Results

The logistic map $f(x)=\lambda x(1-x)$ for various values of $\lambda$ exhibits most of the behaviors observed in one dimensional maps. For all $\lambda\geq 1$ any initial x outside the range [0,1] ends up at $-\infty$, while the behavior of points within this range depend of $\lambda$ as follows: For $0\leq\lambda\leq1$, the point x=0 is a stable fixed point, marginally so at $\lambda=1$, and then unstable for $\lambda>1$. For $1\leq\lambda\leq3$, the fixed point $x=1-1/\lambda$ is stable, and then bifurcates to a stable cycle of period 2. This cycle in turn becomes unstable, bifurcating to a 4-cycle, then an 8-cycle, and so on, to $\lambda\approx 3.57$ at which point a chaotic attractor forms. The period doubling cascade in the presence of weak noise may be described by the renormalization approach of Ref. [13]. At larger values of $\lambda$ more stable cycles are created, including a 3-cycle which is stable at $\lambda=3.84$, leading to a pattern of alternating stable ``windows'' surrounded by non-attracting unstable cycles and chaotic attractors containing many unstable cycles. At $\lambda=4$ the attractor fills the interval [0,1], and in this case, the Ulam map, the dynamics is exactly solvable. For $\lambda>4$ almost all initial conditions leave the interval, but infinitely many unstable cycles remain, forming a fractal repeller, with a well defined escape rate.

Imposing additive noise to the logistic map leads to escape for all $\lambda>0$, although this may be very unlikely if $\sigma$ is small. At $\lambda=2$, for example, every point (except the endpoints) is attracted to the stable fixed point at x=1/2, and the noise must move the trajectory out of the interval to escape. In cases like this, the stochastic behavior is analogous to quantum tunneling, and is exponentially suppressed for small $\sigma$. At bifurcation points, including $\lambda=1$, the stability of the relevant cycles is marginal, leading to intermittency. Marginal cycles are difficult to treat using cycle expansions, and it is one of the goals of this work to understand how this poor convergence is modified by the presence of noise.

The results of numerical evaluation of ${\rm tr}{\cal L}^n$ up to n=5 are shown in Tab. I. The spectral determinant is evaluated using (10), and C₅, the coefficient of z⁵ is noted. Since for the parameters shown the first zero of the determinant is close to 1, $-\log_{10}\vert C_5\vert$ gives roughly the number of significant digits of z, and hence the escape rate, evaluated to n=4. It also gives the approximate range of z over which the n=4 approximation is valid.

It is seen that, for the trivial case $\lambda=0$, corresponding to pure noise, and for strong noise $\sigma=1$, the calculation is limited by the double precision arithmetic: evaluation of the trace beyond n=4 is superfluous at this level of precision. Almost as precise is the case $\lambda=5$ which has a repeller with complete binary symbolic dynamics in the absence of noise, and hence is an ideal candidate for cycle expansion methods. Nine significant digits are obtained at n=4, corresponding to just 8 cycles. The presence of noise makes methods based on enumerating these cycles more difficult [10,11], but convergence is rapid at any noise level.

The other cases, where escape is induced by the presence of noise, do rather poorly for small noise. The significance of $\lambda=1,2,3,3.57,3.84,4$ are discussed above; the other values in Tab. I are $\lambda=3.5$ which contains a stable 4-cycle, and $\lambda=3.72$ which is not near any large stable window, and numerically exhibits a chaotic attractor, although mathematical proof is difficult. The nature of the underlying attractor seems to have little effect on the rate of convergence, except that the intermittent case ($\lambda=3$ and particularly $\lambda=1$) is divergent at $\sigma=0.01$ to this level of approximation; the escape rate probably converges at impossibly large n, either for the current numerical approach, or for standard cycle expansion techniques. In the other cases, particularly towards larger $\lambda$ the expansion appears to be converging, albeit slowly.

In conclusion: What are the optimal methods for determining the long time properties of stochastic systems? The strong noise case is best treated by numerical evaluation of the trace, described here, requiring little knowledge of the underlying dynamics. The elements of periodic orbit theory, traces and determinants indeed survive strong noise, and converge rapidly, without reference to periodic orbits. The weak noise case depends on this underlying dynamics: for the hyperbolic case ($\lambda>4$), the cycle perturbation theory of [10,11] or numerical evaluation; for noise induced escape from a strongly chaotic attractor ( $\lambda\approx4$), the analytic methods of [15]; and for tunneling from a stable fixed point, analytic approaches analogous to quantum mechanics. The intermittent case with weak noise remains an open problem; the results here show that weak noise does not substantially regularize cycle expansions of intermittent systems, at least with respect to the rate of convergence.

The author is grateful for helpful discussions with V. Baladi and P. Cvitanovic.

Links to references: [2] [11] [12]

 

Table: Convergence of the spectral determinant, as measured by $-\log_{10}\vert C_5\vert$, where C₅ is the coefficient of z⁵ in the cumulant expansion (8) for various types of dynamics of the logistic map (2). Larger numbers imply faster convergence, giving roughly the number of converged digits in the escape rate calculated to n=4.

				$\sigma$
$\lambda$	Type	0.01	0.03	0.1	0.3	1
0	Pure noise	12.7	12.7	12.4	12.7	12.6
1	Intermittent	-2.3	-0.8	1.2	3.8	8.5
2	Stable 1-cycle	2.5	2.2	2.1	5.9	11.8
3	Bifurcation	-0.3	0.7	2.8	7.4	13.2
3.5	Stable 4-cycle	0.3	1.4	3.4	7.8	13.2
3.57	$\infty$-cycle	0.4	1.1	3.6	7.8	13.3
3.72	Chaos	1.5	1.4	4.1	8.0	13.4
3.84	Stable 3-cycle	1.6	2.4	4.6	8.1	13.4
4	Ulam map	2.2	2.9	4.9	8.2	13.8
5	Repeller	9.2	9.1	8.4	9.1	13.3