Unstable Periodic Orbits

In a mathematical space whose coordinates represent the state of adynamical system (i.e., a state space), periodic orbits are the set ofequilibrium states. If all of the periodic orbits in this abstractdynamical landscape are unstable, the system’s temporal evolution willnever settle down to any one of them. Instead, system’s behaviorwanders incessantly in a sequence of close approaches to these orbits.The more unstable an orbit, the less time the system spends near it.Unstable Periodic Orbits (UPOs) form the “skeleton” of nonlineardynamics, and one can build a model of a system by counting andcharacterizing its UPOs in a hierarchy of orbits with increasingperiodicity. Model’s accuracy can be improved by progressivelly addinglonger-period orbits to the hierarchy. The dynamical landscape can thenbe tesselated into regionsof the state space centered around these UPOs. Orbit locations andstability can also offer short-term predictions for the system’s futurestates. This type of predictive model can be used for parametriccontrol of nonlinear systems, whether they are chaotic or not. However,rigorous identification of UPOs from noisy experimental data is adifficult task.

There’s a straightforward methodfor identification of UPOs which relies on the recurrence of patternsin state space, though that’s a very rare event (i.e., a staterepeatedly returning near an orbit) in the reality of short andnonstationary datasets. There’s another method based on a localdynamics data transformation, which acts as a dynamical lens so thatthe new datasets are concentrated about distinct UPOs, helping tooffset the usual scarcity of trajectories near UPOs. With theadditional ability to identify complex higher period orbits by usingonly fragments of trajectories near those orbits, identification ofUPOs was successfully achieved in various experimental settings,including epileptiform activities from the human cortex.

Tracking “parameter” changes from the inherently nonstationary dataof, e.g., neurological systems, with UPOs has also been accomplished,as this is a strong requirement for UPO-based control of nonlinearsystems. Furthermore, this tracking could be used to detect changes insystem state due to intrinsic “parameter” variations (e.g., thetransition to epileptic seizures), extrinsic effects (e.g., due toelectromagnetic fields), or even perceptual discrimination.

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