Deformation of invariant tori under perturbation

Abstract Invariant tori (flux surfaces) reside in the ordered regions of phase space of a dynamical system and represent the well-confined region of a magnetic confinement fusion (MCF) device, which may be threatened in non-axisymmetric cases such as tokamaks under resonant magnetic perturbation and stellarators. In MCF devices, the structure of nested closed flux surfaces governs radial transport and thus plays a critical role in confinement performance. Using the method of variation as a mathematical foundation (the vector field itself as a spatial function is considered as an argument of the geometry of these tori), this paper derives the formulae that describe <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:mrow> <mml:mtable columnalign="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mtext>how invariant tori (flux surfaces)</mml:mtext> <mml:mtext> </mml:mtext> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">T</mml:mi> </mml:mrow> <mml:mtext> </mml:mtext> <mml:mtext>deform under perturbation</mml:mtext> <mml:mtext> </mml:mtext> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">B</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> which can calculate the deformation in tens of seconds for stellarator configuration optimisation by Julia programming language without delicate hardware acceleration and almost instantly for tokamaks.