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The global dynamics of a cosmological model based on Hořava-Lifshitz gravity in the presence of curvature is described by using the qualitative theory of differential equations.

Using the qualitative theory, the global dynamics of the cosmological model based on Hořava-Lifshitz gravity is described.

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The nonlinear orbital dynamics of a class of the perturbed restricted three-body problem is studied. The two primaries considered here refer to the binary system HD 191408. The third particle moves under the gravity of the binary system, whose triaxial rate and radiation factor are also considered. Based on the dynamic governing equation of the third particle in the binary HD 191408 system, the motion state manifold is given. By plotting bifurcation diagrams of the system, the effects of various perturbation factors on the dynamic behavior of the third particle are discussed in detail. In addition, the relationship between the geometric configuration and the Jacobian constant is discussed by analyzing the zero-velocity surface and zero-velocity curve of the system. Then, using the Poincaré–Lindsted method and numerical simulation, the second- and third-order periodic orbits of the third particle around the collinear libration point in two- and three-dimensional spaces are analytically and numerically presented. This paper complements the results by Singh et al. [Singh et al., AMC, 2018]. It contains not only higher-order analytical periodic solutions in the vicinity of the collinear equilibrium points but also conducts extensive numerical research on the bifurcation of the binary system.

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In this paper, the large-amplitude oscillation of a triple-well non-natural system, covering both qualitative and quantitative analysis, is investigated. The nonlinear system is governed by a quadratic velocity term and an odd-parity restoring force having cubic and quintic nonlinearities. Many mathematical models in mechanical and structural engineering applications can give rise to this nonlinear problem. In terms of qualitative analysis, the equilibrium points and its trajectories due to the change of the governing parameters are studied. It is interesting that there exist heteroclinic and homoclinic orbits under different equilibrium states. By adjusting the parameter values, the dynamic behavior of this conservative system is shifted accordingly. As exact solutions for this problem expressed in terms of an integral form must be solved numerically, an analytical approximation method can be used to construct accurate solutions to the oscillation around the stable equilibrium points of this system. This method is based on the harmonic balance method incorporated with Newton's method, in which a series of linear algebraic equations can be derived to replace coupled and complicated nonlinear algebraic equations. According to this harmonic balance-based approach, only the use of Fourier series expansions of known functions is required. Accurate analytical approximate solutions can be derived using lower order harmonic balance procedures. The proposed analytical method can offer good agreement with the corresponding numerical results for the whole range of oscillation amplitudes.