We present a stochastic numerical model for what can be considered, at the present state-of-the-art, a prototype chemical oscillator, i.e. the Belousov- Zhabotinsky reaction. We mainly focus our attention on properly modifying an usual model based on systems of deterministic differential equations, commonly known as Oregonator. We aim to provide a mathematical description of the Belousov-Zhabotinsky reaction by means of systems of stochastic differential equations of Ito type, together with the corresponding numerical discretization. Together with standard numerical schemes that approximate the whole stochas- tic system, such as the Euler-Maruyama method and its improvements, we also aim to separately treat the deterministic term and the stochastic one, by cou- pling the so-called exponential fitting technique [2] and Montecarlo simulations. This coupling looks particularly suitable to provide numerical approximations of oscillatory problems, since classical methods could require a very small step- size to accurately reproduce oscillatory behaviours. We rather propose a method that is constructed in order to be exact on functions other than polynomials for the approximation of the deterministic part. The coecients of the resulting adapted method are no longer constant as in the classical case, but rely on a parameter linked to oscillatory character, whose value is clearly unknown. The proposed estimation strategy is based on exploiting the informations belonging to known time series of experimental data which are available in the literature [1,3]. Numerical experiments will be provided to show the effectiveness of the presented approach. References 1. D’Ambrosio, R., Moccaldi, M., Paternoster, B., Rossi, F.: On the employ of time series in the numerical treatment of di↵erential equations modelling oscillatory phe- nomena. In: Advances in Artificial Life, Evolutionary Computation, and Systems Chemistry - 11th Workshop, WIVACE 2016, Fisciano, Italy, ed. by F. Rossi, S. Piotto, S. Concilio, Comm. Comput. Inf. Sci., Springer (2017). 2. Ixaru, L.Gr., Vanden Berghe, G.: Exponential Fitting. Kluwer. Boston-Dordrecht- London (2004) 3. Rossi,F., Budroni,M.A., Marchettini,N., CutiettaL., Rustici,M. and Turco Liveri, M. L.: Chaotic dynamics in an unstirred ferroin catalyzed Belousov-Zhabotinsky reaction. Chem. Phys. Lett. 480, 322–326 (2009)
Stochastic numerical modeling of selected oscillatory phenomena
MOCCALDI, MARTINA;PATERNOSTER, Beatrice;ROSSI, FEDERICO
2017-01-01
Abstract
We present a stochastic numerical model for what can be considered, at the present state-of-the-art, a prototype chemical oscillator, i.e. the Belousov- Zhabotinsky reaction. We mainly focus our attention on properly modifying an usual model based on systems of deterministic differential equations, commonly known as Oregonator. We aim to provide a mathematical description of the Belousov-Zhabotinsky reaction by means of systems of stochastic differential equations of Ito type, together with the corresponding numerical discretization. Together with standard numerical schemes that approximate the whole stochas- tic system, such as the Euler-Maruyama method and its improvements, we also aim to separately treat the deterministic term and the stochastic one, by cou- pling the so-called exponential fitting technique [2] and Montecarlo simulations. This coupling looks particularly suitable to provide numerical approximations of oscillatory problems, since classical methods could require a very small step- size to accurately reproduce oscillatory behaviours. We rather propose a method that is constructed in order to be exact on functions other than polynomials for the approximation of the deterministic part. The coecients of the resulting adapted method are no longer constant as in the classical case, but rely on a parameter linked to oscillatory character, whose value is clearly unknown. The proposed estimation strategy is based on exploiting the informations belonging to known time series of experimental data which are available in the literature [1,3]. Numerical experiments will be provided to show the effectiveness of the presented approach. References 1. D’Ambrosio, R., Moccaldi, M., Paternoster, B., Rossi, F.: On the employ of time series in the numerical treatment of di↵erential equations modelling oscillatory phe- nomena. In: Advances in Artificial Life, Evolutionary Computation, and Systems Chemistry - 11th Workshop, WIVACE 2016, Fisciano, Italy, ed. by F. Rossi, S. Piotto, S. Concilio, Comm. Comput. Inf. Sci., Springer (2017). 2. Ixaru, L.Gr., Vanden Berghe, G.: Exponential Fitting. Kluwer. Boston-Dordrecht- London (2004) 3. Rossi,F., Budroni,M.A., Marchettini,N., CutiettaL., Rustici,M. and Turco Liveri, M. L.: Chaotic dynamics in an unstirred ferroin catalyzed Belousov-Zhabotinsky reaction. Chem. Phys. Lett. 480, 322–326 (2009)I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
