Sliding fast-slow oscillations are interesting oscillation patterns discovered recently in the Duffing system with frequency switching.Such oscillations have been obtained with a fixed 1:2 low frequency ratio in the p...Sliding fast-slow oscillations are interesting oscillation patterns discovered recently in the Duffing system with frequency switching.Such oscillations have been obtained with a fixed 1:2 low frequency ratio in the previous work.The present paper aims to explore composite fast-slow dynamics when the frequency ratio is variable.As a result,a novel route to composite fast-slow dynamics is obtained.We find that,when presented with variable frequency ratios in a 1:n fashion,the sliding fast-slow oscillations may turn into the ones characterized by the fact that the clusters of large-amplitude oscillations of relaxational type are exhibited in each period of the oscillations,and hence the mixedmode fast-slow oscillations.Depending on whether the transition of the trajectory is from the upper subsystem via the fold bifurcation or not,these interesting oscillations are divided into two classes,both of which are investigated numerically.Our study shows that,when the frequency ratio n is increased from n=3,newly created boundary equilibrium bifurcation points may appear on the original sliding boundary line,which is divided into smaller parts,showing sliding and downward crossing dynamical characteristics.This is the root cause of the clusters,showing large-amplitude oscillations of relaxational type,resulting in the formation of mixed-mode fast-slow oscillations.Thus,a novel route to composite fast-slow dynamics by frequency switching is explained.Besides,the effects of the forcing on the mixed-mode fast-slow oscillations are explored.The magnitude of the forcing frequency may have some effects on the number of large-amplitude oscillations in the clusters.The magnitude of the forcing amplitude determines whether the fast-slow characteristics can be produced.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.12272150,12072132,12372093)。
文摘Sliding fast-slow oscillations are interesting oscillation patterns discovered recently in the Duffing system with frequency switching.Such oscillations have been obtained with a fixed 1:2 low frequency ratio in the previous work.The present paper aims to explore composite fast-slow dynamics when the frequency ratio is variable.As a result,a novel route to composite fast-slow dynamics is obtained.We find that,when presented with variable frequency ratios in a 1:n fashion,the sliding fast-slow oscillations may turn into the ones characterized by the fact that the clusters of large-amplitude oscillations of relaxational type are exhibited in each period of the oscillations,and hence the mixedmode fast-slow oscillations.Depending on whether the transition of the trajectory is from the upper subsystem via the fold bifurcation or not,these interesting oscillations are divided into two classes,both of which are investigated numerically.Our study shows that,when the frequency ratio n is increased from n=3,newly created boundary equilibrium bifurcation points may appear on the original sliding boundary line,which is divided into smaller parts,showing sliding and downward crossing dynamical characteristics.This is the root cause of the clusters,showing large-amplitude oscillations of relaxational type,resulting in the formation of mixed-mode fast-slow oscillations.Thus,a novel route to composite fast-slow dynamics by frequency switching is explained.Besides,the effects of the forcing on the mixed-mode fast-slow oscillations are explored.The magnitude of the forcing frequency may have some effects on the number of large-amplitude oscillations in the clusters.The magnitude of the forcing amplitude determines whether the fast-slow characteristics can be produced.
