Bifurcations of phase portraits and chaotic behaviors of the (2+1)-dimensional double-chain DNA system with beta derivative: A qualitative approach

Qualitative analysis in mathematical modeling has become an important research area within the broad domain of nonlinear sciences. In the realm of qualitative analysis, the bifurcation method is one of the significant approaches for studying the structure of orbits in nonlinear dynamical systems. To apply the bifurcation method to the (2 + 1)-dimensional double-chain Deoxyribonucleic Acid system with beta derivative, the bifurcations of phase portraits and chaotic behaviors, combined with sensitivity and multi-stability analysis of this system, are examined. Initially, the bifurcations of phase portraits are visually identified at the obtained equilibrium points of a planar dynamical system via both Hamiltonian and Jacobian algorithms. The obtained results indicate Jacobian algorithm is more efficient in identifying the stability of bifurcations than the Hamiltonian algorithm for this system. Subsequently, by introducing an external perturbation term into the planar dynamical system, the chaotic behavior is effectively identified by using a variety of tools, such as two- and three-dimensional phase portraits, time series, Lyapunov exponents, and Poincare<acute accent> maps. The findings suggest that the perturbed dynamical system deviates from regular patterns and exhibits behavior ranging from periodic to quasiperiodic and from quasi-periodic to chaotic. Finally, the sensitivity and multi-stability of the system are examined using the Runge-Kutta method to assess the model's response to minor variations in initial conditions through numerical solutions, revealing that the model is sensitive and multi-stable. The outcomes of this study will enhance a relationship between applied mathematicians and experimental biologists, helping to explore hidden features of Deoxyribonucleic Acid through the studied model.