Oscillation of Second-order Nonlinear Dynamic Equation on Time Scales

The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati technique, integral averaging technique and the time scales theory, some new sufficient conditions for oscillation of the equation are proposed. These results generalize and extend many known results for second order dynamic equations. Some examples are given to illustrate the main results of this article.

Generalized Chaplygin equations for nonholonomic systems on time scales

The generalized Chaplygin equations for nonholonomic systems on time scales are proposed and studied. The Hamil- ton principle for nonholonomic systems on time scales is established, and the corresponding generalized Chaplygin equa- tions are deduced. The reduced Chaplygin equations are also presented. Two special cases of the generalized Chaplygin equations on time scales, where the time scales are equal to the set of real numbers and the integer set, are discussed. Finally, several examples are given to illustrate the application of the results.

Mei’s symmetry theorem for time scales nonshifted mechanical systems

We focus on Mei symmetry for time scales nonshifted mechanical systems within Lagrangian framework and its resulting new conserved quantities.Firstly,the dynamic equations of time scales nonshifted holonomic systems and time scales nonshifted nonholonomic systems are derived from the generalized Hamilton’s principle.Secondly,the definitions of Mei symmetry on time scales are given and its criterions are deduced.Finally,Mei’s symmetry theorems for time scales nonshifted holonomic conservative systems,time scales nonshifted holonomic nonconservative systems and time scales nonshifted nonholonomic systems are established and proved,and new conserved quantities of above systems are obtained.Results are illustrated with two examples.

Oscillation of Third-order Delay Dynamic Equations on Time Scales

This paper is concerned with the oscillatory behavior of a class of third-order nonlinear variable delay neutral functional dynamic equations on time scale. By using the generalized Riccati transformation and inequality technique, we establish some new oscillation criteria for the equations. Our results extend and improve some known results, but also unify the oscillation of third-order nonlinear variable delay functional differential equations and functional difference equations with a nonlinear neutral term. Some examples are given to illustrate the importance of our results.

Methods of reduction for Lagrange systems on time scales with nabla derivatives

The Routh and Whittaker methods of reduction for Lagrange system on time scales with nabla derivatives are studied.The equations of motion for Lagrange system on time scales are established, and their cyclic integrals and generalized energy integrals are given. The Routh functions and Whittaker functions of Lagrange system are constructed, and the order of differential equations of motion for the system are reduced by using the cyclic integrals or the generalized energy integrals with nabla derivatives. The results show that the reduced Routh equations and Whittaker equations hold the form of Lagrnage equations with nabla derivatives. Finally, two examples are given to illustrate the application of the results.

Bursting oscillations in a hydro-turbine governing system with two time scales

A fast-slow coupled model of the hydro-turbine governing system （HTGS） is established by introducing frequency disturbance in this paper. Based on the proposed model, the performances of two time scales for bursting oscillations in the HTGS are investigated and the effect of periodic excitation of frequency disturbance is analyzed by using the bifurcation diagrams, time waveforms and phase portraits. We find that stability and operational characteristics of the HTGS change with the value of system parameter kd. Furthermore, the comparative analyses for the effect of the bursting oscillations on the system with different amplitudes of the periodic excitation a are carried out. Meanwhile, we obtain that the relative deviation of the mechanical torque mt rises with the increase of a. These methods and results of the study, combined with the performance of two time scales and the fast-slow coupled engineering model, provide some theoretical bases for investigating interesting physical phenomena of the engineering system.

Some integral inequalities on time scales

In this article, we study the reverse HSlder type inequality and HSlder inequality in two dimensional case on time scales. We also obtain many integral inequalities by using HSlder inequalities on time scales which give Hardy＇s inequalities as spacial cases.

Forced Oscillation of Neutral Advanced Dynamic Equations with Positive and Negative Coeffcients on Time Scales

In this paper,we consider the forced neutral advanced dynamic equations with positive and negative coeffcients[x(t)-R(t)x(t+r)]+P(t)x(t+τ)△Q(t)x(t+δ)= f(t),t≥t0 on a time scale T.Several suffcient conditions are obtained for the oscillation of the dynamic equations.The main result is illustrated by an example.

Existence and exponential stability of almost-periodic solutions for MAM neural network with distributed delays on time scales

This paper is concerned with multidirectional associative memory neural network with distributed delays on almost-periodic time scales.Some sufficient conditions on the existence,uniqueness and the global exponential stability of almost-periodic solutions are established.An example is presented to illustrate the feasibility and effectiveness of the obtained results.

HYERS-ULAM STABILITY OF SECOND-ORDER LINEAR DYNAMIC EQUATIONS ON TIME SCALES

We investigate the Hyers-Ulam stability(HUS)of certain second-order linear constant coefficient dynamic equations on time scales,building on recent results for first-order constant coefficient time-scale equations.In particular,for the case where the roots of the characteristic equation are non-zero real numbers that are positively regressive on the time scale,we establish that the best HUS constant in this case is the reciprocal of the absolute product of these two roots.Conditions for instability are also given.

Interdecadal Variations of Phase Delays Between Two Nino Indices at Different Time Scales

Phase delays between two Nino indices-sea surface temperatures in Nino regions 1+2 and 3.4 (1950-2001)-at different time scales are detected by wavelet analysis. Analysis results show that there are two types of period bifurcations in the Nino indices and that period bifurcation points exist only in the region where the wavelet power is small. Interdecadal variation features of phase delays between the two indices vary with different time scales. In the periods of 40-72 months, the phase delay changes its sign in 1977: Nino 1+2 indices are 2-4 months earlier than Nino 3.4 indices before 1977, but 3-6 months later afterwards. In the periods of 20-40 months, however, the phase delay changes its sign in another way: Nino 1+2 indices are 1-4 months earlier before 1980 and during 1986-90, but 1-4 months later during 1980-83 and 1993-2001.

Crystal Growth and Crystallization Time Scales of the Panzhihua Layered Intrusion:Constraint from Crystal Size Distribution

The Panzhihua layered intrusions is generated closely related to the Emeishan LIPs.This paper analyzes the spatial distribution of plagioclase and pyroxene.The quantitative texture analysis of 2209 plagioclase shows that the characteristic length of plagioclase is 0.54 to 0.96 mm,the intercept variation range is large,from-0.67 to 0.96,and the slope is-1.85 to-1.04,the Aspect Ratio shows from 1.84 to 2.59 and fractal dimension D is 1.908–1.933.The quantitative texture analysis of 2342 pyroxene shows that the characteristic length of pyroxene is 0.38–0.64 mm,the intercept shows from 0.46 to 2.26,The slope ranges from-2.6 to-1.47,the Aspect Ratio value varies from 1.53 to 1.71,the fractal dimension D is 0.93 to 1.13.All the CSDs results of the Panzhihua intrusions indicate that plagioclase and pyroxene form in an open magma system and undergo four replenishment of magma injection.The plagioclase crystals do not grow as the lathlike shape,and the fractal growth leads to complex crystal surface.The plagioclase undergoes deformation compaction during the crystal process,and then is oriented.The pyroxene crystals grow along an approximately triaxial ratio and undergo texture adjustment and small crystal dissolution reabsorption.When all crystals in magma system grows up to 2 mm,the pyroxene undergoes cumulation in the Panzhihua layered intrusions.The plagioclase crystallization time scale is 171.23–304.41 years,representing that the crystallization is the more uniform in central part of the melt.The nucleation density continuously increases during the crystallization process of the magma system.The time scale to reach the final maximum crystal nucleation density is 15.28–58.98 years.

Multiple time scales of fluvial processes—theory and applications

Fluvial processes comprise water flow,sediment transport and bed evolution,which normally feature distinct time scales.The time scales of sediment transport and bed deformation relative to the flow essentially measure how fast sediment transport adapts to capacity region in line with local flow scenario and the bed deforms in comparison with the flow,which literally dictates if a capacity based and/or decoupled model is justified.This paper synthesizes the recently developed multiscale theory for sediment-laden flows over erodible bed,with bed load and suspended load transport,respectively.It is unravelled that bed load transport can adapt to capacity sufficiently rapidly even under highly unsteady flows and thus a capacity model is mostly applicable,whereas a non-capacity model is critical for suspended sediment because of the lower rate of adaptation to capacity.Physically coupled modelling is critical for fluvial processes characterized by rapid bed variation.Applications are outlined on very active bed load sediment transported by flash floods and landslide dam break floods.

Discrete Fractional Birkhoff Equations in Terms of Time Scales

In order to study discrete fractional Birkhoff equations for Birkhoffian systems,the method of isochronous variational principle is used in this paper. Discrete fractional Pfaff-Birkhoff principle in terms of time scales is presented. Discrete fractional Birkhoff equations with left and right discrete operators of Riemann-Liouville type are established and some special cases including classical discrete Birkhoff equations,discrete fractional Hamilton equations and discrete fractional Lagrange equations are discussed. Finally,an example is devoted to illustrate the results.

On Conformable Delta Fractional Derivative on Time Scales

In this paper, we introduce and investigate the concept of conformable delta fractional derivative on time scales. By using the theory of time scales, we obtain some basic properties of the conformable delta fractional derivative. Our results extend and improve both the results in [9] and the usual delta derivative.

Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales

This paper focuses on studying the Noether symmetry and the conserved quantity with non-standard Lagrangians, namely exponential Lagrangians and power-law Lagrangians on time scales. Firstly, for each case, the Hamilton prin- ciple based on the action with non-standard Lagrangians on time scales is established, with which the corresponding Euler-Lagrange equation is given. Secondly, according to the invariance of the Hamilton action under the infinitesimal transformation, the Noether theorem for the dynamical system with non-standard Lagrangians on time scales is established. The proof of the theorem consists of two steps. First, it is proved under the infinitesimal transformations of a special one-parameter group without transforming time. Second, utilizing the technique of time-re-parameterization, the Noether theorem in a general form is obtained. The Noether-type conserved quantities with non-standard Lagrangians in both clas- sical and discrete cases are given. Finally, an example in Friedmann-Robertson-Walker spacetime and an example about second order Duffing equation are given to illustrate the application of the results.

OSCILLATION FOR NONAUTONOMOUS NEUTRAL DYNAMIC DELAY EQUATIONS ON TIME SCALES

The article is concerned with oscillation of nonautonomous neutral dynamic delay equations on time scales. Sufficient conditions are established for the existence of bounded positive solutions and for oscillation of all solutions of this equation. Some results extend known results for difference equations when the time scale is the set Z^＋ of positive integers and for differential equations when the time scale is the set IR of real numbers.

Existence of Positive Solutions for Higher Order Boundary Value Problem on Time Scales

In this paper, we investigate the existence of positive solutions of a class higher order boundary value problems on time scales. The class of boundary value problems educes a four-point （or three-point or two-point） boundary value problems, for which some similar results are established. Our approach relies on the Krasnosel＇skii fixed point theorem. The result of this paper is new and extends previously known results.

New results on global exponential stability of competitive neural networks with different time scales and time-varying delays

This paper studies the global exponential stability of competitive neural networks with different time scales and time-varying delays. By using the method of the proper Lyapunov functions and inequality technique, some sufficient conditions are presented for global exponential stability of delay competitive neural networks with different time scales. These conditions obtained have important leading significance in the designs and applications of global exponential stability for competitive neural networks. Finally, an example with its simulation is provided to demonstrate the usefulness of the proposed criteria.

Periodic Solutions to a Delayed Semi-ratio Dependent Diffusion System on Time Scales

In this paper,we investigate the existence of periodic solutions of a semi-ratiodependent predator-prey diffusion system with functional responses and time delays in a two-patch environment on time scales by using a continuation theorem based on coincidence degree theory.