Chaotic heteroclinic tangles with the degenerate Melnikov function

This work establishes the heteroclinic tangles when the Melnikov function is degenerate. By deriving the higher order Melnikov functions, we obtain an explicit formula of the second-order Melnikov function $$E_1(t_0)$$ , which essentially facilitates the evaluation and application to the concrete differential equations. Two examples manifest this efficient method to characterize chaotic heteroclinic tangle.