# Classifying and quantifying basins of attraction.

@article{Sprott2015ClassifyingAQ, title={Classifying and quantifying basins of attraction.}, author={Julien Clinton Sprott and Anda Xiong}, journal={Chaos}, year={2015}, volume={25 8}, pages={ 083101 } }

A scheme is proposed to classify the basins for attractors of dynamical systems in arbitrary dimensions. There are four basic classes depending on their size and extent, and each class can be further quantified to facilitate comparisons. The calculation uses a Monte Carlo method and is applied to numerous common dissipative chaotic maps and flows in various dimensions.

## 61 Citations

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## References

SHOWING 1-10 OF 20 REFERENCES

Scaling behavior of chaotic systems with riddled basins.

- PhysicsPhysical review letters
- 1993

Recently it has been shown that there are chaotic attractors whose basins are such that every point in the attractor's basin has pieces of another attractor's basin arbitrarily nearby (the basin is…

On the Hausdorff dimension of fractal attractors

- Mathematics
- 1981

We consider such mappingsxn+1=F(xn) of an interval into itself for which the attractor is a Cantor set. For the same class of mappings for which the Feigenbaum scaling laws hold, we show that the…

Basin bifurcations of two-dimensional noninvertible maps : Fractalization of basins

- Mathematics
- 1994

Properties of the basins of noninvertible maps of a plane are studied using the method of critical curves. Different kinds of basin bifurcation, some of them leading to basin boundary fractalization…

Simple Chaotic flows with One Stable equilibrium

- PhysicsInt. J. Bifurc. Chaos
- 2013

Using the Routh–Hurwitz stability criterion and a systematic computer search, 23 simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a coexisting…

YET ANOTHER CHAOTIC ATTRACTOR

- Physics
- 1999

This Letter reports the finding of a new chaotic at tractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler at tractors.