Combined Gradient Representations for Generalized Birkhoffian Systems in Event Space and Its Stability Analysis

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1.College of Civil Engineering，Suzhou University of Science and Technology，Suzhou 215011，P.R.China；2.College of Civil and Transportation Engineering，Hohai University，Nanjing 210098，P.R.China

（Received 22 July 2020；revised 18 January 2021；accepted 16 December 2021）

Abstract:The combined gradient representations for generalized Birkhoffian systems in event space are studied.Firstly，the definitions of six kinds of combined gradient systems and corresponding differential equations are given.Secondly，the conditions under which generalized Birkhoffian systems become combined gradient systems are obtained.Finally，the characteristics of combined gradient systems are used to study the stability of generalized Birkhoffian systems in event space.Seven examples are given to illustrate the results.

Key words：generalized Birkhoffian system；event space；combined gradient systems；stability

0 Introduction

In 1892，Lyapunov published his doctoral dissertation“general problems of motion stability”，in which the concepts，research methods and related theories of stability were given.Since the end of the 19th century，Lyapunov stability theory is an important method to study the stability of mechanical system.Sometimes it is not so easy to construct Lyapunov function，and it is not very convenient in practical application［1-2］.In Ref.［3］，two kinds of important systems were studied，in which one was gradient system，and the other is Hamilton system.Gradient system is a kind of mathematical system.The differential equation of the gradient equation is of first-order.Gradient system is especially suitable for studying the stability with Lyapunov function.If a mechanical system can be transformed into a gradient system，the properties of this gradient system can be used to study the behavior of the mechanical system，especially the stability problem［4-9］.Mei et al.［10］studied bifurcation for the generalized Birkhoffian system.Mei et al.［11-13］also studied skew-gradient representation and combined representation for the generalized Birkhoffian system.Two kinds of generalized gradient representations for generalized Birkhoffian systems were derived in Ref.［14］.The stability of the generalized Birkhoffian system was discussed by using the properties of the gradient system in Refs.［15-18］.Triple combined gradient systems representations for autonomous generalized Birkhoffian systems were given in Ref.［19］.A semi-negative definite matrix gradient system representation for non-autonomous generalized Birkhoffian system was given in Ref.［20］.Some progresses have been made on the gradient system representation and stability analysis for generalized Birkhoffian systems in configuration space.Event space is an extension of configuration space and time，and generalized coordinates are in the same position in event space，so parameters can be selected flexibly and relatively simple equations can be established.The parametric equation in event space can get not only the motion equation in the configuration space，but also the energy integral directly.There are few studies on generalized Birkhoffian dynamics in event space.Zhang［21］studied integrating factors and conservation laws of generalized Birkhoffian system dynamics in event space.Wu et al.［22］studied the gradient representation of holonomic system in event space.In this paper，six combined gradient system representations for generalized Birkhoffian systems in event space are discussed.Under certain conditions，a generalized Birkhoffian system in event space can become a combined gradient system，and then its stability can be discussed by using properties of the combined gradient system.

1 Generalized Birkhoff Equations in Event Space

In configuration space，we consider a generalized Birkhoffian system that is determined by 2nBirkhoff’s variablesaμ(μ=1，2，…，2n).Now an event space of (2n＋1) dimensions is constructed.The coordinates of space points aretandaμ.Introduce the notation

Then，all the variablesxα(α=1，2，…，2n＋1)may be given as functions of some parameterτ.Letxα=xα(τ)be some curves of classC2，such that

are not all zero at the same time，and

In configuration space，the Birkhoffian isB=B(t，a)，Birkhoff’s functions areRν=Rμ(t，a)，and the additional terms areΛμ=Λμ(t，a).In event space，Birkhoff’s functionsBβ(xα)(β=1，2，…，2n＋1)are defined by the following equations［23］.

The additional terms are

In event space，the generalized Birkhoff equations are［23］

The generalized Birkhoff equations of Eq.（6）in event space are not independent each other，where the first equation of Eq.（6）is the result of the last 2nequations，and the last 2nequations can be written as

Suppose thatx′μ＋1can be solved from Eq.（6），that is

where

2 Combined Gradient System

2.1 Combined gradient systemⅠ

This kind of combined gradient system is composed of generalized gradient system and generalized skew-gradient system.The differential equations of the system have the form as

where (bij(X)) is the anti-symmetric matrix andbij=－bji.According to Eq.（10），we can obtain

where the second term at the right-hand side is less than zero.IfVis positive definite，andis negative definite，the solution of the system is asymptotically stable.

2.2 Combined gradient systemⅡ

This kind of combined gradient system is composed of generalized gradient system and generalized gradient system with symmetric negative definite matrix.The differential equations of the system have the form as

where (sij(X)) is symmetric negative definite matrix.According to Eq.（12），we can obtain

where the second and third terms at the right-hand side are less than zero.IfVis positive definite，and＜0，the solution of the system is stable.IfV˙is negative definite，the solution of the system is asymptotically stable.

3 Combined Gradient System Representations for Generalized Birkhoffian System in Event Space

In general，a generalized Birkhoffian system in event space is not a combined gradient system.For the system（8），if there are matricesbμν，sμν，aμνand the functionVsatisfies

it can be transformed into combined gradient systemsⅠ，Ⅱ，Ⅲ，Ⅳ，ⅤandⅥ，respectively.

4 Examples

Example 1 A generalized Birkhoffian system in event space is

We try to convert it into a combined gradient system and study the stability of its zero solution.

From Eq.（6），we obtain

From the last two equations，we have

Takingτ=x1，thenx′1=1 and Eq.（29）can be written as

Leta1=x2，a2=x3，then

which is the combined gradient systemⅠ，and the functionVis

Vis positive definite and decreasing in the neighborhood ofa1=a2=0.According to Eq.（32），we get

is positive definite.The solutiona1=a2=0 is asymptotically stable.

Example 2A generalized Birkhoffian system in event space is

We try to convert it into a combined gradient system and study the stability of its zero solution.

From Eq.（6），we obtain

From the last two equations，we have

Takingτ=x1，thenx′1=1 and Eq.（34）can be written as

Leta1=x2，a2=x3，then

which is the combined gradient systemⅡ，and the functionVis

Vis positive definite and decreasing in the neighborhood ofa1=a2=0.According to Eq.（37），we get

is positive definite.The solutiona1=a2=0 is asymptotically stable.

Example 3A generalized Birkhoffian system in event space is

We try to convert it into a combined gradient system and study the stability of its zero solution.

From Eq.（6），we obtain

From the last two equations，we have

Takingτ=x1，thenx′1=1，Eq.（39）can be written as

Leta1=x2，a2=x3，then

which is the combined gradient systemⅢ，and the functionVis

Vis positive definite and decreasing in the neighborhood ofa1=a2=0.Then，the solutiona1=a2=0 is uniformity asymptotically stable.

Example 4A generalized Birkhoffian system in event space is

We try to convert it into a combined gradient system and study the stability of its zero solution.

From Eq.（6），we obtain

From the last two equations，we have

Takingτ=x1，thenx′1=1 and Eq.（44）can be written as

Leta1=x2，a2=x3，then

which is the combined gradient systemⅣ，and the functionVis

Vis positive definite and decreasing in the neighborhood ofa1=a2=0.Then，the solutiona1=a2=0 is uniformity asymptotically stable.

Example 5A generalized Birkhoffian system in event space is

We try to convert it into a combined gradient system and study the stability of its zero solution.

From Eq.（6），we obtain

From the last two equations，we have

Takingτ=x1，thenx′1=1，and Eq.（49）can be written as

Leta1=x2，a2=x3，then

which is combined gradient systemⅤ，and the functionVis

Vis positive definite and decreasing in the neighborhood ofa1=a2=0.According to Eq.（52），we get

V˙is positive definite.The solutiona1=a2=0 is asymptotically stable.

Example 6A generalized Birkhoffian system in event space is

We try to convert it into a combined gradient system and study the stability of its zero solution.

From Eq.（6），we obtain

From the last two equations，we have

Takingτ=x1，thenx′1=1，and Eq.（54）can be written as

Leta1=x2，a2=x3，then

which is the combined gradient systemⅥ，and the functionVis

Vis positive definite and decreasing in the neighborhood ofa1=a2=0.Then，the solutiona1=a2=0 is uniformity asymptotically stable.

Example 7Linear damped oscillator

We try to convert it into a combined gradient system and study the stability of its zero solution in the event space.

In event space，the representation of generalized Birkhoffian system of the linear damped oscillator is

which is the combined gradient systemⅤ，and the functionVis

Whenγ≥6，Vis positive definite and decreasing in the neighborhood ofa1=a2=0.The solutiona1=a2=0 is uniformity asymptotically stable.Whenγ＜6，Vis not the Lyapunov function，then the stability of the solution is analyzed according to the characteristic roots of the linearized system.

5 Conclusions

It is an important and difficult problem to study the stability of constrained systems.The gradient of constrained mechanical systems is a new method for studying the stability of dynamic systems in analytical mechanics.In this paper，combined gradient systems are utilized to study the stability of generalized Birkhoffian systems in event space.If a generalized Birkhoffian system in event space satisfies the conditions（22—27），the generalized Birkhoffian system in event space can be transformed into a combined gradient system.Its dynamic behaviors can be discussed by using the properties of combined gradient systems，and some conclusions are given for the generalized Birkhoffian system in event space.Examples illustrate the application of the results.