Abstract
We consider a stochastic perturbation of the classical Lorenz system in the range of parameters for which the origin is the global attractor. We show that adding noise in the last component causes a transition from a unique to exactly two ergodic invariant measures. The bifurcation threshold depends on the strength of the noise: if the noise is weak, the only invariant measure is Gaussian, while strong enough noise causes the appearance of a second ergodic invariant measure.
Highlights
Which the system exhibits either a chaotic attractor or stable limit cycles, see for example [Tuc99]
For very large values of ( 313), the system admits a stable limit cycle which undergoes a cascade of period-doubling bifurcations as one decreases
The fact that for α > 0 (1.2) admits at most one ergodic invariant measure besides ν0 is the content of Theorem 3.1
Summary
It will be convenient to write (1.2) in such a way that all of the arbitrary constants appear in the equation for Z. This will be convenient since we will be mostly interested in the regime where x2 + y2 1, so that Z is close to a simple Ornstein–Uhlenbeck process. We write μα for the invariant measure on S1 × R for the diffusion with generator L0. Writing λ+(z) for the largest real part of the eigenvalues√of the system (x, y) given in (2.2) (with z frozen), one can verify that λ+(z) = − 1 + z − 11z>1 This suggests that α is the smallest value such that.
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