Ergodic Theory of One-Dimensional Model of Drilling
by
Georgi Chakvetadze
Moscow State Aviation Institute

G. Chakvetadze

ERGODIC THEORY OF ONE-DIMENSIONAL MODEL OF DRILLING

We consider a family of interval maps serving as a model for the process of rock destruction by drilling bit. These maps depend on two parameters - a scalar and a function - and are defined as follows. Let p:R --> R be a 1-periodic function, twice continuously differentiable on [0, 1], p''(r) < 0, p(1-r)=p(r), r in R. We will refer to the graph of p as basic profile. The basic profile is determined by geometry of the drilling bit. Set m=inf_{r in [0, 1]}|p''(r)|, M=sup_{r in [0, 1]}|p''(r)|, a=[ M/m] and assume that 0 < b < 1. Given s in R the graph of the function

hs(r)=p(s)+p'(s+0)(r-s)-  bm 2 (r-s)2,     r >= s,

(1)

intersects the basic profile in a finite number of points with abscissas s < r₁(s) < r₂(s) < ... < r_q(s)(s). Set F(s)=F_{b, p}(s)=r₁(s) mod 1, s in [0, 1).

Lemma 1 There are points s₁, ..., s_n, s_n+1, 0=s_n+1 < s_n < ... < s₁ < [ 1/2] < s₀=1, such that the transformation F a continuously differentiable and strictly decreasing on the set I₁=(s₁, s₀) from 1-s₁ till 0, on the sets I_k=(s_k, s_k-1), k=2, ..., n, from 1-s_k till F(s_k-1-0)=F(1-s_k-1) and on the set I_n+1=(s_n+1, s_n) from F(0+0) till F(s_n-0)=F(1-s_n).

We study the dynamical properties of the system with respect to Lebesgue measure on the unit interval. The next lemma describes the occurence of trivial dynamics in the system.

Lemma 2 Assume that b > [ a/2]. Then all the phase space is attracted to the unique fixed point of the transformation F.

The next theorem provides the sufficient condition for the system to behave chaotically on some large set on the unit interval.

Theorem 1 Assume that p'' has a bounded variation on [0, 1] and the values of the parameters a and b satisfy the inequality (1-b)³/b > (a-b)²+[ 1/8](a-1)(1-b). Then the transformation F has an absolutely continuous invariant probability (acip).

Note, that the measure v=v_{b, p} constructed in the theorem 1 has a property that any acip for F is absolutely continuous with respect to v. In the proof of theorem 1 we derive the expansion property for the map T, induced by F on some set of positive Lebesgue measure. Then we use the well-known results on the existence of acip for piecewise monotonic expanding transformations.

The next theorem describes the set of absolutely continuous invariant probabilities of F.

Theorem 2 Assume that p'' has a bounded variation on [0, 1]. Then the measure v is ergodic if the inequality (1-b)³/b > 2(a-b)² holds.

In the proof of theorem 2 we show that T is ergodic. In fact, all the powers of F are ergodic. This is used in the proof of the following result.

Theorem 3 Under the assumptions of the theorem 2 the measure v is weakly mixing.

One can derive from the theorem 3 that the system < F, v > is exact, isomorphic to some Bernoulli shift and the central limit theorem holds for the functions of bounded p-variation (p >= 1).

Finally, we discuss the stability property of the measure v with respect to stochastic perturbations Y_c, c > 0, introduced as follows. Given c > 0 and s in [0, 1) we replace p(s) (p'(s+0)) in the equality (1) by the stochastic variable distributed with some density on the interval [p(s)-c, p(s)+c] ([p'(s+0)-c, p'(s+0)+c]). Then F(s) is replaced by the stochastic variable distributed on some set including F(s) with the density u_c(s, ·). The Markov chain Y_c has u_c(s, ·) as the densities of transition probabilities. In the proof of the next theorem additionally some technical conditions are used which we omit.

Theorem 4 Assume that p in C³([0, 1]) and the transformation F has no periodic turning points. Then the densities of stationary distributions of the chains Y_c tend to the density of v in L₁-metric as c --> 0.

The dynamical model of drilling was suggested by Lasota and Rusek. The problem of existence of acip for one dimensional maps was stated by Ulam. The stochastic perturbations of dynamical systems was studied by Blank, Keller, Kifer, Sinai etc.

References

Date received: August 6, 1999