89 research outputs found
Robust Chaos
It has been proposed to make practical use of chaos in communication, in
enhancing mixing in chemical processes and in spreading the spectrum of
switch-mode power suppies to avoid electromagnetic interference. It is however
known that for most smooth chaotic systems, there is a dense set of periodic
windows for any range of parameter values. Therefore in practical systems
working in chaotic mode, slight inadvertent fluctuation of a parameter may take
the system out of chaos. We say a chaotic attractor is robust if, for its
parameter values there exists a neighborhood in the parameter space with no
periodic attractor and the chaotic attractor is unique in that neighborhood. In
this paper we show that robust chaos can occur in piecewise smooth systems and
obtain the conditions of its occurrence. We illustrate this phenomenon with a
practical example from electrical engineering.Comment: 4 pages, Latex, 4 postscript figures, To appear in Phys. Rev. Let
A Probabilistic Distance-Based Stability Quantifier for Complex Dynamical Systems
For a dynamical system, an attractor of the system may represent the
`desirable' state. Perturbations acting on the system may push the system out
of the basin of attraction of the desirable attractor. Hence, it is important
to study the stability of such systems against reasonably large perturbations.
We introduce a distance-based measure of stability, called `stability bound',
to characterize the stability of dynamical systems against finite
perturbations. This stability measure depends on the size and shape of the
basin of attraction of the desirable attractor. A probabilistic sampling-based
approach is used to estimate stability bound and quantify the associated
estimation error. An important feature of stability bound is that it is
numerically computable for any basin of attraction, including fractal basins.
We demonstrate the merit of this stability measure using an ecological model of
the Amazon rainforest, a ship capsize model, and a power grid model
Dangerous bifurcation at border collision: when does it occur?
It has been shown recently that border collision bifurcation in a piecewise smooth map can lead to a situation where a fixed point remains stable at both sides of the bifurcation point, and yet the orbit becomes unbounded at the point of bifurcation because the basin of attraction of the stable fixed point shrinks to zero size. Such bifurcations have been named "dangerous bifurcations". In this paper we provide explanation of this phenomenon, and develop the analytical conditions on the parameters under which such dangerous bifurcations will occur
Multi-parametric bifurcations in a piecewise-linear discontinuous map
In this paper a one-dimensional piecewise linear map with discontinuous system function is investigated. This map actually represents the normal form of the discrete-time representation of many practical systems in the neighbourhood of the point of discontinuity. In the 3D parameter space of this system we detect an infinite number of co-dimension one bifurcation planes, which meet along an infinite number of co-dimension two bifurcation curves. Furthermore, these curves meet at a few co-dimension three bifurcation points. Therefore, the investigation of the complete structure of the 3D parameter space can be reduced to the investigation of these co-dimension three bifurcations, which turn out to be of a generic type. Tracking the influence of these bifurcations, we explain a broad spectrum of bifurcation scenarios (like period increment and period adding) which are observed under variation of one control parameter. Additionally, the bifurcation structures which are induced by so-called big bang bifurcations and can be observed by variation of two control parameters can be explained
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