Sensitivity problems related to certain bifurcations in non-linear recurrence relations

Gumowski, I.; Mira, C.

doi:10.1016/0005-1098(69)90073-9

Cited by 14 publications

(2 citation statements)

References 4 publications

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“…2), 36 Edward Lorenz, discovered his chaotic attractor, at MIT, on 1963, 10 and Christian Mira, discovered in 1978 his chaotic attractor in an iterated quadratic map of the plane creating the theory of critical curves for iterated maps. 37,38 These discoveries sounded the death knell for our approach based on differential topology. In 1971-73, new people had arrived in my department (UC Santa Cruz), including John Guckenheimer, a recent Ph.D. with Smale, and veterans of our group, Palais and Mike Shub.…”

Section: The Smale Program By Ralph Abrahammentioning

confidence: 99%

Some elements for a history of the dynamical systems theory

Letellier

Abraham

Shepelyansky

et al. 2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to “reconstruct” some supposed influences. In the 1970s, a new way of performing science under the name “chaos” emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements—which were never published—illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.

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Section: The Smale Program By Ralph Abrahammentioning

confidence: 99%

Some elements for a history of the dynamical systems theory

Letellier

Abraham

Shepelyansky

et al. 2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…In this paper we are concerned with the study of non-invertible, planar maps. The study of such maps goes back to at least the works [Gumowski and Mira, 1965, 1969, Mira, 1964 in which the role of critical lines in the formation of basin boundaries and their bifurcations was studied. Since the 1990s the interest for 2-dimensional endomorphisms has increased tremendously.…”

Section: Motivationmentioning

confidence: 99%

Periodicity and Chaos Amidst Twisting and Folding in Two-Dimensional Maps

Garst¹,

Sterk

2018

Int. J. Bifurcation Chaos

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We study the dynamics of three planar, noninvertible maps which rotate and fold the plane. Two maps are inspired by real-world applications whereas the third map is constructed to serve as a toy model for the other two maps. The dynamics of the three maps are remarkably similar. A stable fixed point bifurcates through a Hopf–Neĭmark–Sacker which leads to a countably infinite set of resonance tongues in the parameter plane of the map. Within a resonance tongue a periodic point can bifurcate through a period-doubling cascade. At the end of the cascade we detect Hénon-like attractors which are conjectured to be the closure of the unstable manifold of a saddle periodic point. These attractors have a folded structure which can be explained by means of the concept of critical lines. We also detect snap-back repellers which can either coexist with Hénon-like attractors or which can be formed when the saddle-point of a Hénon-like attractor becomes a source.

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