Stochastic maps, continuous approximation, and stable distribution

A continuous approximation framework for general nonlinear stochastic as well as deterministic discrete maps is developed. For the stochastic map with uncorelated Gaussian noise, by successively applying the Itô lemma, we obtain a Langevin type of equation. Specifically, we show how nonlinear maps give rise to a Langevin description that involves multiplicative noise. The multiplicative nature of the noise induces an additional effective force, not present in the absence of noise. We further exploit the continuum description and provide an explicit formula for the stable distribution of the stochastic map and conditions for its existence. Our results are in good agreement with numerical simulations of several maps.

Figure 1

Deterministic maps and their continuous approximations. In panel (a) the dynamics of $X_t$ under the linear map $X_{t+1}=X_t-0.5X_t$ ($X_0=1$) is plotted (circles) and the appropriate first (dashed), second (dotted), and third (thick) approximations. Panel (b) describes the dynamics of the Pomeau-Manneville map Eq. (10) (circles) and appropriate first (dashed) and second (thick) order approximations. The parameters of the Pomeau-Manneville map are $a_0=-0.5$, $\alpha =1.5$, and $X_0=0.05$.

Figure 2

Stable distributions of linear stochastic maps. In panel (a) the linear map $X_{t+1}=X_t-\alpha X_t+{\eta }_t$ is plotted (circles), the noise term is zero mean, a Gaussian and $\langle {\eta }_t^2\rangle =1$, $\alpha =0.75$. The appropriate first-order (dashed line) and second-order (thick line) approximations are plotted. Panel (b) is similar to panel (a), except $\alpha =1.5$, $\langle {\eta }_t\rangle =1$, and $\langle {\eta }^2\rangle =2$.

Figure 3

Stable distributions of piecewise linear stochastic maps. In panel (a) the mapping given by Eq. (31) is plotted (circles), the noise term is zero mean, Gaussian and $\langle {\eta }_t^2\rangle =0.2^2$, $p_{-}=1$, and $p_{+}=0.1$. The appropriate first-order (dashed line) and second-order (thick line) approximations are plotted. Panel (b) is similar to panel (a), except $\langle {\eta }_t^2\rangle =1$.

Figure 4

Stable distributions of stochastic maps with the $tanh$ mapping, Eq. (33). In panel (a) the stable distribution for the mapping is plotted (circles), the noise term is zero mean, Gaussian and $\langle {\eta }_t^2\rangle =0.5^2$, $z=0.5$. The first-order (dashed line) and second-order (thick line) approximations are plotted. Panel (b) is similar to panel (a) but plotted on a semilog scale. The decay of the distribution is exponential. Panel (c) is similar to panel (a) except $z=1.5$ and panel (d) is similar to panel (c) only plotted on a semilog scale.

Figure 5

Stable distributions of stochastic maps with oscillating mapping. Panel (a) presents the stable distribution for the map $a\left(X\right)=-0.25tanh\left(X\right)+0.5cos\left(X\right)$ (circles), the noise term is zero mean, Gaussian, and $\langle {\eta }_t^2\rangle =0.4^2$. The second-order (thick line) approximation is plotted. In panel (b) the map has higher local derivatives, i.e., $a\left(X\right)=-0.25tanh\left(X\right)+0.5cos\left(2X\right)$ (circles), the noise term is zero mean, Gaussian, and $\langle {\eta }_t^2\rangle =0.4^2$. The second-order (thick line) approximation is plotted.

Figure 6

Stable distributions of stochastic maps with uniform noise, i.e., ${\eta }_t$ is a random number between $-1$ and 1 multiplied by $\beta$. The second-order approximation for the stable distribution is the thick line while circles present the simulation result. In panel (a) the map is $a\left(X\right)=-1.1X$ and $\beta =\sqrt{3}$. In panel (b) the map is $a\left(X\right)=-0.25X$ and $\beta =\sqrt{3}$. In panel (c) the map is $a\left(X\right)=-0.25tanh\left(X\right)$ and $\beta =\sqrt{3/10}$ and panel (d) is similar to (c) except $\beta =\sqrt{3}$.

Figure 7

Stable distributions of stochastic maps with noise following a zero mean Laplace distribution, i.e., the probability density of ${\eta }_t$ is $P\left({\eta }_t\right)=\frac{1}{2\zeta }exp(-|{\eta }_t|/\zeta )$ with second moment $\langle {\eta }^2\rangle =2{\zeta }^2$. The second-order approximation for the stable distribution is the thick line while circles present the simulation result. In panel (a) the map is $a\left(X\right)=-1.1X$ and $\langle {\eta }^2\rangle =1$. In panel (b) the map is $a\left(X\right)=-0.25X$ and $\langle {\eta }^2\rangle =1/10$. In panel (c) the map is $a\left(X\right)=-0.25tanh\left(X\right)$ and $\langle {\eta }^2\rangle =1$ and panel (d) is similar to (c) except $\langle {\eta }^2\rangle =1$.

Figure 8

Temporal traces of unstable maps with Gaussian noise. Panel (a) present a single trace for $a\left(X\right)=-{X\left|X\right|}^{1/2}$ and $\langle {\eta }^2\rangle =1$. Panel (b) present a single trace for $a\left(X\right)=-{X\left|X\right|}^{3/10}$ and $\langle {\eta }^2\rangle =3$. For both cases the trajectory escapes to infinity in an oscillating fashion.