Narrow Results

We generalize the classical two-body problem from flat space to spherical space and realize much of the complexity of the classical three-body problem with only two bodies. We show analytically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe's radius. We show computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby geodesics to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos.

In this paper, we are interested in studying the initial value problem for parabolic problem associated with the Caputo–Fabrizio derivative. We deal the problem in two cases: linear inhomogeneous case and nonlinearity source term. For the linear case, we derive the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$ under some various assumptions on the initial datum. For the nonlinear problem, we show the existence and uniqueness of the mild solution using Banach fixed point theory. We also prove the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$.