Hyperchaotic set in continuous chaos–hyperchaos transition

• Hyperchaotic invariant sets can be embedded in chaotic attractors with 1 positive LE.
• Hyperchaotic invariant sets can keep existing during chaos–hyperchaos transition.
• This is a new scenario for the continuous transition from chaos to hyperchaos.
• The above phenomenon may generally exist in piecewise-linear or smooth systems.

Topological horseshoes with two-directional expansion imply invariant sets with two positive Lyapunov exponents (LE), which are recognized as a signature of hyperchaos. However, we find such horseshoes in two piecewise linear systems and one smooth system, which all exhibit chaotic attractors with one positive LE. The three concrete systems are the simple circuit by Tamaševičius et al., the Matsumoto–Chua–Kobayashi (MCK) circuit and the linearly controlled Lorenz system, respectively. Substantial numerical evidence from these systems suggests that a hyperchaotic set can be embedded in a chaotic attractor with one positive LE, and keeps existing while the attractor becomes hyperchaotic from chaotic. This paper presents such a new scenario of the continuous chaos–hyperchaos transition.