Since we know that the Lyapunov exponent is the exponential measure of the separation between infinitesimally close phase-space trajectories. One positive exponent specifies the presence of chaos in the solutions of the dynamical system. When there are two positive exponents then the hyper-chaos can be observed in the system. Now, is it possible for a system to have more than two positive Lyapunov exponents? If yes, how that system will behave?
Thanks in advance for your help in this regard.
I appreciate you asking this good question. As I know, a hyperchaotic system has at least two positive LEs. If a system has three or more positive LEs then it will be also a hyperchaotic system.
Regarding the behavior of the system, it depends on the dimensions and nature of nonlinear terms.
Of course it’s possible, A system of n degrees of freedom, for instance described by n ordinary differential equations, has n Lyapunov exponents, that define the way perturbations propagate along the n directions; as well as surface expopnents, volume exponents and so on.
What is most commonly used to characterize chaos is the Lyapunov expopent defined by ||δX(t)||~exp(λt)||δX(0)||, where the vectors have n components.
So just looking at the sign of some of the Lyapunov exponents isn’t useful.
A Hamiltonian chaotic system has positive and negative Lyapunov expopents, whose sum vanishes, since the volume remains constant, during the flow.
Why not? And shift focus to MLE and hyperplanes of bifurcation. Then recall, the pursuit remains...invariant structures.
Literature provides lot of answers concerning this issue. Clearly with more than two positive Lyapunov exponents, such dynamical systems are still called hyperchaotic. Just that its dimension will be large and the degree of disorder of the system will be high. Time delay systems are good examples of systems with more than two positive Lyapunov exponents. More reading can be found in the book of M. Lakshmanan: Book Dynamics of Nonlinear Time-Delay Systems
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