Manifolds on the verge of a hyperbolicity breakdown

We study numerically the disappearance of normally hyperbolic invariant tori in quasiperiodic systems and identify a scenario for their breakdown. In this scenario, the breakdown happens because two invariant directions of the transversal dynamics come close to each other, losing their regularity. On the other hand, the Lyapunov multipliers associated with the invariant directions remain more or less constant. We identify notable quantitative regularities in this scenario, namely that the minimum angle between the two invariant directions and the Lyapunov multipliers have power law dependence with the parameters. The exponents of the power laws seem to be universal.


Introduction
The long term behavior of dynamical systems is organized by the invariant objects. Hence, it is important to understand which invariant objects persist under modifications of the system. We study a mechanism of destruction of invariant manifolds. The mechanism consists in the fact that the attracting directions merge with repelling directions.
This geometric mechanism has spectral implications.
We show that this mechanism satisfies scaling properties.

Set up 2.1 Quasi-periodic maps and invariant tori
Quasi-periodic forced systems are non-autonomous systems in which the external forcing is quasi-periodic.
A quasi-periodic map with frequency vector ω ∈ R d is a skew product in An invariant torus whose dynamics is the rotation ω is given as a solution K : T d → R n of the invariance equation

Cocycles and transfer operators
The cocycle is the linearization around the torus, that is where M (θ) = DF (K(θ), θ).
The transfer operator M acting on bounded vector fields v : T d → R n is given by

Mechanism
For the sake of simplicity: n = 2, d = 1.
We consider a system (1) depending on a parameter ε: F = F ε .
We assume that: • The system posses a smooth invariant torus K ε for 0 ≤ ε < ε c ; • In these ranges of ε, the torus has an invariant splitting as in (5); • As ε approaches ε c , the distance ∆ ε between the invariant bundles goes to zero, but the Lyapunov multipliers Λ ± ε remain different from 1 and from each other.
The splitting becomes zero in a complicated collision set, and the collapse of the spectral subbundles implies the sudden growth of the spectrum.

SNAs in the projective linearization
For a fixed θ, the 1-D space E ± θ is described by an angle α ± θ in [0, π]. The invariant bundles E ± θ are represented by curves in the projective bundle.
The mechanism described above corresponds to the collision of those curves forming a non-smooth object. These are commonly called SNAs.

Two bundle-merging scenarios
a) The splitting corresponds to two stable bundles (slow and fast) (the Lyapunov multipliers do not straddle 1) ⇓ the attracting-node torus is not destroyed and it can be continued beyond ε c .
b) The splitting corresponds to one unstable and one stable bundle (the Lyapunov multipliers straddle 1).
⇓ the saddle torus presumably breaks down at ε = ε c .

Consequences
1) For ε = ε c , the collision set is dense, and has zero measure.
The invariant bundles are discontinuous in ε = ε c , but they are defined is set of full measure and are measurable.
[Oseledec 68] 2) The spectrum of the transfer operator M is a set of annuli centered in the origin of the complex plane, and each spectral annulus has associated an invariant subbundle characterized by growth rates.
[Mather 68][Haro,de la Llave] If the invariant bundles are 1D, the spectrum is a union of circles.
In the present case (n = 2): For ε < ε c the spectrum is just two circles of radii Λ ± ε , but for ε = ε c , the spectrum has to be the full annulus enclosed by these circles.
• a = 0.68, b = 0.1 are the parameters of the Hénon map.
• ε is the forcing parameter; • For ε = 0.530, the torus is attracting and the cocycle is reducible to a constant diagonal matrix diag(0.694546750046480781363, −0.143978789035162504966).
The Lyapunov multipliers are different during the continuation ⇒ The cocycle can not be reducible during the whole continuation!
The bundles have a collapse, and the spectrum grows suddenly.
Remark. The justification can be made purely topological.
The transition e In summary, the formation of an strange non chaotic attractor for the linearized dynamics of an attracting torus produces a sudden growth of the spectrum and it is the prelude of the destruction of the torus and the formation of an strange chaotic attractor for the non linear dynamics.

Bundle merging causing breakdown
Model (conservative): rotating standard map • κ = 0.2 is the parameter of the standard map; • ε is the quasi-periodic parameter; •  In summary, the formation of an strange non chaotic attractor for the linearized dynamics of a saddle type torus produces the sudden growth of the spectrum and the destruction of the torus.