<p>In this work, we investigate the field equations of modified Gauss–Bonnet gravity, namely <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>f</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$f(R,G)$</EquationSource> </InlineEquation> gravity, within the framework of a flat Friedmann–Robertson–Walker (FRW) spacetime. By adopting a dynamical systems approach, we analyze cosmological evolution and determine the corresponding fixed points for a specific functional form of the model, <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>f</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>f</mi> <mn>0</mn> </msub> <msup> <mi>R</mi> <mi>α</mi> </msup> <msup> <mi>G</mi> <mi>β</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$f(R,G)=f_{0} R^{\alpha }G^{\beta }$</EquationSource> </InlineEquation>, which couples the Ricci scalar and the Gauss–Bonnet invariant. Using fixed points, we have constructed the <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$H(z)$</EquationSource> </InlineEquation> model. Also, with the help of Bayesian statistics, specifically the Markov Chain Monte Carlo (MCMC) mechanism was employed to constrain the parameters of the proposed model using observational datasets such as Hubble, Pantheon, DESI BAO and RSD datasets.</p>