Teleparallel based cosmological models provide a description of gravity in which torsion is the mediator of gravitation. Several extensions have been made within the so-called teleparallel equivalent of general relativity which is equivalent to general relativity at the level of the equations of motion where attempts are made to study the extensions of this form of gravity and to describe more general functions of the torsion scalar T. One of these extensions is $$f(T,\phi )$$f ( T , ϕ ) gravity; T and $$\phi $$ϕ respectively denote the torsion scalar and scalar field. In this work, the dynamical system analysis has been performed for this class of theories to obtain the cosmological behaviour of a number of models. Two models are presented here with some functional form of the torsion scalar and the critical points are obtained. For each critical point, the stability behaviour and the corresponding cosmology are shown. Through the graphical representation, the equation of state parameter and the density parameters for matter-dominated, radiation-dominated and dark energy phase are also presented for both the models.

Scalar-tensor theories offer the prospect of explaining the cosmological evolution of the Universe through an effective description of dark energy as a quantity with a non-trivial evolution. In this work, we investigate this feature of scalar-tensor theories in the teleparallel gravity context. Teleparallel gravity is a novel description of geometric gravity as a torsional- rather than curvature-based quantity which presents a new foundational base for gravity. Our investigation is centered on the impact of a nontrivial input from the kinetic term of the scalar field. We consider a number of model settings in the context of the dynamical system to reveal their evolutionary behavior. We determine the critical points of these systems and discuss their dynamics.

In this paper, we perform the dynamical system analysis of the cosmological models framed in the extended teleparallel gravity, the f(T, B) gravity. We use the mapping, $$f(T, B)\rightarrow -T+\tilde{f}(T, B)$$f ( T , B ) → - T + f ~ ( T , B ) , and define the dynamical variables to form the autonomous dynamical system. The critical points are obtained in two well-motivated forms of f(T, B), one that involves the logarithmic form of the boundary term B, and the other one is the non-linear form of the boundary term. The position of critical points is shown in the different evolutionary phases of the Universe such as radiation, matter, and de-Sitter phase. The stability condition of each of the critical points of both the models is derived and the behavior of each point has been obtained mathematically and through the phase portrait. The evolution of standard density parameters such as radiation ( $$\Omega _{r}$$Ω r ), matter ( $$\Omega _{m}$$Ω m ), and dark energy ( $$\Omega _{DE}$$Ω DE ) are also analyzed. Further to connect with the present cosmological scenario, the behavior of deceleration and equation of state parameter both in the dark energy phase ( $$\omega _{DE}$$ω DE ) and total ( $$\omega _{tot}$$ω tot ) are shown from the initial condition of the dynamical variables. The accelerating behaviour has been obtained for both models.

All degrees of freedom related to the torsion scalar can be explored by analysing, the f ( T , T _G ) gravity formalism where, T is a torsion scalar and T _G is the teleparallel counterpart of the Gauss-Bonnet topological invariant term. The well-known Noether symmetry approach is a useful tool for selecting models that are motivated at a fundamental level and determining the exact solution to a given Lagrangian, hence we explore Noether symmetry approach in f ( T , T _G ) gravity formalism with three different forms of f ( T , T _G ) and study how to establish nontrivial Noether vector form for each one of them. We extend the analysis made in S Capozziello, M De Laurentis, and K F Dialektopoulos 2016, “Noether symmetries in gauss–bonnet-teleparallel cosmology,” Eur. Phys. J. C 76 , 629. for the form f ( T , T G ) = b 0 T G k + t 0 T m and discussed the symmetry for this model with linear teleparallel equivalent of the Gauss-Bonnet term, followed by the study of two models containing exponential form of the teleparallel equivalent of the Gauss-Bonnet term. We have shown that all three cases will allow us to obtain non-trivial Noether vector which will play an important role to obtain the exact solutions for the cosmological equations.