Bifurcations of Resonant Solutions and Chaos in Physical Pendulum Equation with Suspension Axis Vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.

High-order rational solutions and resonance solutions for a (3+1)-dimensional Kudryashov–Sinelshchikov equation

We mainly investigate the rational solutions and N-wave resonance solutions for the(3+1)-dimensional Kudryashov–Sinelshchikov equation, which could be used to describe the liquid containing gas bubbles. With appropriate transformations, two kinds of bilinear forms are derived. Employing the two bilinear equations, dynamical behaviors of nine district solutions for this equation are discussed in detail, including bright rogue wave-type solution, dark rogue wave-type solution, bright W-shaped solution, dark W-shaped rational solution, generalized rational solution and bright-fusion, darkfusion, bright-fission, and dark-fission resonance solutions. In addition, the generalized rational solutions, which depending on two arbitrary parameters, have an interesting structure: splitting from two peaks into three peaks.

MULTIPLE SOLUTIONS FOR HAMILTONIAN SYSTEMS WITH PERIODIC NONLINEARITY AND STRONG RESONANCE

I. Introduction In this paper we are looking for solutions of the following Hamiltonian system of second order: where x= （x1, x2） and V satisfies （V. 1） V: R×R2→R is a C1-function, 1-periodic In t, （V.2） V is periodic in x1 with the period T>0, （V. 3） V→O, Vx→O as |x2|→∞, uniformly in （t, x1）.

Soil Phosphorus Composition and Phosphatase Activities along Altitudes of Alpine Tundra in Changbai Mountains,China

Alpine tundra ecosystems have specific vegetation and environmental conditions that may affect soil phosphorus （P） composition and phosphatase activities. However, these effects are poody understood. This study used NaOH-EDTA extraction and solution ^31P nuclear magnetic resonance （NMR） spectroscopy to determine soil P composition and phosphatase activities, including acid phosphomonoesterase （AcP）, phosphodiesterase （PD） and inorganic pyrophosphatase （IPP）, in the alpine tundra of the Changbai Mountains at seven different altitudinal gradients （i.e., 2000 m, 2100 m, 2200 m, 2300 m, 2400 m, 2500 m, and 2600 m）. The results show that total P （TP）, organic P （OP）, OP/TP, NaOH-EDTA extracted P and AcP, PD, and IPP activities over the altitude range of 2500-2600 m are significantly lower than those below 2400 m. The dominant extracted form of P is OP （73%0-83%） with a large proportion of monoesters （65%0-72%）, whereas inorganic P is present in lower proportions （17%-27%）. The activity of AcP is significantly positively correlated with the contents of soil OP, total carbon （TC）, total nitrogen （TN）, and TP （P 〈 0.05）, indicating that the AcP is a more sensitive index for responding P nutrient storage than PD and IPP. Soil properties, P composition, and phosphatase activities decrease with increased altitude and soil pH. Our results indicate that the distribution of soil P composition and phosphatase activities along altitude and AcP may play an important role in P hydrolysis as well as have the potential to be an indicator of soil quality.

Breather molecules and localized interaction solutions in the(2+1)-dimensional BLMP equation

The(2+1)-dimensional Boiti-Leon-Manna-Pempinelli(BLMP)equation is an important integrable model.In this paper,we obtain the breather molecule,the breather-soliton molecule and some localized interaction solutions to the BLMP equation.In particular,by employing a compound method consisting of the velocity resonance,partial module resonance and degeneration of the breather techniques,we derive some interesting hybrid solutions mixed by a breather-soliton molecule/breather molecule and a lump,as well as a bell-shaped soliton and lump.Due to the lack of the long wave limit,it is the first time using the compound degeneration method to construct the hybrid solutions involving a lump.The dynamical behaviors and mathematical features of the solutions are analyzed theoretically and graphically.The method introduced can be effectively used to study the wave solutions of other nonlinear partial differential equations.