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The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear…
We develop a novel method for finding bifurcations for nonlinear systems of equations based on directly finding bifurcations through saddle points of extended quotients. The method is applied to find the saddle-node bifurcation point for…
Following the previous work [1], we investigate the impact of damping on the oscillation of smooth solutions to some kind of quasilinear wave equations with Robin and Dirichlet boundary condition. By using generalized Riccati transformation…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An…
We use some properties of solutions of Riccati equation for establishing boundedness and stability criteria for solutions of second order linear ordinary differential equations. We show that the conditions on coefficients of the equations,…
The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and…
In this paper we have considered higher order two dimensional coupled system of non-linear ordinary differential equations. We have given necessary and sufficient conditions on the non-linear functions such that the solutions pair oscilla
We derive the extended fluctuation theorems in presence of multiple measurements and feedback, when the system is governed by Hamiltonian dynamics. We use only the forward phase space trajectories in the derivation. However, to obtain an…
The need to estimate a positive definite solution to an overdetermined linear system of equations with multiple right hand side vectors arises in several process control contexts. The coefficient and the right hand side matrices are…
Stochastic algebraic Riccati equations, also known as rational algebraic Riccati equations, arising in linear-quadratic optimal control for stochastic linear time-invariant systems, were considered to be not easy to solve. The-state-of-art…
The adiabatic approximation in open systems is formulated through the effective Hamiltonian approach. By introducing an ancilla, we embed the open system dynamics into a non-Hermitian quantum dynamics of a composite system, the adiabatic…
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…
Matrix Riccati differential equations arise in many different areas and are particular important within the field of control theory. In this paper we consider numerical integration for large-scale systems of stiff matrix Riccati…
In this paper, we study the determination of Hamiltonian from a given equations of motion. It can be cast into a problem of matrix factorization after reinterpretation of the system as first-order evolutionary equations in the phase space…
This paper studies diagonal implicit symplectic extended Runge--Kutta--Nystr\"{o}m (ERKN) methods for solving the oscillatory Hamiltonian system $H(q,p)=\dfrac{1}{2}p^{T}p+\dfrac{1}{2}q^{T}Mq+U(q)$. Based on symplectic conditions and order…
A novel integrability condition for the Riccati equation, the simplest form of nonlinear ordinary differential equations, is obtained by using elementary quadrature method. Under this condition, the analytic general solution is presented,…
This paper is devoted to discrete mechanical systems subject to external forces. We introduce a discrete version of systems with Rayleigh-type forces, obtain the equations of motion and characterize the equivalence for these systems.…
This paper considers the robustness of an uncertain nonlinear system along a finite-horizon trajectory. The uncertain system is modeled as a connection of a nonlinear system and a perturbation. The analysis relies on three ingredients.…
This paper proposes a novel lifting method which converts the standard discrete-time linear periodic system to an augmented linear time-invariant system. The linear quadratic optimal control is then based on the solution of the…