This　paper　considers　the　asymptotic　stabilization　problem　of　second-order　linear　time-invariant　(LTI)　autonomous　switched　systems　consisting　of　two　subsystems　with　unstable　focus　equilibrium.　More　precisely,　we　find　a　necessary　and　sufficient　condition　for　the　origin　to　be　asymptotically　stable　under　the　predesigned　switching　law.　The　result　is　obtained　without　looking　for　a　common　Lyapunov　function　or　multiple　Lyapunov　function,　but　studying　the　locus　in　which　the　two　subsystem＇s　vector　fields　are　parallel.　Then　the　＂most　stabilizing＂　switching　laws　are　designed　which　have　translated　the　switched　system　into　a　piecewise　linear　system.　Two　numerical　examples　are　presented　to　show　the　potential　of　the　proposed　techniques.　(c)　2005　Elsevier　Ltd.　All　rights　reserved.