When chaos meets hyperchaos: 4D Rössler model

• The coexistence of chaos and hyperchaos in the 4D Rössler system is proved via Computer-Assisted Proofs techniques.
• A global hyperchaotic invariant set is present giving rise to long hyperchaotic transient behaviors.
• The long transient behaviors make difficult in numerical simulations to distinguish chaos from hyperchaos in some situations.

Chaotic behavior is a common feature of nonlinear dynamics, as well as hyperchaos in high-dimensional systems. In numerical simulations of these systems it is quite difficult to distinguish one from another behavior in some situations, as the results are frequently quite “noisy”. We show that in such systems a global hyperchaotic invariant set is present giving rise to long hyperchaotic transient behaviors. This fact provides a mechanism for these noisy results. The coexistence of chaos and hyperchaos is proved via Computer-Assisted Proofs techniques.