Noise-Induced Hopf-Bifurcation-Type Sequence and Transition to Chaos in the Lorenz Equations

We study the effects of noise on the Lorenz equations in the parameter regime admitting two stable fixed point solutions and a strange attractor. We show that noise annihilates the two stable fixed point attractors and evicts a Hopf-bifurcation-like sequence and transition to chaos. The noise-induced oscillatory motions have very well defined period and amplitude, and this phenomenon is similar to stochastic resonance, but without a weak periodic forcing. When the noise level exceeds certain threshold value but is not too strong, the noise-induced signals enable an objective computation of the largest positive Lyapunov exponent, which characterize the signals to be truly chaotic.

Figure 1

Time series for the Lorenz equations with $r=24.10$, $D=0.1$ (a) and $D=0.2$ (b). The sampling time is $\delta t=0.06$.

Figure 2

(a)–(c) Time series for the Lorenz equations with $r=24.72$, $D=0.001$ (a), $D=0.01$ (b), and $D=0.02$ (c). The sampling time for (a)–(c) is $\delta t=0.06$. In (d), we have plotted, in log-log scale, the variation of the amplitude of the oscillation with the noise strength ($D^2$) for $D$ not too large. The slope of the linear line is 0.51.

Figure 3

The $\Lambda (k)$ curves for the time series of (a) Fig. 2c and (b) Fig. 1b. The numbers 1 to 4 correspond to shells defined by $[2^{-(i+1)/2},2^{-i/2}]$ with $i=9$ to 12. The embedding parameters are $m=4$ and $L=3$.

Figure 4

The power spectral density (PSD) for the time series of (a) Fig. 1, (b) Fig. 1b, (c) Fig. 2a, and (d) Fig. 5. It is in logarithmic scale and normalized by the variance of the noise, to ensure that the value of the PSD corresponding to the sharpest peak in the spectrum gives the signal-to-noise ratio.

Figure 5

The time series for the Hopf bifurcation. The sampling time is $1/30$.