For　thermal　transport　in　one-dimensional　(1D)　systems,　recent　studies　have　suggested　that　employing　different　theoretical　models　and　different　numerical　simulations　under　different　system＇s　parameter　regimes　might　lead　to　different　universality　classes　of　the　scaling　exponents.　In　order　to　well　understand　the　universality　class(es),　here　we　perform　a　direct　dynamics　simulation　for　two　archetype　1D　oscillator　systems　with　quite　different　phonon　dispersions　under　various　system＇s　parameters　and　find　that　there　is　a　crossover　between　the　different　universality　classes.　We　show　that　by　varying　anharmonicity　and　temperatures,　the　space-time　scaling　exponents　for　the　systems　with　different　dispersions　can　be　feasibly　tuned　in　different　ways.　The　underlying　picture　is　suggested　to　be　understood　by　phonons　performing　various　kinds　of　continuous-time　random　walks　(in　most　cases,　be　the　Lévy　walks　but　not　always),　probably　induced　by　the　peculiar　phonon　dispersions　along　with　nonlinearity.　The　results　and　suggested　mechanisms　may　provide　insights　into　controlling　the　transport　of　heat　in　some　1D　materials.　Copyright　©　EPLA,　2016.