Echo　state　networks　(ESNs)　have　wide　applications　in　chaotic　time　series　prediction.　In　the　ESN,　if　the　smallest　singular　value　of　the　reservoir　state　matrix　is　infinitesimal,　the　ill-posed　problem　might　occur　during　the　training　process.　To　overcome　this　problem,　an　adaptive　Levenberg　Marquardt　(LM)　algorithm-based　echo　state　network　(ALM-ESN)　is　developed.　In　the　developed　ALM-ESN,　a　new　adaptive　damping　term　is　introduced　into　the　LM　algorithm.　The　adaptive　factor　is　amended　by　the　trust　region　technique,　furthermore,　convergence　analysis,　and　stability　analysis　are　performed.　Moreover,　to　make　the　inputs　fall　within　the　active　region　of　the　activation　function　and　improve　the　learning　speed,　a　weight　initialization　method　using　linear　algebra　is　deployed　to　determine　the　appropriate　input　weights　and　reservoir　weights.　Simulations　demonstrate　that　the　ALM-ESN　can　overcome　the　ill-posed　problem.　Furthermore,　it　exhibits　better　performance　and　robustness　for　chaotic　time　series　prediction　than　some　other　existing　methods.