Chaos and Wild Chaos in Lorenz-Type Systems

Abstract

This contribution provides a geometric perspective on the type of chaotic dynamics that one finds in the original Lorenz system and in a higher-dimensional Lorenz-type system. The latter provides an example of a system that features robustness of homoclinic tangencies; one also speaks of 'wild chaos' in contrast to the 'classical chaos' where homoclinic tangencies accumulate on one another, but do not occur robustly in open intervals in parameter space. Specifically, we discuss the manifestation of chaotic dynamics in the three-dimensional phase space of the Lorenz system, and illustrate the geometry behind the process that results in its description by a one-dimensional noninvertible map. For the higher-dimensional Lorenz-type system, the corresponding reduction process leads to a two-dimensional noninvertible map introduced in 2006 by Bamón, Kiwi, and Rivera-Letelier [arXiv 0508045] as a system displaying wild chaos. We present the geometric ingredientsin the form of different types of tangency bifurcations-that one encounters on the route to wild chaos.

Matching geometric and expansion characteristics of wild chaotic attractors

“…The numerical results in [17] give a geometric perspective and show that the first backward critical tangency bounds a region in the (a, λ)-plane that contains (1, 1) and has topologically the same backbone structure of the dynamical system. Hence, the authors conjectured that the region of wild chaos is bounded by the curve BWT of (first) backward critical tangency [17,29]. Our computations of the sum Λ 1 + Λ 2 of the two Lyapunov exponents corroborates this hypothesis; see Fig.…”

“…In [2], wild chaos was proven to exist for system (1) provided (a, λ) ≈ (1, 1). However, wild chaos is conjectured to exist for a much larger region in the (a, λ)-plane [17,29]. Note that f is not defined for z = 0.…”

Symbolic coding for noninvertible systems: uniform approximation and numerical computation

“…By the assumptions, the system (30) has exponential dichotomies on every subinterval (red or black in Figure 3) with constants (K,ᾱ s,u ). Our construction shows that every subinterval can be enlarged by its right neighbor, where the dichotomy constant grows at most by I + ∆ ± (N) ≤ 2 due to (28). Therefore, we have exponential dichotomies on the extended subintervals with projectors from (31) and constants (2K,ᾱ s,u ).…”

“…An example of this kind is discussed in Section 6.4. The extra condition in the snap-back case guarantees convergence of the dichotomy projectors at the left and right boundary, see (28). Figure 2 illustrates this choice of intervals in both cases.…”

“…The authors of [5] reduce the flow of a special vector field on the solid torus T n of dimension n ≥ 5 to the dynamics of this map. Detailed investigations of wild chaos for this map can be found in [15,28,16,17]. The authors use Cl MatContM to compute homoclinic orbits and their tangencies by solving suitable boundary value problems.…”