The long-time behavior of the first moment for the output signal of a fractional oscillator with fluctuating frequency subjected to an external periodic force is considered. Colored fluctuations of the oscillator eigenfrequency are modeled as a dichotomous noise. The viscoelastic type friction kernel with memory is assumed as a power-law function of time. Using the Shapiro-Loginov formula, exact expressions for the response to an external periodic field and for the complex susceptibility are presented. On the basis of the exact formulas it is demonstrated that interplay of colored noise and memory can generate a variety of cooperation effects, such as multiresonances versus the driving frequency and the friction coefficient as well as stochastic resonance versus noise parameters. The necessary and sufficient conditions for the cooperation effects are also discussed. Particularly, two different critical memory exponents have been found, which mark dynamical transitions in the behavior of the system.

We introduce a general class of generating functionals for the calculation of quantum-mechanical expectation values of arbitrary functionals of fluctuating paths with fixed end points in configuration or momentum space. The generating functionals are calculated explicitly for harmonic oscillators with time-dependent frequency, and used to derive a smearing formulas for correlation functions of polynomial and nonpolynomials functions of time-dependent positions and momenta. These formulas summarize the effect of thermal and quantum fluctuations, and serve to derive generalized Wick rules and Feyn-man diagrams for perturbation expansions of nonpolynomial interactions.

A broad class of (N+1) -species ratio-dependent predator-prey stochastic models, which consist of one predator population and N prey populations, is considered. The effect of a fluctuating environment on the carrying capacities of prey populations is taken into account as colored noise. In the framework of the mean-field theory, approximate self-consistency equations for prey-populations mean density and for predator-population density are derived (to the first order in the noise variance). In some cases, the mean field exhibits Hopf bifurcations as a function of noise correlation time. The corresponding transitions are found to be reentrant, e.g., the periodic orbit appears above a critical value of the noise correlation time, but disappears again at a higher value of the noise correlation time. The nonmonotonous dependence of the critical control parameter on the noise correlation time is found, and the conditions for the occurrence of Hopf bifurcations are presented. Our results provide a possible scenario for environmental-fluctuations-induced transitions between the oscillatory regime and equilibrium state of population sizes observed in nature.

A symbiotic ecosystem is studied by means of the Lotka-Volterra stochastic model, using the generalized Verhulst self-regulation. The effect of fluctuating environment on the carrying capacity of a population is taken into account as dichotomous noise. The study is a follow-up of our investigation of symbiotic ecosystems subjected to three-level (trichotomous) noise [Phys. Rev. E 65, 051108 (2002)]]. Relying on the mean-field theory, an exact self-consistency equation for stationary states is derived. In some cases the mean field exhibits hysteresis as a function of noise parameters. It is established that random interactions with the environment can cause discontinuous transitions. The dependence of the critical coupling strengths on the noise parameters is found and illustrated by phase diagrams. Predictions from the mean-field theory are compared with the results of numerical simulations. Our results provide a possible scenario for catastrophic shifts of population sizes observed in nature.

An N-species Lotka-Volterra stochastic model of a symbiotic ecological system with the Verhulst self-regulation mechanism is considered. The effect of fluctuating environment on the carrying capacity of a population is modeled as the colored three-level Markovian (trichotomous) noise. In the framework of the mean-field theory an explicit self-consistency equation for stationary states is presented. Stability and instability conditions and colored-noise-induced discontinuous transitions (catastrophic shifts) in the model are investigated. In some cases the mean field exhibits hysteresis as a function of the noise parameters. It is shown that the occurrence of catastrophic shifts can be controlled by noise parameters, such as correlation time, amplitude, and flatness. The dependence of the critical coupling strengths on the noise parameters is found and illustrated by phase diagrams. Implications of the results on some modifications of the model are discussed.

The long-time limit behavior of the variance and the correlation function for the output signal of a fractional oscillator with fluctuating eigenfrequency subjected to a periodic force is considered. The influence of a fluctuating environment is modeled by a multiplicative white noise and by an additive noise with a zero mean. The viscoelastic-type friction kernel with memory is assumed as a power-law function of time. The exact expressions of stochastic resonance (SR) characteristics such as variance and signal-to-noise ratio (SNR) have been calculated. It is shown that at intermediate values of the memory exponent the energetic stability of the oscillator is significantly enhanced in comparison with the cases of strong and low memory. A multiresonancelike behavior of the variance and SNR as functions of the memory exponent is observed and a connection between this effect and the memory-induced enhancement of energetic stability is established. The effect of memory-induced energetic stability encountered in case the harmonic potential is absent, is also discussed.

The colored three-level Markovian noise-driven nonequilibrium dynamics of overdamped Brownian particles in a spatially periodic asymmetric potential (ratchet) is investigated. An explicit second-order linear ordinary differential equation for the stationary probability density distribution is obtained for the process. In the case of a piecewise linear potential with an additive three-level (trichotomous) noise the exact formula for the stationary current is presented. The dependence of the current reversals on the noise parameters is investigated in detail and illustrated by a phase diagram. Asymptotic formulas for the current for various limits of the noise parameters are found and compared with the results of other authors. Applications to the fluctuation-induced separation of particles are also discussed.

Overdamped motion of Brownian particles in an asymmetric double-well potential driven by an additive nonequilibrium three-level noise and a thermal noise is considered. In the stationary regime, an exact formula for the mean occupancy of the metastable state is derived, and the phenomenon of enhancement of stability versus temperature is investigated. It is established that in a certain region of the system parameters the mean occupancy can be either multiply enhanced or suppressed by variations of temperature. We show that this effect is due to the involvement of different time scales in the problem. The necessary conditions for several different behaviors of the mean occupancy as a function of temperature are also discussed. The effect is more pronounced when the kurtosis of the three-level noise tends to -2 , i.e., in the case of dichotomous noise.