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.

Control of oscillations is a practically important problem in many technical applications (see e.g. [

To obtain an asymptotically stable zero solution which attracts all initial conditions in a suitably large region (regulator problem [

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., [

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 [

For example, in [

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 [

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. [

There are several approaches to the topic of vibration control in industrial equipment; for example, in the work [

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., [

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:

The multiarmed pitchfork manifold

Rewriting (

We show that the nonlinear system described by differential equation from (

In-well, small orbit dynamics, where the system state remains within the potential well centered at a stable equilibrium point (center).

Cross-well, large orbit dynamics, whose trajectories surround the

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

Finally, we prove that the solutions

System (

Another way to study the singular limit

Both scalings agree on the level of phase space structure when

The critical manifold

Without loss of generality, we will assume hereafter that

Then, the equations

Consider the total energy functional

Potential profile

We use the level surfaces

Computing the derivative of

To characterize input-output relationships, we determine the areas

Let us denote

Thus, taking into consideration the definition of the sets

The Figures

Numerical solution of

Numerical solution of

Numerical solution of

Let

Then, the solutions of (

Numerical solution of

The selected points of the spiral-shaped sets

On the basis of the Kneser and Fukuhara theory, the sets

for every

Also, as a consequence of the oddness of the function

for every

We note that there also exists for

The phase portrait of system (

Five solutions (

Two of infinitely many solutions of (

Two of infinitely many solutions of (

Two of infinitely many solutions of (

In this section, using the appropriate changes of coordinates, we show that the singular perturbation parameter

From the theory above, it follows that the solution of (

Let us put

Now, let us denote by

The data for Figures

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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