# Complete bifurcation analysis of driven damped pendulum systems/Juhitava sumbumisega pendelsusteemide taielik bifurkatsioonianaluus.

1. INTRODUCTION

Recent research in non-linear dynamics shows, that the so-called rare attractors (RA) exist in all typical and well studied models [1-9], but are unnoticed by the traditional methods of analysis. The systematic research of rare attractors is based on the method of complete bifurcation groups (MCBG) . This method allows conducting more complete global analysis of the dynamical systems. The main idea of the approach is periodic branch continuation along stable and unstable solutions. The method is based on the ideas of Poincare, Birkhoff, Andronov and others [6,7]. It is shown that the MCBG allows to find important rare attractors and new bifurcation groups in different non-linear models. The main features of this method are illustrated in this work by three driven damped pendulum systems.

Our aim is to build complete bifurcation diagrams and to find unknown rare regular and chaotic attractors using complete bifurcation analysis for some important parameters of the model: the amplitudes and the frequency of excitation. For complete global bifurcation analysis we have used the MCBG, Poincare mappings, basins of attraction, etc.

The main results of this work are presented by complete bifurcation diagrams for variable parameters of the driven damped pendulum systems. We consider three pendulum models: a) model with an additional linear restoring moment and with the periodically vibrating in both directions point of suspension, b) model with a linear restoring moment and with the external periodic moment of excitation, c) model with a sliding mass and with the external periodic moment of excitation. By building the complete bifurcation diagrams with stable and unstable periodic solutions, we have found different new bifurcation groups with their own rare regular and chaotic attractors. All results were obtained numerically, using software NLO  and SPRING , created at Riga Technical University.

2. MODELS AND RESULTS

The first pendulum model is shown in Fig. 1a. The system has additional linear restoring moment with the harmonically vibrating in both directions point of suspension. Restoring moment and backbone curves for the system are shown in Figs. 1b,d. The system has three equilibrium positions (Fig. 1c).

The first mathematical pendulum model is described by the following equation of motion:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [phi] is the angle of rotation, read-out from a vertical line, [??] is angular velocity of the pendulum [??] = d[phi]/dt, t is time, m is mass, L is the length of the pendulum, [mu] is the gravitation constant, b is the linear damping coefficient, c is the linear stiffness coefficient, [A.sub.1], [A.sub.2] and [omega] are oscillation amplitudes and frequency of the point of suspension in the horizontal and a vertical direction. The variable parameters of the pendulum system are horizontal and vertical oscillation amplitudes [A.sub.1] and [A.sub.2] of the suspension point. Other parameters are fixed: m = 1, L = 1, b = 0.2, c = 1, [mu] = 9.81, [omega] = 1.5.

[FIGURE 1 OMITTED]

Similar models have been examined in papers [10-15]. But the structures of these pendulum-like models are different, because of the difference in excitations of the pendulum support and in restoring characteristics. In our work, instead of the elliptical excitation, etc, we consider the periodically vibrating in both directions point of suspension and the external periodic moment of excitation. Moreover, in this paper we investigate two models with an additional linear restoring moment and a third model with two degrees of freedom and with a sliding mass. The main difference is in the aim of these investigations. Our aim was to build complete bifurcation diagrams and to find new bifurcation groups and unknown rare regular and chaotic attractors using complete bifurcation analysis in these pendulum systems. Some results of complete bifurcation analysis for the first pendulum model were presented in several previous papers [16,17]. According to , there is a difference in the used methods. Our method of complete bifurcation groups is intended for the construction of bifurcation diagrams, which, as a rule, follow up the stable regimes only (the so-called brute-force). The basic feature of this method is following up of all stable and unstable regimes in the parameter space together with their stability characteristics and the origin of unstable periodic infinity [6,7]. As was mentioned above, the method is based on the ideas of Poincare, Birkhoff and Andronov and used together with Poincare mappings, mappings from a line and from a contour, basins of attraction, etc.

The results of bifurcation analysis of the model (1) are represented in Figs. 2 to 5. Five different 1T bifurcation groups and one 2T bifurcation group have been found (Fig. 2). The symbols 1T and 2T mean the characteristic of the period of free oscillations dependent on period [T.sub.[omega]] of excitation force: T = n[T.sub.[omega]], n = 1, 2,3,.... If n > 2, there are subharmonic oscillations.

Two of these groups are topologically similar and have rare attractors of a tip kind [P1.sub.1] RA and [P1.sub.3] RA. The symbol P1 means the characteristic of the period of oscillation regime. For this case it is a regime of period 1. Subscripts 1, 2, 3 etc mean the classification numbers of bifurcation groups, because of the existence of many bifurcation groups with the same period of oscillations. The stable solutions are plotted by solid lines and unstable--by red lines .

Two period one brunches near [A.sub.2] = 4 are not completed, because of problems of sufficiently high instability with maximal value less than 2500. Other three 1T bifurcation groups have their own rare attractors [P1.sub.4] RA and [P1.sub.5] RA, which are stable in small parameter regions.

Some cross-sections ([A.sub.2] = const) of bifurcation diagrams with dynamical characteristics from Fig. 2 are represented in Figs. 3 and 4. All attractors are of the tip kind so each of them has not only periodic attractors, but also chaotic attractors as well. The examples of coexistence of period 1 (P1) stable solutions and P1 RA rare attractors for three cross-sections [A.sub.2] = 0.44, [A.sub.2] = 0.526 and [A.sub.2] = 3.44 on bifurcation diagram (Fig. 2a) with the time histories and phase projections are shown in Fig. 4. Oscillation amplitudes of rare attractors in some cases are tenfold bigger than oscillating amplitudes of stable P1 regimes. The examined system has also another bifurcation group of higher order with rare attractors, for example, 2T bifurcation group with P2 RA rare attractor with large oscillation amplitudes.

Each bifurcation group has its own unstable periodic infinity (UPI) [1-5] with corresponding chaotic attractors. UPI is a sub-bifurcation group based on Poincare and Birkhoff ideas with infinite unstable periodic solutions nT, 2nT, 4nT, ... . The existence of the UPI due to the complete cascade of nT-period doubling and the crisis is a necessary part of the typical bifurcation group with chaotic behaviour. It is a well-known fact that the presence of the UPI characterizes the parameter region with chaotic attractors and/or chaotic transients (see Fig. 2). It is known that the system may have several different UPIs simultaneously, and so the resulting chaotic behaviour depends in this case on each bifurcation group with UPI . The example of globally stable chaotic attractor for cross-section with parameters [A.sub.1] = 0.5, [A.sub.2] = 4.9, obtained using the contour mapping [1-9], is shown in Fig. 3a.

The system also has other nT subharmonic bifurcation groups with n = 3, not shown.

Comparison of two different methods for building domains of attraction of tip kind rare attractor [P1.sub.1] RA is represented in Fig. 5. These methods show the same results of building domains of attraction.

The second mathematical pendulum model, shown in Fig. 6a, is described by the following equation of motion:

[??] + b[??] + (a[phi] + [a.sub.1] sin 2[pi][phi]) = [h.sub.1] cos [omega]t, (2)

where [phi] is the angle, read out from a vertical, [??] is the angular velocity of the pendulum, a is the linear stiffness coefficient, [a.sub.1] is a coefficient, which includes pendulum length and gravitational constant, and [h.sub.1] cos [omega]t is harmonical moment, enclosed at a point of support. The variable parameter of the pendulum system is frequency [omega] of the external periodic excited moment. Other parameters are fixed: b = 0.1, a = 1, [a.sub.1] = 0.1, [h.sub.1] = 1.

For the given system with one equilibrium position the complete bifurcation analysis was made. Results of the analysis are shown in Fig. 7. Special feature of this system is the unexpected isolated P1 isle, amplitudes of vibrations of which are much greater than ones of the usual P1 regime. Also for these three attractors P1 the domains of attraction were constructed (Fig. 8) for cross-section [omega] = 0.347 of the bifurcation diagram in Fig. 7.

Equations of motion for the third mathematical pendulum model with a sliding mass and with the external periodic excited moment (Fig. 9) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where y is the displacement of the sliding mass, read out from a quiescent state, y is velocity of the sliding mass, [m.sub.1] is the pendulum mass, l is length of the pendulum, [m.sub.2] is second mass, [l.sub.0] is the quiescent state of the sliding mass, [mu] is gravitational constant, [b.sub.1] and [b.sub.2] are linear damping coefficients of the pendulum and the sliding mass, [c.sub.2] is the linear stiffness coefficient of the sliding mass, M([[omega].sub.1]\t) = [h.sub.1] cos [[omega].sub.1]t is the external periodically excited moment and [h.sub.1] and [[omega].sub.1] are the amplitude and frequency of excitation. The variable parameters of the pendulum system are amplitude [h.sub.1] and frequency [[omega].sub.1] of excitation. Other parameters are fixed: [m.sub.1] = 1, [m.sub.2] = 0.1, l = 1, [l.sub.0] = 0.25, [b.sub.1] = 0.2, [b.sub.2] = 0.1, [c.sub.2] = 2, [mu] = 10.

Similar models have been studied in . Results for this system are represented in Figs. 10 and 11. One 1T and one 5T bifurcation group with Andronov-Hopf bifurcations, several symmetry breakings, period doublings and rare attractor P5 RA (Fig. 10a) have been found in the third model.

In Fig. 10b there are 1T and 2T bifurcation groups with several symmetry breakings, period doublings, folds and tip type rare attractors. Global chaos ChA-1 has been found (Fig. 11) in this pendulum system using the Poincare mapping Cm 4 x 50Q x (400-500)T from a contour. Figures 10 and 11 show that the method of complete bifurcation groups allows finding new unnoticed before regimes also in the system with two degrees of freedom. Thus, the application of this method for global analysis of forced oscillations is also possible for systems with several degrees of freedom.

3. CONCLUSIONS

The pendulum systems are widely used in engineering, but their qualitative behaviour has not been investigated enough. Therefore in this work the new non-linear effects in three driven damped pendulum systems, which are sufficiently close to the real models used in dynamics of the machines and mechanisms, were shown. These results were obtained using the method of complete bifurcation groups. Only the method of complete bifurcation groups allows finding rare periodic and chaotic regimes systematically. These regimes can lead to small breakages of machines and mechanisms and to global technical catastrophes, because they are unexpected and usually have large amplitudes.

Some of the obtained new effects can be used for the parametric stabilization of unstable oscillations in technological processes. Authors hope to attract attention of scientists and engineers to solving the important problems of non-linear oscillations analysis by the MCBG. Rare dangerous or useful attractors may find application in the pendulum-like dynamical systems.

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doi: 10.3176/eng.2011.1.08

ACKNOWLEDGEMENTS

This work has been supported by the European Social Fund within the project "Support for the implementation of doctoral studies at Riga Technical University" 2009-2010 and by Latvian scientific grants R7329, R7942, 06.1965, and 09.1073 (2005-2010).

Received 1 October 2010, in revised form 30 November 2010

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Mikhail Zakrzhevsky, Alex Klokov, Vladislav Yevstignejev, Eduard Shilvan Institute of Mechanics, Riga Technical University, 6 Ezermalas Str., LV-1658 Riga, Latvia; alex_klokov@inbox.lv