We introduce a new version of Hill's problem that incorporates the effects of radiation of the primary and oblateness of the secondary and study the basic dynamical features of this new model-problem. This formulation is more appropriate for some astronomical applications as an approximation to the corresponding restricted three-body problem. We use iterative methods for deriving approximate expressions of the equilibrium point locations and study their stability properties by using a linear stability analysis. All equilibrium points are unstable. We also employ singular perturbations methods for obtaining approximate expressions of the Lyapunov families emanating from equilibrium points, in both coplanar and spatial case, and numerical techniques for their continuation.
A new SEIRS epidemic model with nonlinear incidence rate and nonpermanent immunity is presented in the present paper. The fact that the incidence rate per infective individual is given by a nonlinear function and product of rational powers of two state variables, as well as the introduction of an epidemic-induced death rate, leads to a more realistic modeling of the physical problem itself. A stability analysis is performed and the features of Hopf bifurcation are investigated. Both the corresponding critical regions in the parameter space and their stability characteristics are presented. Furthermore, by using algorithms based on a new symbolic form as regards the restriction of an -dimensional nonlinear parametric system to the center manifold and the normal forms of the corresponding Hopf bifurcation, as well, the associated bifurcation diagram is derived, and finally various emerging limit cycles are numerically obtained by appropriate implemented methods.
The restriction of ann-dimensional nonlinear parametric system on the center manifold is treated via a new proper symbolic form and analytical expressions of the involved quantities are obtained as functions of the parameters by lengthy algebraic manipulations combined with computer assisted calculations. Normal forms regarding degenerate Hopf bifurcations up to codimension 3, as well as the corresponding Lyapunov coefficients and bifurcation portraits, can be easily computed for any system under consideration.
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