The　nature　of　the　shock　wave,　governed　by　Burgers　equation,　is　presented　from　viewpoint　of　saddle-node　bifurcations.　First,　the　inviscid　Burgers　equation　is　studied　in　detail,　and　the　solution　of　the　system　with　a　certain　initial　condition　is　obtained.　The　solution　in　vector　form　is　reduced　into　a　Map　in　order　to　investigate　the　stability　and　bifurcation　in　the　system.　It　has　been　proved　that　there　exists　a　thin　spatial　zone　where　a　saddle-node　bifurcation　occurs　as　time　goes　on,　and　the　velocity　of　the　fluid　behaves　as　jumping,　namely,　the　characteristic　of　shock　wave.　Further,　the　period-doubling　bifurcation　is　captured,　that　means　there　exist　multiple　states　as　time　increases,　and　the　complicated　spatio-temporal　pattern　is　formatted.　In　addition　to　above,　the　viscous　Burgers　equation　is　further　studied　to　extend　to　dissipative　systems.　By　traveling　wave　transformation,　the　governing　equation　is　reduced　into　an　ordinary　differential　equation.　Then,　the　instability　or　bifurcation　condition　is　obtained,　and　it　is　proved　that　there　are　three　singular　points　in　the　system　as　the　bifurcation　condition　is　satisfied.　More,　the　velocity　distribution　is　obtained　with　a　certain　initial　condition.　The　results　show　that　the　discontinuity　resulting　from　saddle-node　bifurcation　is　removed　with　the　introduction　of　viscousity,　and　another　kind　of　velocity　change　with　strong　gradient　is　obtained.　However,　the　change　of　velocity　is　continuous　with　sharp　slopes.　As　a　conclusion,　all　results　can　provide　a　fundamental　understanding　of　the　nonlinear　phenomena　relevant　to　shock　wave　and　other　complicated　nonlinear　phenomena,　from　viewpoint　of　nonlinear　dynamics.　©　2012　IEEE.