In this work we investigate how the behavior of a continuous stirred tank reactor (CSTR) sustaining a zeroth-order reaction is modified when one of the parameters varies stochastically. We start with a brief discussion of the theoretical basis which helps us analyze the effect of stochastic variations in one parameter on the dynamic behavior of one-dimensional nonlinear systems. The terminal behavior of dissipative, 'open', deterministic one-dimensional systems is the time-invariant steady state. In the presence of nonlinearities, the system can possess multiple steady solutions. In the presence of noise the system state is now described probabilistically in terms of probability density functions. This describes the frequency with which the state variable attains values in an interval. The steady state of the deterministic system is now replaced by the stationary probability distribution (if it exits). The stationary probability distribution characterizing the system behavior is obtained when the noise is idealized as a white noise. The distributions predicted theoretically are verified using numerical simulations on a zeroth-order reaction occurring adiabatically in a CSTR. For this system it is found that noise has a deleterious effect because it results in an increase in the regions of instability. The stationary probability distribution function is either unimodal or bimodal depending on the choice of the parameters. This is consistent with the fact that the deterministic system has either one or two stable steady-state solutions. In particular, noise does not induce new instabilities which would be reflected as new peaks in the distribution function.
In this work we investigate how the behavior of a continuous stirred tank reactor (CSTR) sustaining a zeroth-order reaction is modified when one of the parameters varies stochastically. We start with a brief discussion of the theoretical basis which helps us analyze the effect of stochastic variations in one parameter on the dynamic behavior of one-dimensional nonlinear systems. The terminal behavior of dissipative, 'open', deterministic one-dimensional systems is the time-invariant steady state. In the presence of nonlinearities, the system can possess multiple steady solutions. In the presence of noise the system state is now described probabilistically in terms of probability density functions. This describes the frequency with which the state variable attains values in an interval. The steady state of the deterministic system is now replaced by the stationary probability distribution (if it exists). The stationary probability distribution characterizing the system behavior is obtained when the noise is idealized as a white noise. The distributions predicted theoretically are verified using numerical simulations on a zeroth-order reaction occuring adiabatically in a CSTR. For this system it is found that noise has a deleterious effect because it results in an increase in the regions of instability. The stationary probability distribution function is either unimodal or bimodal depending on the choice of the parameters. This is consistent with the fact that the deterministic system has either one or two stable steady-state solutions. In particular, noise does not induce new instabilites which would be reflected as new peaks in the distribution function.