TY - JOUR
T1 - Distribution of return point memory states for systems with stochastic inputs
AU - Amann, A.
AU - Brokate, M.
AU - Rachinskii, D.
AU - Temnov, G.
PY - 2011
Y1 - 2011
N2 - We consider the long term effect of stochastic inputs on the state of an open loop system which exhibits the so-called return point memory. An example of such a system is the Preisach model; more generally, systems with the Preisach type input-state relationship, such as in spin-interaction models, are considered. We focus on the characterisation of the expected memory configuration after the system has been effected by the input for sufficiently long period of time. In the case where the input is given by a discrete time random walk process, or the Wiener process, simple closed form expressions for the probability density of the vector of the main input extrema recorded by the memory state, and scaling laws for the dimension of this vector, are derived. If the input is given by a general continuous Markov process, we show that the distribution of previous memory elements can be obtained from a Markov chain scheme which is derived from the solution of an associated one-dimensional escape type problem. Formulas for transition probabilities defining this Markov chain scheme are presented. Moreover, explicit formulas for the conditional probability densities of previous main extrema are obtained for the Ornstein-Uhlenbeck input process. The analytical results are confirmed by numerical experiments.
AB - We consider the long term effect of stochastic inputs on the state of an open loop system which exhibits the so-called return point memory. An example of such a system is the Preisach model; more generally, systems with the Preisach type input-state relationship, such as in spin-interaction models, are considered. We focus on the characterisation of the expected memory configuration after the system has been effected by the input for sufficiently long period of time. In the case where the input is given by a discrete time random walk process, or the Wiener process, simple closed form expressions for the probability density of the vector of the main input extrema recorded by the memory state, and scaling laws for the dimension of this vector, are derived. If the input is given by a general continuous Markov process, we show that the distribution of previous memory elements can be obtained from a Markov chain scheme which is derived from the solution of an associated one-dimensional escape type problem. Formulas for transition probabilities defining this Markov chain scheme are presented. Moreover, explicit formulas for the conditional probability densities of previous main extrema are obtained for the Ornstein-Uhlenbeck input process. The analytical results are confirmed by numerical experiments.
UR - http://www.scopus.com/inward/record.url?scp=79952676020&partnerID=8YFLogxK
U2 - 10.1088/1742-6596/268/1/012001
DO - 10.1088/1742-6596/268/1/012001
M3 - Article
AN - SCOPUS:79952676020
SN - 1742-6588
VL - 268
JO - Journal of Physics: Conference Series
JF - Journal of Physics: Conference Series
IS - 1
M1 - 012001
ER -