The　research　on　existence,　bifurcation,　and　number　of　periodic　solutions　is　closely　related　to　Hilbert＇s　16th　problem.　The　main　goal　of　this　chapter　is　to　investigate　the　nonlinear　dynamic　response　and　periodic　vibration　characteristic　of　a　simply　supported　concave　hexagonal　honeycomb　sandwich　plate　with　negative　Poisson＇s　ratio.　The　plate　is　subjected　to　its　in-plane　and　transverse　excitation.　The　curvilinear　coordinate　frame,　Poincare　map,　and　improved　Melnikov　function　are　proposed　to　detect　the　existence　and　number　of　the　periodic　solutions.　The　theoretical　analyses　indicate　the　existence　of　periodic　orbits　and　can　guarantee　at　most　four　periodic　orbits　under　certain　conditions.　Numerical　simulations　are　performed　to　verify　the　theoretical　results.　The　relative　positons　as　well　as　the　vibration　characteristics　can　also　be　clearly　found　from　the　phase　portraits.　The　periodic　motion　for　the　equation　is　closely　related　to　the　amplitude　modulated　periodic　vibrations　of　the　plate.　The　results　will　provide　theoretical　guidance　to　nonlinear　vibration　control　for　the　metamaterial　honeycomb　sandwich　structures.