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Sample-based approach in strongly non-linear multistable systems with uncertainties
Last modified: 2017-05-22
Abstract
In this paper we compare two numerical methods for the analysis of stability of multistable nonlinear dynamical systems and assess them by experimental
investigations. The first method is a classical path-following analysis with Auto--07p, while the second one is a modified basin stability approach [1] called a sample-based appraoch [2]. We study the dynamics of a double pendulum forced kinematically. First, we investigate the system's dynamics with the path-following method. We calculate the ranges of stability for different solutions of the system in a two parameter space, namely the amplitude and the frequency of excitation. We investigate $9$ different periodic solutions. Then we perform the experimental
investigations for the same purpose. Finally,
we apply the extended basin stability analysis to determine the ranges
of stability and two parameters basin stability maps.
Our results from both numerical methods are in a remarkably good agreement
with the experimental data. Hence, we claim that the sample-based
approach ensures the level of accuracy comparable with the classical
path-following method. The advantage of the presented method is that
it enables to analyse the influence of infinite number of parameters
simultaneously, which is impossible for classical methods of analysis.
Also, contrary to classical methods, the computational effort does
not increase with the dimensions of the system. Moreover, the method
enables to detect hidden attractors and solutions with rather meagre
basins of attraction. Apart from that, the method is straight forward
and does not require any specialized knowledge.
Another advantage of the presented approach is that the collected
data allow to quantify the stability of each attractor and prepare
the two parameter basin stability maps. Such diagrams enable to detect
the ranges in parameter space for which the solution is more likely
to appear due to a large volume of the basin of attraction and have
strong practical significance.
In the last part of the work we repeat the basin stability analysis
under uncertainties in the parameter values. The obtained results
does not differ significantly from the original data, hence we claim
that the proposed method can be applied even without the precise knowledge
of the parameters values which is a unique feature.
The presented results are the first broad comparison between the path-following
method, the basin stability approach and the experimental investigation.
We show that the sample-based methods are a reliable tool for the
analysis of complex dynamical systems. Moreover, we prove that the
extended basin stability method has significant advantages which make
it robust and appropriate for many applications in which classical
analysis methods are difficult to apply.
[1] P. Brzeski, M. Lazarek, T. Kapitaniak, J. Kurths, and P. Perlikowski. Basin stability approach for quantifying responses of multistable systems with parameters mismatch. Meccania, 51(11), 2713--2726, 2016.
[2] P. Brzeski, J. Wojewoda, T. Kapitaniak, J. Kurths, and P. Perlikowski. Can sample-based approach outperform the classical dynamical anaysis?
- experimental confirmation of the basin stability method. In preparation.
investigations. The first method is a classical path-following analysis with Auto--07p, while the second one is a modified basin stability approach [1] called a sample-based appraoch [2]. We study the dynamics of a double pendulum forced kinematically. First, we investigate the system's dynamics with the path-following method. We calculate the ranges of stability for different solutions of the system in a two parameter space, namely the amplitude and the frequency of excitation. We investigate $9$ different periodic solutions. Then we perform the experimental
investigations for the same purpose. Finally,
we apply the extended basin stability analysis to determine the ranges
of stability and two parameters basin stability maps.
Our results from both numerical methods are in a remarkably good agreement
with the experimental data. Hence, we claim that the sample-based
approach ensures the level of accuracy comparable with the classical
path-following method. The advantage of the presented method is that
it enables to analyse the influence of infinite number of parameters
simultaneously, which is impossible for classical methods of analysis.
Also, contrary to classical methods, the computational effort does
not increase with the dimensions of the system. Moreover, the method
enables to detect hidden attractors and solutions with rather meagre
basins of attraction. Apart from that, the method is straight forward
and does not require any specialized knowledge.
Another advantage of the presented approach is that the collected
data allow to quantify the stability of each attractor and prepare
the two parameter basin stability maps. Such diagrams enable to detect
the ranges in parameter space for which the solution is more likely
to appear due to a large volume of the basin of attraction and have
strong practical significance.
In the last part of the work we repeat the basin stability analysis
under uncertainties in the parameter values. The obtained results
does not differ significantly from the original data, hence we claim
that the proposed method can be applied even without the precise knowledge
of the parameters values which is a unique feature.
The presented results are the first broad comparison between the path-following
method, the basin stability approach and the experimental investigation.
We show that the sample-based methods are a reliable tool for the
analysis of complex dynamical systems. Moreover, we prove that the
extended basin stability method has significant advantages which make
it robust and appropriate for many applications in which classical
analysis methods are difficult to apply.
[1] P. Brzeski, M. Lazarek, T. Kapitaniak, J. Kurths, and P. Perlikowski. Basin stability approach for quantifying responses of multistable systems with parameters mismatch. Meccania, 51(11), 2713--2726, 2016.
[2] P. Brzeski, J. Wojewoda, T. Kapitaniak, J. Kurths, and P. Perlikowski. Can sample-based approach outperform the classical dynamical anaysis?
- experimental confirmation of the basin stability method. In preparation.