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 电子与信息学报 , 1999, Abstract: The concern here is retrieval of multiple tone harmonics observed in complex-valued multiplicative noises with nonzero mean. Cyclic mean statistics have proved to be useful for harmonic retrieval in the presence of complex-valued multiplicative noises with nonzero mean of arbitrary colors and distributions. Performance analysis of cyclic estimators is carried through and large sample variance expressions of the cyclic estimators are derived. Simulations validate the large sample performance analysis.
 电子与信息学报 , 1999, Abstract: The concern here is retrieval of multiple tone harmonics observed in complex-valued multiplicative noises with zero mean. Cyclic statistics have proved to be useful for harmonic retrieval in the presence of complex-valued multiplicative noises with zero mean of arbitrary colors and distributions . Performance analysis of cyclic estimators is carried through and large sample variance expressions of the cyclic estimators are derived. Simulations validate the large sample performance analysis.
 Physics , 2013, DOI: 10.1088/1751-8113/47/1/012001 Abstract: The Langevin equation with multiplicative noise and state-dependent transport coefficient has to be always complemented with the proper interpretation rule of the noise, such as the Ito and Stratonovich conventions. Although the mathematical relationship between the different rules and how to translate from one rule to another are well-established, it still remains controversial what is a more {\it physically} natural rule. In this communication, we derive the overdamped Langevin equation with multiplicative noise for Brownian particles, by systematically eliminating the fast degrees of freedom of the underdamped Langevin equation. The Langevin equations obtained here vary depending on the choice of the noise conventions but they are different representations for an identical phenomenon. The results apply to multi-variable, nonequilibrium, non-stationary systems, and other general settings.
 Discrete Dynamics in Nature and Society , 2009, DOI: 10.1155/2009/781976 Abstract: This paper contrastively researches the structural characteristic and the fission-evolution law of four different kinds of generalized Julia set (generalized J set in short) with different parameter , which includes the generalized J set without any perturbation, the generalized J set perturbed by additive noises, the generalized J set perturbed by multiplicative noise, and the generalized J set perturbed by composing additive and multiplicative noises, analyzes the effect of random perturbation to the generalized J set, and illuminates the stability of the generalized J set.
 Hideo Hasegawa Physics , 2006, Abstract: We have studied the synchronization in finite $N$-unit FitzHugh-Nagumo neuron ensembles subjected to additive and multiplicative noises, by using the augmented moment method (AMM) which is reformulated with the use of the Fokker-Planck equation. It has been shown that for diffusive couplings, the synchronization may be enhanced by multiplicative noises while additive noises are detrimental to the synchronization. In contrast, for sigmoid coupling, both additive and multiplicative noises deteriorate the synchronization. The synchronization depends not only on the type of noises but also on the kind of couplings.
 Physics , 1999, DOI: 10.1142/S0217984998001414 Abstract: We analyze the effect of the simultaneous presence of correlated additive and multiplicative noises on the stochastic resonance response of a modulated bistable system. We find that when the correlation parameter is also modulated, the system's response, measured through the output signal-to-noise ratio, becomes largely independent of the additive noise intensity.
 Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/717504 Abstract: The paper deals with Kalman (or H2) smoothing problem for wireless sensor networks (WSNs) with multiplicative noises. Packet loss occurs in the observation equations, and multiplicative noises occur both in the system state equation and the observation equations. The Kalman smoothers which include Kalman fixed-interval smoother, Kalman fixedlag smoother, and Kalman fixed-point smoother are given by solving Riccati equations and Lyapunov equations based on the projection theorem and innovation analysis. An example is also presented to ensure the efficiency of the approach. Furthermore, the proposed three Kalman smoothers are compared.
 A. N. Vitrenko Physics , 2006, DOI: 10.1016/j.physa.2005.04.036 Abstract: An overdamped system with a linear restoring force and two multiplicative colored noises is considered. Noise amplitudes depend on the system state $x$ as $x$ and $|x|^{\alpha}$. An exactly soluble model of a system is constructed due to consideration of a specific relation between noises. Exact expressions for the time-dependent univariate probability distribution function and the fractional moments are derived. Their long-time asymptotic behavior is investigated analytically. It is shown that anomalous diffusion and stochastic localization of particles, not subjected to a restoring force, can occur.
 Mathematics , 2013, DOI: 10.1016/j.sysconle.2013.01.011 Abstract: In this paper, the consensus problems of the continuous-time integrator systems under noisy measurements are considered. The measurement noises, which appear when agents measure their neighbors' states, are modeled to be multiplicative. By multiplication of the noises, here, the noise intensities are proportional to the absolute value of the relative states of agent and its neighbor. By using known distributed protocols for integrator agent systems, the closed-loop {system is} described in the vector form by a singular stochastic differential equation. For the fixed and switching network topologies cases, constant consensus gains are properly selected, such that mean square consensus and strong consensus can be achieved. Especially, exponential mean square convergence of agents' states to the common value is derived for the fixed topology case. In addition, asymptotic unbiased mean square average consensus and asymptotic unbiased strong average consensus are also studied. Simulations shed light on the effectiveness of the proposed theoretical results.
 Hideo Hasegawa Physics , 2006, Abstract: We have studied the dynamical properties of finite $N$-unit FitzHugh-Nagumo (FN) ensembles subjected to additive and/or multiplicative noises, reformulating the augmented moment method (AMM) with the Fokker-Planck equation (FPE) method [H. Hasegawa, J. Phys. Soc. Jpn. {\bf 75}, 033001 (2006)]. In the AMM, original $2N$-dimensional stochastic equations are transformed to eight-dimensional deterministic ones, and the dynamics is described in terms of averages and fluctuations of local and global variables. The stochastic bifurcation is discussed by a linear stability analysis of the {\it deterministic} AMM equations. The bifurcation transition diagram of multiplicative noise is rather different from that of additive noise: the former has the wider oscillating region than the latter. The synchronization in globally coupled FN ensembles is also investigated. Results of the AMM are in good agreement with those of direct simulations (DSs).
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