My inputs are:
Since the sum of all Lyapunov exponents is zero, it means the system is conservative (i.e., there is no dissipation). Thus, a volume element of the phase space will stay the same along a trajectory. That is, some sort of steady state mode.
Also since at least one exponent is zero, it means that the trajectory does not converge to a single point.
But also in most cases, if the largest Lyapunov exponent is zero, one is usually faced with periodic motion.
Dear Jay
Lyapunov exponent is quantitative tool which indicate how the dynamical system shrink or expand in the phase space. If your system is of 4th order, to the best of my knowledge, you can only have three of them been positive. ie LE1=0, LE2=0, LE3=0. Meaning that the system is in a 3D torus state. If only two are positive, then the system is in 2D torus state. Finally, if only LE1=0, then the system évolve periodically. But in all cases, at least one of the Lyapunov exponent must be negative, ie the sum of all the Lyapunov exponent must be negative meaning that the dynamical system is bounded or dissipative.
@Vikash Pandey
Thank you, Sir four answer.
Sir, if possible can you please attach some references or example.
@Theophile Fozin
Thank you Sir for your answer.
Sir, i have a 4-D system and i am getting all the Lyapunov exponents zero for some value of parameter. Thank you again Sir.
@Leonard A Smith
Thank You Sir for your answer.
Yes Sir, I have a 4-D system, I calculated the Lyapunov exponents whose natures are (0, 0, 0, 0).
@ Jay,
Most of the problems I have worked out myself have at least one non zero Lyapunov exponent. Your case is new to me.
In the field of statistical physics systems with many degrees of freedom, having only zero Lyapunov exponents are very well-known. These include integrable systems, such as harmonic oscillator chains and Toda chains, but also non-integrable ones. Notably the latter class contains systems of single particles moving among polygonal or polyhedral fixed scatterers, or in general dimension D, fixed scatterers with D-1 dimensional flat faces connected by sharp edges. Well-known examples are hard point particles of alternating mass in one dimension and elastically colliding hard polyhedra in e.g. 3 dimensions (equivalent, for N particles to a single particle in 3N dimensional configuration space with properly shaped scatterers). In general these systems are dissipative. In the mathematical literature they are commonly referred to as non-dispersing billiards.
consider the movement of a particle on a closed compact constant negatively curved surface which particle is itself curving to the left say with a constant geodesic curvature A] stricly less than B] equal to C] strictly more than the surface curvature.
case A has nonzero exponents and chaotic behaviour which is equivalent to that of the geodesic flow on the unit tangent space to the surface.
case C is periodic motion equivalent to the circle action rotating unit tangent vectors.
case B is NOT quasiperiodic ,integrable type motion but rather the famous horocyclic motion [ the prbits are horocyles] but all exponents are zero
the expansion/contraction is there but subexponential
this flow has every orbit dense and is uniquely ergodic.
it is a totally new creature.
If the LE value is zero, this implies that the chaotic trajectory is falling down in a cycle. It is better to have at least positive 2 LE values and negative 2 LE values to generate a strange attractor with two unstable fixed points.
Jay Prakash Singh if all the LEs are zero and no other - or + LES is there than.. recalculate LEs with some different parameter and initial condition.
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