A general approach for treating the spatially extended stochastic systems with the nonlinear damping and correlations between additive and multiplicative noises is developed. Within the modified cumulant expansion method, we derive an effective Fokker-Planck equation with stationary solutions that describe the character of the ordered state. We find that the fluctuation cross-correlations lead to a symmetry breaking of the distribution function even in the case of zero-dimensional system. In a general case, continuous, discontinuous and reentrant noise induced phase transitions take place. It appears that the cross-correlations play the role of bias field which can induce a chain of phase transitions of different nature. Within the mean field approach, we give an intuitive explanation of the system behavior by an effective potential of the thermodynamic type. This potential is written in the form of an expansion with coefficients defined by the temperature, intensity of spatial coupling, autocorrelation and cross-correlation times and intensities of both additive and multiplicative noises.

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