Abstract
In this paper we present how sample based analysis can complement classical methods for analysis of dynamical systems. We describe how sample based algorithms can be utilized to obtain better understanding of complex dynamical phenomena, especially in multistable dynamical systems that are difficult for analytical investigations. Relying on the simple, direct numerical integration algorithms we are able to detect all possible solutions including hidden and rare attractors; investigate the ranges of stability in multiple parameters space; analyse the influence of parameters mismatch or model imperfections; assess the risk of dangerous or unwanted behaviour and reveal the structure of multidimensional phase space. For each mentioned application we present methodology, example on paradigmatic non-linear dynamical system and discuss practical applications. The presented methods of analysis can be applied to solve numerous of scientific problems originating from different disciplines. Moreover, their robustness and efficiency will grow with the upcoming increase of computational power.
Highlights
The stability of dynamical systems have focused the interest of scientist for more than hundred years
By white colour we mark areas where only Duffing system is oscillating in 1:1 resonance with frequency of excitation and the pendulum is in stable equilibrium position, i.e., hanging down pendulum (HDP) state
In this case the dynamics of the system is reduced to the oscillations of summary mass (M þ m)
Summary
The stability of dynamical systems have focused the interest of scientist for more than hundred years. Kepller assumed that planets move along the perfect ellipses, after full revolution they follow the same trajectory [93]. This beautiful simplicity has been destroyed be the Newtons law of universal gravitation [70] which proofs that interactions occur between the Sun and all objects that orbit it. In the 18th century Lagrange and Laplace correctly formulated the equations of motion and proofed that Newton’s law is the universal explanation for the motion of the celestial bodies [68]. The breakthrough work has been published by Poincare [4, 87] where he showed that it is not possible to integrate the equations of motion of three
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