Symmetries of Itô and Stratonovich Dynamical Systems and Their Conserved Quantities

A new definition for the symmetries of Itô and Stratonovich dynamicalsystem is given. Determining systems of symmetries for Itô andStratonovich systems have been obtained, and their relation has beendiscussed. It has been shown that some of the Lie point symmetries ofthe Fokker–Planck equation can be constructed using the symmetries ofItô dynamical systems. Conserved quantities can be found from thesymmetries of stochastic dynamical systems which do not arise from aHamiltonian. The results have been applied to an example.     

Approximate symmetries and conservation laws for Itô and Stratonovich dynamical systems

A new definition for the approximate symmetries of Itô dynamical system is given. Determining systems of approximate symmetries for Itô and Stratonovich dynamical systems have been obtained. It has been shown that approximate conservation laws can be found from the approximate symmetries of stochastic dynamical systems which do not arise from a Hamiltonian. The results have been applied to an example.     

Approximate First Integrals of Weakly Nonlinear,Damped-Driven Oscillators with One Degree of Freedom

First-order approximate symmetries of weaklynonlinear, damped-driven oscillators have been determined.First-order approximate first integrals have been obtained byemploying an approximate version of Noether's theorem for the conservativecase. Furthermore, approximate first integrals of the damped case have been obtained based on the first integrals of the conservative case.Approximate first integrals enabled us to identify three types of genericbifurcations. Analytical results have been verified by numerical experiments.     

Exact linearization of one-dimensional jump-diffusion stochastic differential equations

Necessary and sufficient conditions for the linearization of the one-dimensional Itô jump-diffusion stochastic differential equations (JDSDE) are given. Stochastic integrating factor has been introduced to solve the linear JDSDEs. Exact solutions to some linearizable JDSDEs have been provided.     

Exact linearization of one dimensional Itô equations driven by fBm: Analytical and numerical solutions

Necessary and sufficient conditions for the linearization of the one-dimensional Itô stochastic differential equations driven by fractional Brownian motion (fBm) are given. Stochastic integrating factor has been introduced. A modified Milstein method has been developed to obtain numerical solutions. Analytical solutions have been compared with the numerical solutions for linearizable equations.     

Comparison of mechanical properties of normal and malignant thyroid cells

Cancer is a disease of uncontrolled cell proliferation causing approximately 13% of deaths worldwide. Cancer cell mechanics is currently an important topic of investigation in cancer diagnostics as a possible tool to distinguish malignant cells from normal cells in addition to increasing our understanding of pathophysiology of the disease. Our study, based on Atomic Force Microscopy (AFM) measurements on cells, shows that malignant thyroid cells are 3- to 5-fold softer in comparison to primary normal thyroid cells depending on duration between cell seeding and AFM experiments. These results reveal cultivation period as an important factor that influences cell mechanics and which must be considered when comparing cells. Investigation of actin cytoskeleton by fluorescent labelling revealed differences in organization of actin between malignant and normal thyroid cells, which may be directly contributing to alteration of cell mechanics in cancer cells.     

Symmetries and Conserved Quantities of Stochastic Dynamical Control Systems

A new definition is given for both exact and quasi symmetries of Itô and Stratonovich dynamical control systems. Determining systems of symmetries for these systems have been obtained and their relation is discussed. It is shown that conserved quantities can be found from both exact and quasi symmetries of stochastic dynamical control systems, which includes Hamiltonian control systems as a special case. Systems which can be controlled via conserved quantities have been investigated. Results have been applied to the control of an N-species stochastic Lotka—Volterra system.