Time delay Duffing’s systems: chaos and chatter control
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- Rusinek, R., Mitura, A. & Warminski, J. Meccanica (2014) 49: 1869. doi:10.1007/s11012-014-9874-4
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The effect of a delay feedback control (DFC), realized by displacement in the Duffing oscillator, for parameters which generate strange chaotic Ueda attractor is investigated in this paper. First, the classical Duffing system without time delay is analysed to find stable and especially unstable periodic orbits which can be stabilized by means of displacement delay feedback. The periodic orbits are found with help of the continuation method using the AUTO97 software. Next, the DFC is introduced with a time delay and a feedback gain parameters. The proper time delay and feedback gain are found in order to destroy the chaotic attractor and to stabilize the periodic orbit. Finally, chatter generated by time delay component is suppressed with help of an external excitation.
KeywordsDuffing’s system Time delay Chaos control Chatter control
The first proponent of the chaos theory was Henri Poincaré, who in the 1880s, studying a three-body problem, found nonperiodic orbits which are not forever increasing nor approaching a fixed point. Since that time chaos has been observed in a number of experiments although it has not been defined. Yoshisuke Ueda on November 27, 1961 at Kyoto University was experimenting with analog computers and noticed “randomly transitional phenomena” in the specific Duffing’s oscillator. Currently scientists are looking for efficient methods to avoid that interesting, but in many cases—from technical point of view—dangerous phenomena. On the other hand, not only the Duffing but also other nonlinear systems exhibit instabilities and chaos. Therefore great attention has been paid to stabilize them. Time delay effect is one of the ideas to achieve this aim. Time delays are common feature of many physical, biological and engineering systems. There are systems where time delay is present intrinsically due to processing time, mechanical properties etc., for instance in technological cutting processes. On the other hand, there are systems where time delay is introduced externally in order to stabilize unstable periodic orbits (UPO) and unstable steady states (USS). Various methods of controlling unstable and chaotic systems have been developed in the past 20 years and applied to real systems in physics, chemistry, biology, and medicine . Pyragas  was the first who introduced delay feedback control (DFC) to stabilize UPO embedded in a chaotic attractor. This method, known as time delay auto-synchronization (TDAS), bases on constructing the control force from the difference of the current state to the state one period in the past, so that the control signal vanishes when the stabilization of the target orbit is attained. Next, the TADS method was improved by introducing multiple delays (extended TADS or ETADS) .
DFC is applied usually for stabilization of the UPO [5, 7, 14, 23, 25] and also to control unstable fixed points (FP) [3, 7, 9, 14, 33]. Separate class of delay differential equations (DDE) are the neutral delay differential equations (NDDE). Its stability and asymptotic properties are described in [2, 13].
DFC method proposed by  works very well in case of the UPO if time delay is precisely equal to the period of the UPO and the feedback gain is strong enough. Note that only the stability properties of the orbit are changed, while the orbit itself and its period remain unaltered. If the orbit has an odd number of the real Floquet multipliers greater than unity, the delayed feedback can never stabilize it .
Usually in the literature DFC is applied to stabilize the Rössler system [1, 7, 14, 23] but only selected papers are devoted to the Duffing’s oscillator [8, 10, 22] which in its various forms is used to describe many nonlinear systems. The Ueda’s oscillator can be considered as a special case of the Duffing’s system which natural frequency equals to zero. Chaotic vibrations for this system exist for some set of parameters and initial conditions.
In general, dynamical systems with time delay are important class of systems, especially in the control theory, but on the other hand time delay is very often encountered in various technical systems, such as electric, pneumatic and hydraulic networks, chemical and machining processes, etc. These problems are described by DDE which indeed are a type of differential equations where time derivatives at the current moment of time depend on the solution and possibly on its derivatives at previous moments. For instance, in cutting processes the effect of time delay generates harmful vibrations—a so called regenerative chatter [6, 11, 12, 15, 26, 27, 30, 31, 32]. This one and another kind of chatter in real processes are very often investigated by means of various methods of nonlinear time series analysis [16, 17, 18, 19, 20, 21, 28].
The DDE exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and lead to chaotic motions. Therefore, this is very important from practical point of view to control chaotic systems by means of DFC method and to control the systems which are time delayed by nature, e.g. cutting processes with regenerative chatter [27, 32].
In this paper, at the beginning we consider the effect of time delay in the Ueda’s oscillator in order to stabilize the chaotic attractor by a proper selection of the DFC parameters: the time delay and the feedback gain. Next, the problem, how the chatter vibrations generated by the time delay system can be suppressed by external excitation is presented.
2 Ueda’s attractor
Increasing the external force amplitude (f) from zero (Fig. 3) leads to the saddle-node bifurcation occurrence. One unstable and two stable solutions exist in the range of excitation amplitude from 0.12 to 0.43, depending on initial conditions. For f = 2, the first period doubling bifurcation (PDB) appears. Simultaneously, the 1 T period solution losses its stability (dashed line) and the new solutions of period 1 and 2 T occur. The next PDBs lead to 2, 4 and 8 T period solutions, that are marked by the colour lines. Finally, the cascade of PDBs leads to chaos which cannot be observed in Fig. 3 but is visible in Fig. 1, where the positive largest Lyapunov exponent (LLE) is shaded. At the external force amplitude equal ca. 8 some of the solutions regain stability. Then, the stable periodic orbits (solid line) coexist with the UPO (dashed lines, Fig. 3).
According to the DFC theory, the Ueda’s chaotic attractor can be stabilized with the help of time delay which corresponds to the period of the selected UPO (1, 2, 4 or 8 T). The choice of time delay seems to be quite obvious, but the feedback gain should be discussed. The analysis of chaos control is presented thoroughly in the next section.
3 Chaos control
The unstable T-periodic orbit can be successfully stabilized (Fig. 4a), when the feedback gain α is greater than 0.065 and lesser than 0.13. When α > 0.13 various scenarios of motion are possible, depending on initial conditions, that is periodic of period 3 T or quasi-periodic motion. This situation lasts till α = 0.215, and next the DFC system works only partially, because a chaotic attractor exists together with a quasi-periodic motion for some initial conditions. Stabilization of 2 T period orbit with the time delay τ = 2 T looks promisingly, as well (Fig. 4b). In the region when 0.25 < α < 0.4 chaos is fully suppressed and the system response frequency is exactly equal to the external force frequency λ = 1. It has been expected, that time delay τ = 2 T results in stabilization of the orbit with period 2 T while the system response period is 1 T.
The stabilization test of the 4 T period solution leads to motion 3 T period (Fig. 4c) in spite of the fact, that stabilization of the 4 T period orbit was expected here. Comparing the bifurcation diagrams in Fig. 4a, c, one can notice that at the feedback gain α ≈ 0.15 the same 3 T period solution exists, both for the time delay τ = 1 T and τ = 4 T. When the time delay τ = 8 T there is no stabilization of any periodic orbits in the analysed range of the feedback gain α. This orbit has an odd number of the real Floquet multipliers greater than unity. This problem will be studied in the future.
On the other hand, chaos can be controlled by adjusting time delay τ. This is a second key parameter of the DFC, which influences the system dynamics. According to a typical approach to the problem only the time delay which is equal to the period of UPO is able to stabilize the orbit and avoid chaos. Therefore, now the value of the time delay τ is investigated at the feedback gain α = 0.1. In this case, the solution of period T is expected to be stable when τ = T. This is of course true but, the regular vibrations periodic or quasi-periodic) are also possible for other time delays in the places when the Largest Lyapunow Exponent LLE equals zero (see Fig. 5). However the widest region of synchronization and periodic motion appears when τ ≈ 1/3 T and τ ≈ 3 T for the analysed system. All these phenomena are illustrated in the bifurcation diagram presented in Fig. 5, where for convenience, the upper scales of the period T corresponding to time delay τ is added.
In the case of τ ≈ 1/3 T, a periodic solution also exists, even though the vibrations’ amplitude is the smallest one. This can have applicable aspects in mechanical systems which should be controlled in order to avoid chaos and large amplitudes of vibrations.
4 Chatter control
However, CCS has a fault because it is difficult to match the external excitation (λ) and the frequency (f) in order to suppress chatter vibrations. In the analysed case the chatter frequency fc is read out from the time series when CCS is off (Fig. 11). The value of chatter frequency is 2π/3 at rotational speed Ω = 2.5 whereas, the external excitation frequency (f), which fulfil assumption of chatter suppression is about π/2 (λ = 1.6010).
Delay feedback is one of control methods commonly used in engineering. There are although systems where time delay is present due to their natural properties and then delay is a source which generates vibrations as well. Here, DFC is applied in order to destroy chaotic attractor and to stabilize the UPO. The classical approach says, that time delay corresponding to the period of unstable orbit can stabilize it and result in the avoidance of chaotic solutions. The results presented in this paper point out other possibilities. It has been shown, that the time delay equal to 1/3 T can successfully destroy strange attractor which transforms into periodic solution. Then, the periodic orbit is more beneficial because of its smaller amplitude.
Additionally, the idea of chatter vibrations avoidance generated by the time delay effect in a manufacturing process is discussed here. It is demonstrated, that by introducing external excitation with proper frequency, amplitude and initial conditions, chatter in regenerative model of cutting can be suppressed. However, a selection of the external excitation parameters is not a simple task. This can be done numerically, as it is presented in the paper, but more general conclusions can be drawn after the analysis of periodic orbits stability.
The work is financially supported under the project of National Science Centre according to the decision no. DEC-2011/01/B/ST8/07504.
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