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the first time expansion of universe discovered in type ia supernova ( sne ia ) by riess et al @xcite , and then perlmutter et al @xcite redemonstrated this issue by @xmath1 supernovae . also accelerated expansion of universe reconfirmed by cosmic microwave background ( cmb ) @xcite and large scale structure ( lss ) @xcite .
thus , the result of these literature confirms the existence of a mysterious energy called dark energy .
one of the biggest challenges of modern theoretical physics is understanding the nature of dark energy .
in fact , dark energy is a hypothetical form of energy in space , and constitutes three  quarters of the total energy of the universe .
dark energy requires strong negative pressure to explain the observed accelerated expansion of the universe . by using the einstein field equation ,
the accelerated expansion is described by a small positive cosmological constant .
also , the discovery illustrates that geometry of the universe is very close to flat space - time @xcite . in order to describe dark energy model ,
numerous studies have been performed in isotropic space - time such as scalar fields @xcite , modified gravity models @xcite , holographic models @xcite , interacting models @xcite , bouncing model @xcite and braneworld models @xcite .
now we intend to describe dark energy model with modified gravity theory .
this issue studied in numerous literatures by topics of @xmath2 gravity @xcite , @xmath3 gravity @xcite , @xmath4 gravity @xcite , @xmath5 gravity @xcite , @xmath6 gravity @xcite and @xmath7 gravity @xcite , in which @xmath8 , @xmath9 , @xmath10 and @xmath11 are ricci scalar curvature , torsion scalar , trace of the matter energy - momentum tensor and the gauss - bonnet term , respectively .
we note that cosmological solutions of these models can provide an alternative explanation for the accelerated expansion of universe .
in general relativity , the einstein - hilbert action includes the curvature term that describes gravity , but in teleparallel gravity , the corresponding action includes torsion term that first proposed by einstein @xcite .
he has been able to introduce the mathematical structure of distant parallelism by a tetrad or vierbein field for unification of electromagnetism and gravity .
this issue is caused to replace the levi - civita connection in the framework of general relativity with the weitzenbck connection in teleparallelism @xcite .
this means that the @xmath8 in the general relativity changes to @xmath9 in teleparallelism .
its modified form has been done by changing @xmath9 to @xmath12 with an arbitrary function in the teleparallelism action so - called @xmath3 gravity theory .    a model recently proposed by subject of @xmath13 gravity , that one has some benefits in comparison with other models of modified gravity theory @xcite .
one of its benefits is much simpler calculations of numerical solutions , and also is compatible with recent observational data to describe accelerated expansion of the universe . since the @xmath13 gravity can be a good candidate for a source of dark energy , then one is a good alternative for the standard gravity model .
hence these are a good motivation for the present job .
one of the other interesting models for the description of dark energy is chaplygin gas , so one is introduced as a fluid with negative pressure .
chaplygin gas model extended to generalized chaplygin gas and modified chaplygin gas , in which the modified chaplygin gas model is the combination of barotropic model and chaplygin gas model .
the advantage of the modified chaplygin gas model is consistent with observational data , and also unify dark matter and dark energy @xcite .
now , we would like to construct a new model based on interacting @xmath13 gravity with the modified chaplygin gas .
the corresponding model may be a development of several previous jobs that also corresponds with observational data .
this paper is organized as the following :    in sec . [ s2 ] , we review the general form of @xmath13 gravity in the frw metric . in sec .
[ s3 ] , we consider the interaction between @xmath0 gravity model and the modified chaplygin gas , and also obtain the friedmann equations and equation of state ( eos ) . in sec .
[ s4 ] , we investigate the stability of our model by phase plane analysis . finally , in sec .
[ s5 ] we will give result and conclusion for our model .
we start with a novel theory of modified gravity in which there is an arbitrary function of torsion scalar @xmath9 and trace of the matter energy - momentum tensor @xmath10 . in that case
, we can write the corresponding action by @xmath14 d^4x,\ ] ] where @xmath15 , and @xmath16 is the matter lagrangian density .
it should be mention that vierbein field @xmath17 to be related with metric tensor by relationship @xmath18 in which @xmath19 .    in this paper , universe
is considered as the homogenous and isotropic in the frw metric in the following form @xmath20 where @xmath21 is scale factor .
the vierbein field can be written in accordance with metric as @xmath22 .
we note that the levi - civita connection is as @xmath23    ricci tensor @xmath24 , asymmetry tensor @xmath25 and torsion tensor @xmath26 are written as follows , respectively @xmath27 where @xmath28    by varying the action with respect to vierbein field , we can obtain the field equation as follows : @xmath29\left ( 1+{{f}_{t } } \right)+\frac{1}{4}\delta _ { \mu } ^{\nu } \,t \\    = { { s}_{\mu } } ^{\nu \rho } { { f}_{tt}}\,{{\delta } _ { \rho } } \mathcal{t } + { { f}_{\mathcal{t } } } \left ( \frac{\theta _ { \mu } ^{\nu } + \delta _ { \mu } ^{\nu } p}{2 } \right)-\frac{1}{4}\delta _ { \mu } ^{\nu } \,f(\mathcal{t } ) + \frac{{{k}^{2}}}{2}\theta _ { \mu } ^{\nu } , \\ \end{aligned}\end{aligned}\ ] ] where indices denote the derivative with respect to the corresponding parameters , and symbol @xmath30 is the energy- momentum tensor of the matter .
tension scalar and trace of energy - momentum tensor are obtained as the follows :    [ ttheta ] @xmath31    where @xmath32 , @xmath33 and @xmath34 are the hubble parameter , matter energy density and matter pressure , respectively .
a recent cosmological model has been designed using a kind of perfect fluid ( chaplygin gas ) , that the universe is assumed to be filled with . in this paper ,
the modified chaplygin gas model was used and its mode equation can be expressed as follows : @xmath35 where @xmath36 , @xmath37 and @xmath38 are positive constant , energy density and pressure of modified chaplygin gas , respectively and also we have @xmath39 . in here we consider the universe dominates by components of matter , dark energy and chaplygin gas . in that case , the effective energy density and the effective pressure of universe are written in terms of the aforesaid three components as    [ rhoeffpeff ] @xmath40    where indices @xmath41 , @xmath42 and @xmath43 are components of cold mater , dark energy and modified chaplygin gas , respectively .    as we know , universe consists a perfect fluid ,
then the corresponding continuity equation can be written in the following form @xmath44    since there is an energy flow between contents of universe , thus we separately write the continuity equations of components of universe as    [ conti2 ] @xmath45    where @xmath46 and @xmath47 are interaction terms between the universe components .
let us consider that there is an energy flow only between the components of dark energy and modified chaplygin gas , in this way @xmath48 .
also , the amount of this interaction terms is considered as @xmath49 in which @xmath50 is the coupling parameter or transfer strength . in this case , the solutions of eqs . and are as follows : @xmath51 @xmath52}^{\frac{1}{\alpha + 1}}},\ ] ] where @xmath53    in this paper , in order to solve the friedmann equations and to avoid its complexity , we consider a simple particular model of @xmath0 gravity as a linear function of @xmath9 and @xmath10 in the form @xmath54 . therefore , friedmann equation is deduced as    [ fri3 ] @xmath55    now , the above equations can be written in friedmann standard as follows .
[ fri4 ] @xmath56    from eqs . and
, we can find the density energy and pressure of dark energy in the following form    [ rhopde ] @xmath57    where @xmath58 is the eos of matter .
also , we define effective eos of a perfect fluid , and eos of dark energy as @xmath59 @xmath60    therefore , friedmann obtained equations are written based on the redshift parameter ( @xmath61 ) , in that case , the eqs . and
are rewritten by redshift parameter as    [ rhopde1 ] @xmath62}^{\frac{1}{\alpha + 1}}},\label{rhopde1 - 1}\\ &    \begin{aligned }    & { { p}_{de}}=\frac{\gamma } { 2}\left ( 1 - 3{{\omega } _ { m } } \right){{\rho } _ { { { m}_{0}}}}{{\left ( \frac{{{a}_{0}}}{1+z } \right)}^{-3(1+{{\omega } _ { m}})}}- a{{\left [ \frac{b}{\eta } + { { c}_{0}}\,{{\left ( \frac{{{a}_{0}}}{1+z } \right)}^{-3\eta ( \alpha + 1 ) } } \right]}^{\frac{1}{\alpha + 1}}}\\ &    + \frac{b}{{{\left [ \frac{b}{\eta } + { { c}_{0}}\,{{\left ( \frac{{{a}_{0}}}{1+z } \right)}^{-3\eta ( \alpha + 1 ) } } \right]}^{\frac{\alpha } { \alpha + 1}}}}.\label{rhopde1 - 2 }    \end{aligned}\end{aligned}\ ] ]    by substituting eqs . into the eq . , and plotting @xmath63 versus @xmath64 . in that case , we see the @xmath63 at the present time @xmath65 equals to @xmath66 , which one indicates universe expanding at an accelerating rate , and also the issue corresponds to work @xcite .    by drawing figures of energy density @xmath67 and pressure @xmath68 of dark energy with respect to redshift ( fig .
[ fig1 ] ) , we can see that the variations of @xmath67 versus @xmath64 is positive , and the variations of @xmath68 versus @xmath64 is negative , so that these issues confirm the accelerated expansion of the universe .
in this section , we intend to investigate the stability conditions of the @xmath0 gravity with modified chaplygin gas by the method of phase plane analysis . in that case , we introduce the following new variables as @xmath69 by making derivative the parameters @xmath70 , @xmath71 and @xmath72 with respect to @xmath73 , and by using the continuity equations , we can obtain them in the following form    [ xyprime ] @xmath74    where the prime denotes derivative with respect to @xmath73 .
we can find term @xmath75 by eq .
as @xmath76    substituting into , the below relationships are deduced    [ xyprime1 ] @xmath77    we note that the first friedmann equation is rewritten by aforesaid new variables as @xmath78    now we describe the stability conditions of the model to obtain the critical points that so - called fixed points .
the corresponding fixed points have the asymptotical behavior , and they are dependent on the @xmath13 model and modified chaplygin gas parameters . in that case , we can obtain the fixed points in terms of the corresponding parameters by setting @xmath79 and @xmath80 , and these results are demonstrated in table [ table1 ] .
as we can see , two fixed points are obtained by names @xmath81 and @xmath82 .
the significance of the fixed point is to study the stability of theory and phase transition .
.the fixed points ( critical points ) . [ cols="^,^,^ " , ]     [ table3 ]
in this paper , we studied the @xmath0 model by frw background , in which @xmath9 and @xmath10 were considered as torsion scalar and trace of the matter energy - momentum tensor , respectively .
then , we interacted the model with modified chaplygin gas , this means that the effective energy density and the effective pressure of universe were written in terms of three components of matter , chaplygin gas and dark energy .
since universe considered as a perfect fluid , hence the effective continuity equation was separated by three continuity equations of universe components . in that case , we supposed to exist only an energy flow between the components of dark energy and modified chaplygin gas by interaction term as @xmath83 . in what follows , the arbitrary function @xmath0 was considered as a linear function of @xmath9 and @xmath10 , and then friedmann equation has been written in terms of redshift .
nonetheless , the dark energy cosmological parameters such as energy density , pressure and eos parameter have drawn with respect to redshift .
the corresponding figures showed that the universe is undergoing accelerated expansion , and also eos parameter is consistent with observational data .
next , we investigated stability conditions for the model by using of phase plane analysis . also , we obtained fixed points that these depended on parameters of the model and chaplygin gas .
we considered linear perturbations for the fixed points , and then we obtained the condition of accelerated expansion by calculating corresponding eigenvalues .
in this paper , the model of interaction is studied between @xmath0 gravity and modified chaplygin gas in frw - flat metric . 
 we obtain the friedmann equations in the framework of teleparallel gravity by vierbein field . 
 we consider that universe dominates by components of cold mater , dark energy and modified chaplygin gas . in 
 what follows we separately write the corresponding continuity equations for components of universe . 
 also dark energy eos and effective eos are obtained with respect to redshift , thereafter the corresponding cosmological parameters are plotted in terms of redshift , thereinafter the accelerated expansion of the universe is investigated . 
 finally , the stability of the model is discussed in phase plane analysis .