Splitting Frequencies for Resonant Solutions in Central Fields

The behavior of a mass point moving in a plane under the effect of a central field and an ex-ternal periodic excitation in resonance with the natural frequency is studied. The asymptotic perturbation method is used in order to determine the nonlinear modulation equations for the amplitude and the phase of the oscillation. Firstly, we calculate the second order approximate solution of the unforced system. It is well known that generally the solution is two period quasiperiodic, but we find some new cases of periodic solutions. If appropriate Diophantine equations are satisfied, the motion is periodic with a frequency depending on the nonlinear terms. Subsequently, the forced system is considered and external force-response curves are shown and moreover jump phenomena are also observed. In certain cases we observe a frequency splitting and a third frequency appears in addition to the forcing frequency Stable three period quasi-periodic motions are present with amplitudes depending on the initial conditions

Isochronous Behavior and Diophantine Relations for Complex-Valued Nonlinear Systems

Isochronous systems are not rare in dynamical systems. Three complex-valued nonlinear systems (quadratic and cubic nonlinearity, van der Pol, gyroscopic oscillator) are investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. Approximate solutions are obtained and their stability is discussed. We find that in the first two cases the motion is periodic, while in the third case the motion is periodic only if appropriate Diophantine relations are satisfied. Analytic approximate solutions are checked by numerical integration

Jerk Dynamics and Vibration Control for the parametrically excited van der Pol system

In parametrically excited van der Pol system, dangerous vibrations can be controlled and governed by Jerk dynamics. We choose a non-local force for the vibration control and a third or-der nonlinear differential equation (jerk dynamics) is necessary for the control method implementation. Two slow flow equations on the amplitude and phase of the response describe the oscillator motion and we are able to check the control strategy performance. The stability and response of the system is connected to the feedback gains. The dangerous excitations amplitude peak can be reduced adequately picking feedback gains. The new method is successfully checked by numerical simulatio

Vibration Control for Liènard Systems

Self-excitations can be dangerous in many nonlinear systems and can produce catastrophic failures, that is a sudden and complete failure that cannot be put right. We extend the nonlocal vibration control to the suppression of the self-excited vibrations of the Liènard system. We introduce a non local control force that yields a third order non-linear differential equation and use a nonlocal active control to mitigate the amplitude peak in the self-excitations. The nonlocal parameters can be carefully adjusted, in order to avoid undesirable behavior and dynamical nonlinear excitations.  We consider the effects of changing the nonlocal parameters on the stability and the value of the response of the system under control. We demonstrate that our method can successfully improve the self-excitation active control, studying a Liènard system through the (AP) asymptotic reduction method. A nonlocal force can be used to suppress self-excitations and put under control the oscillator behavior

Fractal Oscillators

We consider a weakly nonlinear oscillator with a fractal forcing, given by the Weierstrass function, and use the asymptotic perturbation (AP) method to study its behavior.  Being this function nowhere differentiable we can only use adequate approximations. We find that while in the linear case the resulting motion is a simple superposition between the fractal forcing and the standard oscillation, on the contrary in the nonlinear case the oscillator phase and its frequency also become fractal. We obtain the Poincarè sections in various cases and all theoretical findings are corroborated with numerical simulation

Atomic Absorption Spectroscopy Analysis of Heavy metals in water at Daura Gypsum Mining Site, Yobe State, Nigeria

This study was designed to detect heavy metals level in water collected from Daura gypsum mining site, Yobe State, Nigeria. Samples were collected and analyzed using Atomic Absorption Spectroscopy. The level of Pb, Ni, Cd, As, Cu and Zn, were assessed. Also the conductivity of the water samples detected using Conductivity meter. The result shows significant level of As at 0.0382 mg/l, Cd at 0.06-0.18 mg/l and 0.9852 mg/l for Ni which exceeded the WHO limit (0.01 mg/l). Cu, Zn, and Pb were detected at 0.95mg/l, 1.77 mg/l and 0.244 mg/l respectively. Cu, Pb, and Zn were found below the WHO, USEPA limits. This may bring kidney related risk to the people over a long period of time, Therefore, incorporation of ion exchange, reverse osmosis or adsorption in water sources will help reduce the heavy metals burden of the public in the area

The formation of a single star planetary system and heat balance of the planets

This paper considers the principles of planetary formation under gravitational compression of a calm, isolated, rarefied gas-dust cloud, different from the compression of a dense gas-dust cloud of a star cluster. The distinctive feature of such compression is the instantaneous formation at a certain stage of the evolution of a gas-dust cloud of a dense, high-energy plasma that makes up a protoplanetary nebula, in which heavy chemical elements are generated. A hypothesis of the scheme of a possible nucleosynthesis is presented. The new conditions for the formation of a protoplanetary cloud, regardless of the hypothesis put forward, entail a fundamentally new concept of planetary evolution. Almost from the very first moments of the formation of a protoplanetary cloud, the proto-nuclei of future planets with their intrinsic magnetic field and complex structure are formed even before the formation of protoplanets. Formed in a same spatial zone within a same process the proto-nuclei are the progenitors of all types of planets of stellar system, i.e., cores are not formed within planets due to gravitational differentiation and high pressure generated by gravitational forces, but rather the planets emerge from proto-nuclei. After a certain, relatively short period of time, the protoplanetary cloud becomes flat and splits into two rings with a gap between them. The type of planet that is formed depends on the place where it is formed. According to physical laws, only young planets can cool down. Therefore, the Earth, which is no longer a young planet, does not cool down

Explanation of the observed dynamics of matter in the Universe in the framework of classical gravity

The paper shows the prematurity of introduction of a new category of "dark energy" into the theory. The observed dynamics of matter in the Universe can be explained within the framework of classical gravity, if the assumption about the existence of a primary plasma cloud at a certain evolutionary stage of the Universe is considered false. Comments are given on why this step is correct. The expansion of the Universe is not associated with the pressure that existed at any stage of its development; therefore, under the influence of gravitational forces, it must turn into compression, despite the observed accelerated motion of galaxies in the direction away from the terrestrial observer. In general, the Universe is not cooling down, but heating up. The brief review of astronomical observations, presented in this paper, confirms the need to develop a new view of cosmology

Ab initio study of effect of pressure on structural and elastic properties of CaX, X = {O, S, Se}

We present density function theory study of effect of pressure on structural, elastic and electronics properties of compounds CaX (X=O, S and Se) within the generalized gradient approximation. The results presented for transition pressure, elastic parameters and band structures are in good agreement with the available literature. We also present the effect of pressure on these parameters. The generalized stability criteria show that CaSe is not stable above pressure of 29GPa and all the material CaX are not stable at B2 phase. The materials are brittle at equilibrium but this changes with pressure change. They are also generally anisotropic; CaO(B1) was found to be Isotropic at pressure of 12.5GPa. Finally, the band-gap of all the material around (Γ - X)  decreased with pressure, all the material became indirect band-gap semiconductor at high pressure and CaSe undergoes a semiconductor-metal transition at pressure of 68 GPa

Time Delay Control and Frequency Splitting in the forced Kadomtsev-Petviashvili Equation

A time delay control is applied to the forced Kadomtsev-Petviashvili (KP) equation Using an appropriate perturbation method, we derive nonlinear equations describing amplitude and phase of the response and discuss in some detail external force-response and frequency-response curves for the fundamental resonance. For the uncontrolled system, we find a frequency splitting, a second frequency aappears in addition to the forcing one. Saddle-center bifurcation, jumps and hysteresis phenomena are observed together with closed orbits of the slow flow equations. There are stable two-period quasi-periodic modulated motion for the KP equation with amplitudes depending on the initial conditions. Subsequently, we study the controlled system finding sufficient conditions for a periodic behavior. We can accomplish a successful control because the amplitude peak of the fundamental resonance can be reduced and the saddle-center bifurcations and two-period quasi-periodic motions can be removed by adequate choices for delay and feedback gains