Several prototype models are introduced here which are designed to elucidate the interaction between heteroclinic low-dimensional chaos in the projected nonlinear dynamics and intrinsic stochasticity induced by energy exchange with a bath of fast variables. These models are built by coupling a four-dimensional ODE with known analytical properties including heteroclinic cycles with a suitable de...

We study functions from reals to reals which are uniformly degree-invariant from Turing-equivalence to many-one equivalence, and compare them “on a cone.” We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin’s conjecture for uniformly invariant functions from Turingto Turing-equivalence. Our proof works in the gene...

For reducibilities r and r′ such that r is weaker than r′, we say that the r-degree of A, i.e., the class of sets which are r-equivalent to A, collapses to the r′-degree of A if both degrees coincide. We investigate for the polynomial-time bounded many-one, bounded truth-table, truth-table, and Turing reducibilities whether and under which conditions such collapses can occur. While we show that...

The Turing degrees 3d were introduced by Kleene and Post ([9], 1954) to isolate and study those properties of the subsets of the natural numbers N which are expressed purely in terms of relative computability. Intuitively, we form 3d by identifying any pair of subsets of N which are mutually computable and ordering the resulting equivalence classes by relative computability. The natural hierarc...

Spector [l3] has proven that the hyperarithmetic sets H Ax) and H Ax) have the same Turing degree iff \a\ = |è|. Y. Moschovakis has bat the sets /_a proven that H {x) under many-one reducibility for \a\ = y and a e Q have nontrivial reducibility properties if y is not of the form a + 1 or a + cu for any ordinal a. In particular, he proves that there are chains of order type o and incomparable m...

As a parametric polynomial curve family, Bézier curves are widely used in safe and smooth motion design of intelligent robotic systems from flying drones to autonomous vehicles manipulators. In such planning settings, the critical features high-order as length, distance-to-collision, maximum curvature/velocity/acceleration either numerically computed at high computational cost or inexactly appr...