Hypergraphic Functions and Bifurcations in Recurrence Relations

Allwright, D. J.

doi:10.1137/0134057

Cited by 32 publications

(26 citation statements)

References 6 publications

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“…Then the first hypothesis for V is satisfied. Now assume that ~ and ~' are the points of E v closest to c. Then f(4) = f(4') since ~ e E v and f(~)= f(4') imply f"(~')e I -U for all n > 0 and since U satisfies (1). Assume that there is no stable fixed point in U.…”

Section: Let F Ecg and Let U Be A Neighborhood Of C If Ev= { X L F "mentioning

confidence: 95%

“…Then 6 > 0 since it is the distance between two disjoint compact sets. Suppose that J C E v is a nontrivial interval which is a component of E v. We shall denote K , 3 J a maximal interval with the properties that (1) (tl. ) or f~(x)~aU for some j < i < n .…”

Section: Proposition 28 Suppose F ~ Cz(i) and That U C I Is A Finitmentioning

confidence: 99%

“…We shall work with the class of functions C defined by 1) f : I~I with f ( 0 ) = f ( 1 ) = 0 and feC3(I).…”

Section: Introductionmentioning

confidence: 99%

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Sensitive dependence to initial conditions for one dimensional maps

1979

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Abstract. This paper studies the iteration of maps of the interval which have negative Schwarzian derivative and one critical point. The maps in this class are classified up to topological equivalence. The equivalence classes of maps which display sensitivity to initial conditions for large sets of initial conditions are characterized.There has been recent interest in the relationship between the "chaotic" asymptotic behavior of complicated solutions to ordinary differential equations and physically unstable phenomena such as those encountered in fluid flow [21]. This has led to numerical studies of a variety of systems of differential and difference equations which appear to have large sets of initial conditions yielding complicated asymptotic behavior. The mathematical theory of Axiom A "strange attractors" provides a satisfying description of some systems which do have complicated asymptotic behavior, but there is little overlap between the numerical studies referred to above and the class of systems with Axiom A attractors. Perhaps the "Lorenz system" [16], is the only example of an explicit system of equations used in a physical problem for which there is a convincing argument that a large set of its solutions have complicated asymptotic behavior. Nonetheless, the numerical studies of such examples as the "Henon" map [9], the mechanical systems studied by Holmes and Moon [11], the "strange attractor" of Spiegel [26], and the density dependent population models of [8] all provide evidence for the prevalence of complicated solutions in systems near those with homoclinic tangencies. From a practical point of view the distinction between trajectories tending to complicated periodic orbits with very long periods and trajectories with aperiodic asymptotic behavior may be slight, but we would like to understand the extent to which numerical computations of strange attractors reflect the mathematical properties of the systems being studied. This paper aims to study some of these mathematical questions in the simplest situation in which they arise : the iteration of a single real valued function of"quadratic" type. In this context we give a reasonably complete topological picture of what is involved in chaotic behavior and sensitive dependence to initial conditions while failing to answer the outstanding question about their prevalence.

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Section: Let F Ecg and Let U Be A Neighborhood Of C If Ev= { X L F "mentioning

confidence: 95%

Section: Proposition 28 Suppose F ~ Cz(i) and That U C I Is A Finitmentioning

confidence: 99%

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Sensitive dependence to initial conditions for one dimensional maps

1979

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“…The importance of this notion for investigating one-dimensional maps has been early recognized by Guckenheimer [21], and Misiurewicz [22]. A related notion of the cross-ratio has been introduced by Allwright [23]. In Sec.…”

Section: Introductionmentioning

confidence: 99%

On the creation of Wada basins in interval maps through fixed point tangent bifurcation

Breban

Nusse

2005

Physica D: Nonlinear Phenomena

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Basin boundaries play an important role in the study of dynamics of nonlinear models in a variety of disciplines such as biology, chemistry, economics, engineering, and physics. One of the goals of nonlinear dynamics is to determine the global structure of the system such as boundaries of basins. A basin having the strange property that every point which is on the boundary of that basin is on the boundary of at least three different basins, is called a Wada basin, and its boundary is called a Wada basin boundary. Here we consider maps on the interval. We present a sufficient and necessary condition guaranteeing that three Wada basins are emerging from a tangent bifurcation for certain one dimensional maps having negative Schwarzian derivative, two fixed point attractors on one side of the tangent bifurcation, and three fixed point attractors on the other side of the tangent bifurcation. All the conditions involved are numerically verifiable.

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“…Consequently, for the well-studied one-parameter family of maps {fa) with fa defined by fa(x) = ax(l -x) on the unit interval and parameter a in the interval [1,4], the conditions (1) "/a has a (one-sided) asymptotically stable periodic orbit" and (2) "the orbit of the critical point of fa converges to a (one-sided) asymptotically stable periodic orbit of /a" are equivalent. Now we will state our second result.…”

Section: Introductionmentioning

confidence: 99%

Chaotic maps with rational zeta function

Nusse¹

1987

Trans. Amer. Math. Soc.

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ABSTRACT. Fix a nontrivial interval X C R and let / e C1(X,X) be a chaotic mapping. We denote by Aoo (/) the set of points whose orbits do not converge to a (one-sided) asymptotically stable periodic orbit of / or to a subset of the absorbing boundary of X for /.A. We assume that / satisfies the following conditions: (1) the set of asymptotically stable periodic points for / is compact (an empty set is allowed), and (2) AoAf) is compact, / is expanding on Aoc(f).Then we can associate a matrix Aj with entries either zero or one to the mapping / such that the number of periodic points for / with period n is equal to the trace of the matrix [Ay]"; furthermore the zeta function of / is rational having the eigenvalues of Af as poles.B. We assume that / 6 C3(X, X) such that: (1) the Schwarzian derivative of / is negative, and (2) the closure of Aoo(/) is compact and f'(x) ^ 0 for all x in the closure of Acx)(/). Then we obtain the same result as in A.

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Hypergraphic Functions and Bifurcations in Recurrence Relations

Cited by 32 publications

References 6 publications

Sensitive dependence to initial conditions for one dimensional maps

Sensitive dependence to initial conditions for one dimensional maps

On the creation of Wada basins in interval maps through fixed point tangent bifurcation

Chaotic maps with rational zeta function

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