Sensitive dependence on parameters

Abstract

Abstract. For a family of dynamical systems we define sensitive dependence on parameters in a way resembling Guckenheimer's definition of sensitive dependence on initial conditions. While sensitive dependence on initial conditions tells us that if we know the initial condition only approximately then we cannot make deterministic predictions, sensitive dependence on parameters tells us that if we know the parameter value only approximately then we cannot make statistical predictions. We show that the family of logistic maps has sensitive dependence on parameters. Moreover, the long-time response of the system to the change of the parameter is practically independent of the immediate response.

Singular limits of absolutely continuous invariant measures for families of transitive maps

“…On the other hand, W a;1=2;1=4 ð½0; 1Þ ¼ ½0; a and on ½0; a this map is transitive. Therefore, it has an acim with the support ½0; a. Keller conjectured that for continuous maps of the interval, The acim-instability of a dynamical system is closely related to sensitive dependence on parameters defined by the second author in [6]. It is shown in [6] that the popular class of logistic maps has sensitive dependence on parameters which implies that they are not acim-stable.…”

“…Therefore, it has an acim with the support ½0; a. Keller conjectured that for continuous maps of the interval, The acim-instability of a dynamical system is closely related to sensitive dependence on parameters defined by the second author in [6]. It is shown in [6] that the popular class of logistic maps has sensitive dependence on parameters which implies that they are not acim-stable. However, there the acim-instability is based on the fact that for most of the maps there is no acim, and instead we consider Sinai -Ruelle -Bowen (or physical) measures, that are often concentrated on attracting periodic orbits.…”

Towards understanding the theoretical challenges of numerical modeling of dynamical systems

“…Thus, for almost all parameters λ, the maps of the family f λ possess a unique SRB measure µ λ supported on an attractor A λ whose structure is well understood, and which attracts typical (in every reasonable sense) orbits of f λ . One potential cause of concern is the instability of the statistical properties of orbits with respect to λ. M. Misiurewicz showed in [25], that there exists a test function φ : [0, 1] → R, a set S ⊂ [0, 1] of positive measure, and > 0, such that for any λ ∈ S and a neighborhood U ⊂ [0, 1] of λ, there is a set Z ⊂ U of positive measure with φdµ λ − φdµ > for all ∈ Z, the property which he called strong structural instability, and which is likely common in other natural families of examples.…”

“…While we have not proved that it is so, it is very likely to be the case. In fact, the proof of Theorem 5.1 has little to do with the parameter λ itself, but rather is based on a very precise description of the instability of the dependence λ → µ λ , similar in spirit to the strong structural instability of [25]. Instability does therefore become a theoretical barrier to computing the limiting distribution.…”