The emergence of magnetic chaotic lines and layers in tokamaks represents an important field of research in nuclear fusion. Their analysis is based in general on the notion of maps and mappings, which are practical in analyzing the correlation between stable and unstable manifolds and the chaotic layers. Various maps representing the Poincaré section of a continuous magnetic field line system have been introduced in the literature, yet the most renewed one is the tokamap introduced by Balescu et al. [Phys. Rev. Vol. E58, p. 951 (1998)]. However, this tokamap requires additional constraints to make it symplectic.

To remedy this problem, a new technique based on the Hamilton-Jacobi method has been recently introduced. However, several constraints must be imposed as well in the mapping model. In the present study, we analyze this problem based on the notion of nonstandard Hamiltonians, which have proved their relevance in the theory of differential equations and complex systems. Nonstandard tokamaps have been introduced where their Poincaré sections, bifurcation diagrams, and Lyapunov exponents have been obtained and discussed. We observed the emergence of thin area-filling sections dominated by chaotic magnetic field lines with symmetric and nonsymmetric structures near the island separatrix. Small islands emerge, proving the emergence of large chaotic regions.