^{1}

^{*}

^{1}

^{*}

We develop in this text a probabilistic approach to deterministic nonlinear systems. By studying the spectral properties of the Frobenius-Perron operator of a Duffing oscillator, relevant dynamic properties of the system are identified. Using the characteristics of the Dirac operator, the evolution of the probability density function is obtained for the Duffing oscillator, allowing important aspects of this system to be investigated analytically. A comparison with numerical simulation is carried out in order to validate the results obtained by the analytical approach as well as to verify the nonsymmetric features of the oscillator response.

It is well known that nonlinear dynamic systems operating under certain conditions can display unstable be- havior and, in some cases, present bifurcations and deterministic chaos [

These characteristics are sometimes difficult to analyze through time histories generated by means of numeric simulations from specific initial conditions. More efficient numerical methods are required for this task de- pending on the nonlinear behavior of the system, and even then it may be very time consuming to inspect all possible combinations of different control parameters along with different initial conditions. In this scenario, a probabilistic approach can simplify the system analysis, since it can give an overview of all possible occurrences for the whole phase space at once, and for every possible initial condition [

In Section 2, we develop the main theory of the Frobenius-Perron operator, emphasizing the physical inter- pretation of the PDF evolution, and obtaining the evolution equation for a dynamic system described by a sys- tem of ordinary differential equations.

In Section 3, we apply the method to a classical non-linear dynamic system: a Duffing oscillator. Through the use of the Dirac operator, we obtain analytical functions for the PDF evolutions and the resulting expression for the Duffing oscillator is shown in that section.

Finally, in Section 4, other results are shown and comparisons with numerical results are carried out.

The probabilistic approach to the investigation of dynamic systems of the form

It is well known [

The evolution of the above map depends, therefore, on its initial condition. For example, for

trajectories can be seen in

For a small change in the initial condition

seen in

In other words, this map exhibits high sensitivity to the initial conditions; as we know, this feature is a signature of chaotic behavior.

For a map like this, just looking at individual trajectories yields little information about the global dynamics of the system. The trajectories behave so erratically that no pattern can be easily identified. However, it is

possible to take the phase space and divide it into n partitions, allowing us to count the number of times the system passes through each partition.

The result of this counting procedure can be shown in a histogram. One example can be seen in

These plots do not appear to be as highly sensitive to the initial conditions as the times histories presented before. In other words, this form of presenting the dynamic properties of a system seems to show a more well defined pattern, even though the system behaves in a chaotic way.

The use of probability density functions to study the dynamics of a system relies on properties of the Frobenius-Perron operator, [

Considering the initial and final density states [

One may define

So that for another initial condition one has:

The main goal here is to define the relation between

The counter image of an interval

Since

And therefore:

Since up to this point both

This is the relation between

And differentiate with respect to x, giving:

This equation can be recast in terms of an operator as:

The equation above describes the so-called Frobenius-Perron operator.

Making use of properties of the Markov operator, as well as infinitesimal operator properties, [

can be described by:

where

The Duffing oscillator is a well known nonlinear system having as its main feature the presence of a nonlinear restoring force, [

Rewriting the above equation in terms of phase space equations:

One has:

Now, using the Frobenius-Perron infinitesimal operator one can write the above system in terms of its pro- bability density function:

Equation (7) represents the PDF evolution with respect to the state variables

Let us then assume that a solution for the PDF equation can be given in the following form:

where

The function

Taking

For the above equation to hold the terms multiplied by

And therefore the sum of the other terms on the same side of the equation must also be zero:

Using the above equation along with Equation (9) one has:

This procedure has therefore yielded a system of two uncoupled partial differential equations defined by Equations (9) and (11). One possible solution for these equations can be formed by two exponential terms, one dependent on

Additionally, we note that the solution

In other words, the function

Therefore, one may choose a function

Considering this equation, the PDF (r) takes the form:

And with

Equation (12) represents the PDF evolution for the Duffing oscillator.

Observing the behavior of the PDF over a large set of system parameters gives much information about the oscillator, and since we have a closed-form expression for the PDF evolution over time, the behavior of the Duffing oscillator can be easily mapped out. However, in order to analyze the behavior of a system in terms of its PDF we must familiarize ourselves with a new form of presenting results. These results are better understood if contrasted with classical views, such as those granted by potential well analysis or by numerical simulation.

Firstly, we vary the shape of the potential well through the changing of the parameter

In

In the third case (

Using Equation (12), the PDFs for the three cases listed above were plotted in Figures 8-10. Since the value of the PDF oscillates in time (according to Equation (12)), one can chose t to be arbitrary when plotting the function. Moreover, the PDF must be normalized by its integral, making the choice of any specific time t to be irrelevant.

In

As mentioned above, the results for the PDF can also be compared with those obtained by numerical simu- lation. Here have we solved the equations of motion numerically for a finite number of initial conditions. The phase space was divided into n partitions, and a histogram was built based on the number of times that the time- sampled system state fell into each of the n partitions. The results as displayed in Figures 11-13.

It is clear that the same shapes noted before for the PDF curve reappear here. For

Although the solution does not show a straightforward relation between the PDF and the system phase velocity, such relation can be observed in the PDF plots. This can be interpreted as the fact that the lower the phase velocity of the system at a specific phase space area, the higher the likelihood of finding the system there. The system will spend a relatively short period of time over an area where it presents a higher phase velocity. That was the reasoning behind the simplification adopted in the solution, and it has turned out to be in good agreement with the numerical results. Another example of that can be shown by observing the occurrences histogram obtained by numeric simulation in a three-dimensional graph, including also the velocity as another axis. Such a plot can be seen in

A probabilistic approach for the analysis of the well-known deterministic Duffing oscillator was described. Application of the Frobenius-Perron operator, and the use of suitable properties of the Dirac operator, yielded an analytical expression for the evolution of the system probability density function (PDF). The results of such approach were compared with conventional numerical simulations as well as potential well analysis.

The probabilistic approach described in this work turned out to be a very expedient method of analysis. Because it furnished an analytical expression for the PDF, different scenarios of parameters could be addressed easily, and the system could be mapped out as a whole in a very robust way. Another advantage of the method is that one single PDF represents the likelihood of finding the system for all initial conditions in the chosen phase

space domain. In this way, the PDF evolution can be seen as a simple way of producing a first overall picture of the behavior of a dynamic system in one single graph.

The area of applicability of this method in dynamic analysis seems to be extensive, especially for non-linear systems. As it was showed, even for systems which present high instability in their time trajectories (and in some cases, even chaos) this approach offers an efficient way of analysis, something that would not be so easily achieved by simple numerical simulation.

Jessé Rebello de SouzaJunior,Claudio GomesFernandes, (2015) Probabilistic Approach in the Investigation of the Dynamics of a Duffing Oscillator. Applied Mathematics,06,1282-1293. doi: 10.4236/am.2015.68122