On the stability of steady-states of a two-dimensional system of ferromagnetic nanowires

Abstract

Abstract We investigate the stability features of steady-states of a two-dimensional system of ferromagnetic nanowires. We constitute a system with the finite number of nanowires arranged on the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>(</m:mo><m:msub><m:mover><m:mi>e</m:mi><m:mo>→</m:mo></m:mover><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mover><m:mi>e</m:mi><m:mo>→</m:mo></m:mover><m:mn>2</m:mn></m:msub><m:mo>)</m:mo></m:mrow></m:math> {(\vec{e}_{1},\vec{e}_{2})} plane, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>(</m:mo><m:msub><m:mover><m:mi>e</m:mi><m:mo>→</m:mo></m:mover><m:mn>1</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mover><m:mi>e</m:mi><m:mo>→</m:mo></m:mover><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mover><m:mi>e</m:mi><m:mo>→</m:mo></m:mover><m:mn>3</m:mn></m:msub><m:mo>)</m:mo></m:mrow></m:math> {(\vec{e}_{1},\vec{e}_{2},\vec{e}_{3})} is the canonical basis of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mi>ℝ</m:mi><m:mn>3</m:mn></m:msup></m:math> {\mathbb{R}^{3}} . We consider two cases: in the first case, each nanowire is considered to be of infinite length, whereas in the second case, we deal with finite length nanowires to design the system. In both cases, we establish a sufficient condition under which these steady-states are shown to be exponentially stable.