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  • 1
    Electronic Resource
    Electronic Resource
    On noninvertible mappings of the plane: Eruptions (1996)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 6 (1996), S. 108-120 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: In this paper we are concerned with the dynamics of noninvertible transformations of the plane. Three examples are explored and possibly a new bifurcation, or "eruption,'' is described. A fundamental role is played by the interactions of fixed points and singular curves. Other critical elements in the phase space include periodic points and an invariant line. The dynamics along the invariant line, in two of the examples, reduces to the one-dimensional Newton's method which is conjugate to a degree two rational map. We also determine, computationally, the characteristic exponents for all of the systems. An unexpected coincidence is that the parameter range where the invariant line becomes neutrally stable, as measured by a zero Lyapunov exponent, coincides with the merging of a periodic point with a point on a singular curve. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.166158
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  • 2
    Electronic Resource
    Electronic Resource
    On the iteration of a rational function: Computer experiments with Newton's method (1983)
    Springer
    Communications in mathematical physics 91 (1983), S. 267-277 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Using Newton's method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function. This article describes the behavior of Newton iteration for cubic polynomials. After a change of variables, these polynomials can be parametrized by a single complex parameter, and the Newton transformation has a single critical point other than its fixed points at the roots of the polynomial. We describe the behavior of the orbit of the free critical point as the parameter is varied. The Julia set, points where Newton's method fail to converge, is also pictured. These sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01211162
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  • 3
    Electronic Resource
    Electronic Resource
    A generalized Lorenz system (1978)
    Springer
    Communications in mathematical physics 60 (1978), S. 193-204 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A 14-dimensional generalized Lorenz system of ordinary differential equations is constructed and its bifurcation sequence is then studied numerically. Several fundamental differences are found which serve to distinguish this model from Lorenz's original one, the most unexpected of which is a family of invariant two-tori whose ultimate bifurcation leads to a strange attractor. The strange attractor seems to have many of the gross features observed in Lorenz's model and therefore is an excellent candidate for a higher dimensional analogue.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01612888
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  • 4
    Electronic Resource
    Electronic Resource
    On the Hénon transformation (1979)
    Springer
    Communications in mathematical physics 68 (1979), S. 129-140 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In [4] Hénon studied a transformation which maps the plane into itself and appears to have an attractor with locally the structure of a Cantor set cross an interval. By making use of the characteristic exponent, frequency spectrum, and a theorem of Smale, our numerical experiments provide evidence for the existence of two distinct strange attractors for some parameter values, an exponential rate of mixing for the parameter values studied by Hénon, and an argument that there is a Cantor set in the trapping region of Hénon.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418124
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  • 5
    Electronic Resource
    Electronic Resource
    On computing the entropy of the Henon attractor (1981)
    Springer
    Journal of statistical physics 26 (1981), S. 683-695 
    Details
    ISSN: 1572-9613
    Keywords: Characteristic exponent ; entropy ; partition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In a recent article D. Ruelle [inLecture Notes in Physics, No. 80 (Springer, Berlin, 1978)] has conjectured that for the Hénon attractor its measure theoretic entropy should be equal to its characteristic exponent. This result is known to be true for systems which satisfy Smale's Axiom A. In this article we report the results of our computations which suggest that Ruelle's conjecture may be true for the Hénon attractor. Further, in our study we are confronted with fundamental questions which suggest that certain existence theorems from ergodic theory are not sufficient from a computational point of view.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01010933
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