Synchronization Sets in Drive-Response Systems

Invariant Manifolds


A recent article in Chaos [Barreto et al., Chaos, 13(1):151-164, 2003] describes a drive-response system coupling a parametrized family of noninvertible modified baker maps with a nonlinear response. The article describes the effect of noninvertibility on the synchronization set (i.e. the attractor). Namely, at the bifurcation parameter, the synchronization set has a cusp. For every parameter after the bifurcation, there is a multivalued synchronization set with self-intersections, or loops. See the first animation. This example is a special case in which the noninvertibility occurs at a critical point of the synchronization set. In contrast, the synchronization set generically continues to be a smooth nonintersecting curve even after the onset of noninvertibility. See Animations 2 and 3. This is described more fully in an upcoming article by Josic and Sander.


\(u\) versus \(y\) in the exceptional case of Barreto et al. As the parameter varies, a smooth manifold forms a cusp, which turns into a loop.


\(u\) versus \(y\) in the typical case; the noninvertibility does not occur at a critical point. Therefore the synchronization set becomes multivalued as a graph but retains its smooth manifold structure.


A closeup view of the main loop in the previous animation.

Author Institutional AffiliationGeorge Mason University, University of Houston
Author Email
Author Postal MailDepartment of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, Fairfax, VA 22030
Keywordsdrive-response system, synchronization

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