Non-singular bouncing model in energy momentum squared gravity

This work is concerned to study the bouncing nature of universe for an isotropic configuration fluid $\mathcal{T}_{\alpha\beta}$ and Friedmann-Lema\^{i}tre-Robertson-Walker metric scheme. carried out under novel $f(\mathcal{G},\mathcal{T}_{\alpha \beta} \mathcal{T}^{\alpha \beta})$ gravitation by assuming a specific model i.e, $f(\mathcal{G},\mathcal{T}^2)=\mathcal{G}+\alpha \mathcal{G}^2+2\lambda \mathcal{T}^2$ with $\alpha$ $\lambda$ are constants, serving as free parameters. {The terms $\mathcal{G}$ $\mathcal{T}^2$ served Gauss-Bonnet invariant square energy-momentum trace term inclusion in gravitational action respectively, proportional $\mathcal{T}^2=\mathcal{T}_{\alpha \beta}$.} A functional form Hubble parameter taken provide evolution cosmographic well known equation state parameter, $\omega(t)=-\frac{k \log (t+\epsilon )}{t}-1$ used represent dynamical behavior energy density, matter pressure conditions. detailed graphical analysis also provided review bounce. Furthermore, all parameters set way, make supposed act solution ensure viability Conclusively, necessary conditions checked.