Tag Archives: chaos

The geometric principles behind discrete dynamical systems based on the generalized Witch of Agnesi

Consider the family of curves defined by the equation following parametric equation , where and It defines a family of probability distribution functions (PDFs): This can be seen from the above equations because Figure 1 Examples of these PDFs are … Continue reading

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The remarkable behavior of a map displaying derived from a simple model for a biological conflict

One of the simplest yet profound mathematical models for biological growth emerged sometime in the middle of the 1800s due to the work of Verhulst. It describes population growth thus: let be the population of the organism at time . … Continue reading

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Pattern formation in coupled map lattices with the circle map, tanh map, and Chebyshev map

The coupled map lattices (CMLs), first defined by Kunihiko Kaneko around the same time Wolfram was beginning to explore cellular automata, combine features of cellular automata with chaotic maps. The simplest CMLs are defined on a one dimensional lattice with … Continue reading

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The Satija-Ketoja system

Satija and Ketoja discovered an interesting dynamical system in course of the study of the Schrödinger equation for one electron in a two dimensional periodic lattice on a uniform magnetic field. While this equation and its variants have several uses … Continue reading

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Some simple maps specifying strange attractors

This note may be read a continuation of: Some reminiscences of our study of chaotic maps-2 While the story of the chaotic 2D attractors began with the simple-looking maps of Henon and Lozi, by the early 1990s the high-point was … Continue reading

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Cobwebs on the golden hyperbola and parabola

The material presented here is rather trivial to those who have spent even a small time looking at chaotic systems. Nevertheless, we found it instructive when we first discovered it for ourselves while studying conics. Hence, as part of recording … Continue reading

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Deliberations on richness and beauty: discovery of some multi-parameter iterative maps

As we have explained in the earlier notes (1, 2, 3), the second major factor in our exploration of 2D strange attractors maps, IFS and other fractals was the aesthetic experience they produced. Around that time we came across a … Continue reading

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