In this paper, fundamental relationships are established between convergence of solutions, stability of equilibria, and arc length of orbits. More specifically, it is shown that a system is convergent if all of its orbits have finite arc length, while an equilibrium is Lyapunov stable if the arc length (considered as a function of the initial condition) is continuous at the equilibrium, and sem...

by minimizing the total potential energy function and deploying the virtual work principle, a higher-order stiffness matrix is achieved. this new tangent stiffness matrix is used to solve the frame with geometric nonlinear behavior. since authors’ formulation takes into account the higher-order terms of the strain vector, the convergence speed of the solution process will increase. in fact, bot...

The forced Duffing oscillator is investigated by intentional time-delayed displacement feedback by fast Fourier transform based incremental harmonic balance method along with continuation technique (FFT-IHBC). FFT-IHBC can efficiently develop frequency response curves with all stable and unstable solutions and solution branches. Appreciable reduction in peak value of response and gradual reduct...

We develop a stable continuation method for the computation of positive bound states of time-independent, m-coupled nonlinear algebra equation (DNLS) which describe a m-coupled nonlinear Schrödinger equation. The solution curve with respect to some parameter of the DNLS is then followed by the proposed method. For a one-component DNLS, we develop a iterative method for computation of positive g...

introduce an apparatus that completely describes all the relevant properties of a curve. After that, we move on to surfaces and their properties. We shall study the first and second fundamental forms, geodesics, and curvature – an apparatus that completely describes a surface. We shall formulate Gauss’s Theorema Egregium (Remarkable Theorem) that allows the concept of curvature to be generalize...

In this paper we study k-noncrossing RNA structures with minimum arc-length 4 and at most k − 1 mutually crossing bonds. Let T [4] k (n) denote the number of k-noncrossing RNA structures with arc-length ≥ 4 over n vertices. We prove (a) a functional equation for the generating function P n≥0 T [4] k (n)z n and (b) derive for k ≤ 9 the asymptotic formula T [4] k (n) ∼ c k n −((k−1) 2 +(k−1)/2) γ...