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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 650354, 8 pages
Research Article

Active Control of Oscillation Patterns in the Presence of Multiarmed Pitchfork Structure of the Critical Manifold of Singularly Perturbed System

Faculty of Materials Science and Technology, Slovak University of Technology in Bratislava, Hajdoczyho 1, 917 01 Trnava, Slovakia

Received 2 September 2013; Revised 22 September 2013; Accepted 23 September 2013

Academic Editor: Hamid Reza Karimi

Copyright © 2013 Robert Vrabel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We analyze the possibility of control of oscillation patterns for nonlinear dynamical systems without the excitation of oscillatory inputs. We propose a general method for the partition of the space of initial states to the areas allowing active control of the stable steady-state oscillations. Furthermore, we show that the frequency of oscillations can be controlled by an appropriately positioned parameter in the mathematical model. This paper extends the knowledge of the nature of the oscillations with emphasis on its consequences for active control. The results of the analysis are numerically verified and provide the feedback for further design of oscillator circuits.

1. Introduction

Control of oscillations is a practically important problem in many technical applications (see e.g. [1, 2]), which can generally be stated in terms of two different objectives. (a)To obtain an asymptotically stable zero solution which attracts all initial conditions in a suitably large region (regulator problem [3]).(b)To obtain an asymptotically stable periodic solution with desired properties (such as oscillation at the given amplitudes and frequencies) and which attracts all initial states in a suitably large region (oscillator problem).

In this context, usually two-well potential of an unperturbed system was considered by using analytic methods and numerical simulations, (see, e.g., [46] and the references therein).

Also, in many biological systems, some form of oscillation control is needed to track the constantly changing resonant frequency or to tune it to a specified frequency while keeping the amplitude constant [79].

For example, in [10], the oscillation patterns were analyzed by the symmetric Hopf bifurcation theory applying group theory. Recently, in [11], the oscillation patterns of the bifurcating periodic oscillations of three coupled Van der Pol oscillators were studied.

To our knowledge, no paper exists addressing the question of active control of oscillation patterns for the dynamical systems with potential with multiwell and multibarrier structure. For example, the two-armed pitchfork bifurcation in the presence of Hopf bifurcation in the context of double magneto convection was numerically studied in the work [12]. In the paper [13], the problem of output stability for systems under the oscillatory excitation via Lyapunov approach was analyzed.

The analysis of behavior patterns is of high importance to uncover real-time threats in the industrial systems and to plan a predictive maintenance program. Recently, Yin et al. [14] proposed two online schemes for the fault-tolerant architecture for the purpose of fault-tolerant control. One is a gradient based iterative tuning scheme for the online optimization of the system performance, and the other is an adaptive residual generator scheme for the online identification of the abnormal change of the system parameters.

There are several approaches to the topic of vibration control in industrial equipment; for example, in the work [15], the symbolic computation systems are presented with the purpose of analyzing the spectrum and waveform vibration allowing bearing defects in low-speed machines via dynamic modeling of patterns of behavior. The paper [16] provides a comparison study on the basic data-driven methods for process monitoring and fault diagnosis. Most recently, in the paper by Palacios-Quinonero et al. [17] by using an approach based on the connected control method, active-passive structural vibration control strategies for seismic protection of multibuilding systems are designed and applied on the proposed linear mathematical model of connected multistructure mechanical systems.

The method we propose could provide the mechanical engineers the mathematical tool for an accurate diagnosis in the vibration analysis and thus prevent or predict future failures of industrial systems. The main advantage of our approach—an analysis of the time-reverse differential equations associated with the problem under consideration—over methods mentioned above lies in its straightforward extension to nonlinear models (continuous and discrete) with time-delay; for the topic, see, e.g., [1820].

In this paper, we focus our attention on the proof of the existence and the possibility of active control of stable steady-state nonlinear oscillations in the dynamical system describing the singularly perturbed undamped oscillator with a continuous nonlinear restoring force and without the excitation of oscillatory inputs: where is an initial state, is a direct output, is a positive continuous function on ,  ,  ,  ,  and  ,   is a singular perturbation parameter. It is instructive for future reference to keep in mind the symmetric manifold (Figure 1). The parameter is a constant determining the distance between pitchfork arms for , and without loss of generality, we will assume that .

Figure 1: The multiarmed pitchfork manifold .

Rewriting (1) to an equivalent system of three first order autonomous equations, we obtain

We show that the nonlinear system described by differential equation from (1) can basically exhibit two distinct types of steady-state oscillations, namely.(i)In-well, small orbit dynamics, where the system state remains within the potential well centered at a stable equilibrium point (center).(ii)Cross-well, large orbit dynamics, whose trajectories surround the equilibrium points (the saddles between centers).

The results of the analysis will be numerically verified and provide valuable insights into dynamics of the control system under consideration.

Further, we show that the singular perturbation parameter plays role modeling tool for the frequency control of the nonlinear oscillations arising in these systems. Another objective of this paper is to divide the space of initial states to the areas of stability ,,   such that for solution of (1) oscillate around ,   and for .

Finally, we prove that the solutions of (1) will rapidly oscillate for with the frequency of the oscillations increasing without bound as .

System (3) is an example of a singularly perturbed system, because in the limit , it does not reduce to a differential equation of the same type, but to an algebraic-differential reduced system

Another way to study the singular limit is by introducing the new independent variable which transforms (3) to the system Taking the limit , we obtain the so-called associated system [21] in which plays the role of a parameter.

Both scalings agree on the level of phase space structure when but offer very different perspectives since they differ significantly in the limit when . The main goal of singular perturbation theory is to use these limits to understand structure in the full system when .

The critical manifold is defined as a solution of the reduced system; that is, which corresponds to a set of equilibria for the associated system (8)–(10).

2. Multiarmed Pitchfork Bifurcation of Associated System at

Without loss of generality, we will assume hereafter that .

Then, the equations have one solution for if and five if ; that is, the critical manifold is a multiarmed pitchfork manifold. Due to the fact that the eigenvalues of the Jacobian are the associated system (8), (9), and has, for , one equilibrium (degenerate center with a double zero eigenvalue) for and five equilibria (two saddles and three centers) for . Thus, the two pieces of critical manifold corresponding to the saddles of associated system (8)–(10) are the normally hyperbolic submanifolds [21].

Consider the total energy functional with potential with multibarrier and multiwell structure (for , see Figure 2).

Figure 2: Potential profile with ,  , and for .  .

We use the level surfaces of on to characterize the trajectories of (3). These surfaces in -space are defined by extending it as long as remains real. In our case, such trajectories, lying on the surface , are bounded for every small . On the interval , there is a motion in a single potential well and on the interval , triple well with a barrier in between.

Computing the derivative of along the solution of (3) (, for , we proceed analogously), we obtain for For potential profile (), we have As follows from computation above, is a sufficient condition for the existence of nonlinear steady-state oscillations (constant amplitude). In this case, and for .

To characterize input-output relationships, we determine the areas ,  ,   of initial states ,   at the point by solving associated time-reverse initial value problem allowing a specific switching strategy between oscillation patterns.

Let us denote the real roots of the six-order algebraic equation .

Thus, taking into consideration the definition of the sets and the profile of potential at , we obtain

The Figures 3, 4, and 5 demonstrate all types of oscillations patterns for (1) with the initial values from ,  , respectively.

Figure 3: Numerical solution of and with and ,  ,  ,  ,  , and  .
Figure 4: Numerical solution of and with and ,  ,  ,  ,  ,  and .
Figure 5: Numerical solution of , and with and ,  ,  ,  ,  , and  .


Then, the solutions of (1) with surround all pitchfork arms for (Figure 6). For analysis of this type of oscillations for double-well potential profile in more detail, see [22]. The selected points of the sets ,  ,  ,  and    are depicted on the Figure 7.

Figure 6: Numerical solution of , and with and ,  ,  ,  ,  , and  .
Figure 7: The selected points of the spiral-shaped sets ,  ,  , and for ,  ,  ,  ,  , and   (from (a) to (d)).

On the basis of the Kneser and Fukuhara theory, the sets ,  , are continua (i.e., the bounded and connected sets), and is connected in where the overline denotes the closure of a set in the standard topology of . Obviously,(1)(2)for every .

Also, as a consequence of the oddness of the function in the variable for every fixed , we have(1),  (2) and are symmetric about the origin for every .

3. Nonoscillatory Solutions

We note that there also exists for a family of the nonoscillatory solutions of (1) for with where for In order to facilitate the understanding of the dynamics of dynamical system (3) after the time , we drew the Figures 8, 9, 10, 11, and 12.

Figure 8: The phase portrait of system (3) for with and . The solutions of (1) lying in the time on the orbits with arrows are nonoscillatory, and or .
Figure 9: Five solutions () of (1) with     and ( on ),      and  ,  and     and   = .
Figure 10: Two of infinitely many solutions of (1) with and .
Figure 11: Two of infinitely many solutions of (1) with and either ,   or ,  .
Figure 12: Two of infinitely many solutions of (1) with and either ,   or ,  .

4. Frequency Control of Nonlinear Oscillations

In this section, using the appropriate changes of coordinates, we show that the singular perturbation parameter plays role modeling tool for control of frequencies of nonlinear oscillations. We will analyze the case only. For the other cases, we proceed in a similar way.

From the theory above, it follows that the solution of (1), will oscillate around the half-line for (Figure 5).

Let us put and for . Due to requirement of steady-state oscillations, , and consequently, for . Thus, Multiplying these equations by ,  , respectively, and after algebraic manipulations, we finally obtain the following differential equation for : where Denote by the root of equation such that (see Figure 2) and let where is a continuum depending on the constant amplitude of oscillations. Obviously, these nonzero values exist because of positivity of function on . Hence,

Now, let us denote by the spacing between two successive zeros of on , where is a solution of (1) with . Then, integrating the inequality (26) with respect to the variable between two successive zeros of , we immediately obtain Hence, that is, the frequency of oscillations increases without bound for tending to zero. As a consequence of the inequality (28) and the definition of the constant , we note that for a fixed value of parameter , the greater amplitude of oscillations leads to the lower frequency of oscillations.

The data for Figures 312 have been worked out with the computer system MAXIMA [23, 24]. The computer code is available from the authors upon request.

5. Conclusions

In this paper, a novel technique to active control of the oscillation patterns and their frequencies for nonlinear dynamical systems with initial condition is proposed. The method is based on solving the time-reverse differential equation on a particular interval. The analytical results are numerically simulated by employing the computer system MAXIMA. The ideas of this paper may be naturally extended and adapted to wide class of the systems to find the basins for different patterns of systems behavior and may be employed without any principal limitation.


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