Models with a time delay often occur, since there is a naturally occurring delay in the transmiss... more Models with a time delay often occur, since there is a naturally occurring delay in the transmission of information. A model with a delay can be noninvertible, which in turn leads to qualitative dierences between the dynamical properties of a delay equation and the familiar case of an ordinary dieren tial equation. We give specic conditions for the existence of
Journal of Fixed Point Theory and Applications, 2009
. The Hénon family has been shown to have period-doubling cascades. We show here that the same o... more . The Hénon family has been shown to have period-doubling cascades. We show here that the same occurs for a much larger class: Large perturbations do not destroy cascades. Furthermore, we can classify the period of a cascade in terms of the set of orbits it contains, and count the number of cascades of each period. This class of families extends
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013
A period-doubling cascade is often seen in numerical studies of those smooth (one-parameter famil... more A period-doubling cascade is often seen in numerical studies of those smooth (one-parameter families of) maps for which as the parameter is varied, the map transitions from one without chaos to one with chaos. Our emphasis in this paper is on establishing the existence of such a cascade for many maps with phase space dimension 2. We use continuation methods to show the following: under certain general assumptions, if at one parameter there are only finitely many periodic orbits, and at another parameter value there is chaos, then between those two parameter values there must be a cascade. We investigate only families that are generic in the sense that all periodic orbit bifurcations are generic. Our method of proof in showing there is one cascade is to show there must be infinitely many cascades. We discuss in detail two-dimensional families like those which arise as a time-2π maps for the Duffing equation and the forced damped pendulum equation.
Period-doubling cascades are among the most prominent features
of many smooth one-parameter famil... more Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, F : R×M → M, where M is a locally compact manifold without boundary, typically RN. In particular, we investigate F(μ, ·) for μ ∈ J = [μ1, μ2], when F(μ1, ·) has only finitely many periodic orbits while F(μ2, ·) has exponential growth of the number of periodic orbits as a function of the period. For generic F, under additional hypotheses, we use a fixed point index argument to show that there are infinitely many “regular” periodic orbits at μ2. Furthermore, all but finitely many of these regular orbits at μ2 are tethered to their own period-doubling cascade. Specifically, each orbit ρ at μ2 lies in a connected component C(ρ) of regular orbits in J ×M; different regular orbits typically are contained in different components, and each component contains a period-doubling cascade. These components are one-manifolds of orbits, meaning that we can reasonably say that an orbit ρ is “tethered” or “tied” to a unique cascade. When F(μ2) has horseshoe dynamics, we show how to count the number of regular orbits of each period, and hence the number of cascades in J ×M. As corollaries of our main results, we give several examples, we prove that the map in each example has infinitely many cascades, and we count the cascades.
Models with a time delay often occur, since there is a naturally occurring delay in the transmiss... more Models with a time delay often occur, since there is a naturally occurring delay in the transmission of information. A model with a delay can be noninvertible, which in turn leads to qualitative dierences between the dynamical properties of a delay equation and the familiar case of an ordinary dieren tial equation. We give specic conditions for the existence of
Journal of Fixed Point Theory and Applications, 2009
. The Hénon family has been shown to have period-doubling cascades. We show here that the same o... more . The Hénon family has been shown to have period-doubling cascades. We show here that the same occurs for a much larger class: Large perturbations do not destroy cascades. Furthermore, we can classify the period of a cascade in terms of the set of orbits it contains, and count the number of cascades of each period. This class of families extends
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013
A period-doubling cascade is often seen in numerical studies of those smooth (one-parameter famil... more A period-doubling cascade is often seen in numerical studies of those smooth (one-parameter families of) maps for which as the parameter is varied, the map transitions from one without chaos to one with chaos. Our emphasis in this paper is on establishing the existence of such a cascade for many maps with phase space dimension 2. We use continuation methods to show the following: under certain general assumptions, if at one parameter there are only finitely many periodic orbits, and at another parameter value there is chaos, then between those two parameter values there must be a cascade. We investigate only families that are generic in the sense that all periodic orbit bifurcations are generic. Our method of proof in showing there is one cascade is to show there must be infinitely many cascades. We discuss in detail two-dimensional families like those which arise as a time-2π maps for the Duffing equation and the forced damped pendulum equation.
Period-doubling cascades are among the most prominent features
of many smooth one-parameter famil... more Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, F : R×M → M, where M is a locally compact manifold without boundary, typically RN. In particular, we investigate F(μ, ·) for μ ∈ J = [μ1, μ2], when F(μ1, ·) has only finitely many periodic orbits while F(μ2, ·) has exponential growth of the number of periodic orbits as a function of the period. For generic F, under additional hypotheses, we use a fixed point index argument to show that there are infinitely many “regular” periodic orbits at μ2. Furthermore, all but finitely many of these regular orbits at μ2 are tethered to their own period-doubling cascade. Specifically, each orbit ρ at μ2 lies in a connected component C(ρ) of regular orbits in J ×M; different regular orbits typically are contained in different components, and each component contains a period-doubling cascade. These components are one-manifolds of orbits, meaning that we can reasonably say that an orbit ρ is “tethered” or “tied” to a unique cascade. When F(μ2) has horseshoe dynamics, we show how to count the number of regular orbits of each period, and hence the number of cascades in J ×M. As corollaries of our main results, we give several examples, we prove that the map in each example has infinitely many cascades, and we count the cascades.
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of many smooth one-parameter families of maps, F : R×M → M,
where M is a locally compact manifold without boundary, typically RN.
In particular, we investigate F(μ, ·) for μ ∈ J = [μ1, μ2], when F(μ1, ·)
has only finitely many periodic orbits while F(μ2, ·) has exponential
growth of the number of periodic orbits as a function of the period. For
generic F, under additional hypotheses, we use a fixed point index argument
to show that there are infinitely many “regular” periodic orbits at
μ2. Furthermore, all but finitely many of these regular orbits at μ2 are
tethered to their own period-doubling cascade. Specifically, each orbit
ρ at μ2 lies in a connected component C(ρ) of regular orbits in J ×M;
different regular orbits typically are contained in different components,
and each component contains a period-doubling cascade. These components
are one-manifolds of orbits, meaning that we can reasonably say
that an orbit ρ is “tethered” or “tied” to a unique cascade. When F(μ2)
has horseshoe dynamics, we show how to count the number of regular
orbits of each period, and hence the number of cascades in J ×M.
As corollaries of our main results, we give several examples, we
prove that the map in each example has infinitely many cascades, and
we count the cascades.
of many smooth one-parameter families of maps, F : R×M → M,
where M is a locally compact manifold without boundary, typically RN.
In particular, we investigate F(μ, ·) for μ ∈ J = [μ1, μ2], when F(μ1, ·)
has only finitely many periodic orbits while F(μ2, ·) has exponential
growth of the number of periodic orbits as a function of the period. For
generic F, under additional hypotheses, we use a fixed point index argument
to show that there are infinitely many “regular” periodic orbits at
μ2. Furthermore, all but finitely many of these regular orbits at μ2 are
tethered to their own period-doubling cascade. Specifically, each orbit
ρ at μ2 lies in a connected component C(ρ) of regular orbits in J ×M;
different regular orbits typically are contained in different components,
and each component contains a period-doubling cascade. These components
are one-manifolds of orbits, meaning that we can reasonably say
that an orbit ρ is “tethered” or “tied” to a unique cascade. When F(μ2)
has horseshoe dynamics, we show how to count the number of regular
orbits of each period, and hence the number of cascades in J ×M.
As corollaries of our main results, we give several examples, we
prove that the map in each example has infinitely many cascades, and
we count the cascades.