixed probability per unit time (for given well depth and noise amplitude) is similar to the law of radioactive decay, and means that given an ensemble of *N* wells, the number of wells, Δ*N*, in which an escape would occur within the next time interval Δ*t* would obeyand so(5.2)gives the decreasing number of wells that have not yet experienced a noise activated escape (the analogue of the decreasing number of undecayed nuclei).

Suppose a simulation run is started from a known periodic attractor, say U1. As discussed earlier, the system will remain on that attractor for some time, then escape by noise activation. Several typical departures are depicted in the frames of figure 6.

The moment when the system switches from the periodic state U1 to the chaotic state *C* defines the escape time for that trial. By restarting the simulation many times, data can be collected which yield the distribution of lifetimes, as shown in figure 7.

Each column in a histogram represents the number of escape events that occurred within that time interval, where time is measured in forcing cycles. From the perspective of equation (5.2), the number of decay events taking place *between* times *t* and *t*+*δ* has the form(5.3)where *k* is a constant, so the histograms share the same functional exponential decay as *N*(*t*).

Three noise levels were tested with 10^{5} trials in each case. Figure 7 clearly shows the anticipated exponential behaviour associated with noise activation theory. When the vertical axis is converted to a natural log scale, the resulting plots become linear as shown in figure 8.

For a distribution of the form (5.3), the slope equals *P* and the average escape time is simply the reciprocal of this slope. The mean escape times implied by the linear fits in figure 8 are: 619.4 forcing cycles (*σ*=0.050), 1324.9 forcing cycles (*σ*=0.045), and 3848.2 forcing cycles (*σ*=0.040). According to equation (5.1), a lower noise level will result in a lower probability of escape, as is evident in the figure.

## 6. Concluding remarks

The stabilized inverted pendulum is a rich and interesting system that provides many insights into the complex features of nonlinear dynamics. In this work, the effect of added noise on the vertically oscillating pendulum has been considered. It has been found that, whereas in the absence of noise several periodic attractors coexist with the ghost of a strange attractor and the pendulum can remain indefinitely in any of these periodic modes, with noise the system hops endlessly among the periodic attractors. It is shown that the central feature in this repetitive process is noise driven escape from the effective potential wells associated with each periodic state. Interestingly, two of these wells are dynamically induced by the up and down motion of the point of suspension—they would vanish without the excitation; yet the model of activated escape applies to them as well. To the observer of the resulting dynamics, the system would exhibit noise driven intermittent chaos, with laminar times corresponding to intervals of residence in any of the three periodic states. Finally, it is found that the system is more likely to drop into either of the two symmetry broken inverted states than to the non-inverted (regular) configuration.

## Acknowledgments

Financial support was provided by the Natural Sciences and Engineering Research Council of Canada.

## Footnotes

- Received July 8, 2005.
- Accepted November 29, 2005.

- © 2006 The Royal Society