STABILIZING THE PERIODIC ORBITS IN A CHAOTIC MAPPING DESCRIBING THE DISCRETE HEALTH SYSTEMS VIA PREDICTION-BASED CONTROL

In the paper the problem of location and stabilization of unstable periodic orbits (UPOs) in discrete systems is investigated via the prediction-based control (PBC). It involves using the state of the free system one period ahead as reference for the control signal. Two types of control gains are tested, the first requiring the knowledge of the UPO to be stabilized and the second depending only on the actual state of the trajectory. The effectiveness of PBC is demonstrated on a chaotic mapping describing the malignant tumor growth. When the results obtained with the two control laws are compared with each other, it is found that the second variant is qualitatively superior, both in terms of convergence and the number of stabilized UPOs, especially for long-period orbits.

Chaotic Entanglement: Entropy and Geometry

In chaotic entanglement, pairs of interacting classically-chaotic systems are induced into a state of mutual stabilization that can be maintained without external controls and that exhibits several properties consistent with quantum entanglement. In such a state, the chaotic behavior of each system is stabilized onto one of the system’s many unstable periodic orbits (generally located densely on the associated attractor), and the ensuing periodicity of each system is sustained by the symbolic dynamics of its partner system, and vice versa. Notably, chaotic entanglement is an entropy-reversing event: the entropy of each member of an entangled pair decreases to zero when each system collapses onto a given period orbit. In this paper, we discuss the role that entropy plays in chaotic entanglement. We also describe the geometry that arises when pairs of entangled chaotic systems organize into coherent structures that range in complexity from simple tripartite lattices to more involved patterns. We conclude with a discussion of future research directions.

Unstable periodic orbits are faithful biomarker for the onset of epileptic seizure

EEG signals of healthy individuals and epileptic patients, when treated as time series of evolving dynamical systems, are found to display characteristic differences in the behavior of the unstable periodic orbits (UPO), marking the transition from regular periodic variations to self-similar dynamics. The UPO, manifesting as broad resonances in the Fourier power spectra, are quite prominent in their presence in the normal signals and are either absent or considerably weakened with a shift towards lower frequency in the epileptic condition. The weighted average and visibility power computed for the UPO region are found to distinguish epileptic seizure from healthy individuals’ EEG. Remarkably, the unstable periodic motion for healthy ones is well described by damped harmonic motion, the orbits displaying smooth dynamics. In contrast, the epileptic cases show bi-stability and piecewise linear motion for the larger orbits, exhibiting large sudden jumps in the ‘velocity’ (referred to the rate of change of the EEG potentials), characteristically different from the healthy cases, highlighting the efficacy of the UPO as biomarkers. For both the regions, 8-14Hz UPO and 40-45Hz resonance, we used data driven analysis to derive the system dynamics in terms of sinusoidal functions, which reveal the presence of higher harmonics, confirming nonlinearity of the underlying system and leading to quantification of the discernible differences between the healthy and epileptic patients. The gamma wave region in the 40-45Hz range, connecting the conscious and the unconscious states of the brain, reveals well-structured coherence phenomena, in addition to the prominent resonance, which potentially can be used as a biomarker for the epileptic seizure. The wavelet scalogram analysis for both UPO and 40-45Hz region also clearly differentiates the healthy condition from epileptic seizure, confirming the above dynamical picture, depicting the higher harmonic generation, and intermixing of different modes in these two regions of interest.SignificanceUnstable periodic orbits are demonstrated as faithful biomarkers for detecting seizure, being prominently present in the Fourier power spectra of the EEG signals of the healthy individuals and either being absent or significantly suppressed for the epileptic cases, showing distinctly different behavior for the unstable orbits, in the two cases. A phase space study, with EEG potential and its rate of change as coordinate and corresponding velocity, clearly delineates the dynamics in healthy and diseased individuals, demonstrating the absence or weakening of UPO, that can be a reliable bio-signature for the epileptic seizure. The phase-space analysis in the gamma region also shows specific signatures in the form of coherent oscillations and higher harmonic generation, further confirmed through wavelet analysis.

Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

Feedback control modulation for controlling chaotic maps

Based on existing feedback control methods such as OGY and Pyragas, alternative new schemes are proposed for stabilization of unstable periodic orbits of chaotic and hyperchaotic dynamical systems by suitable modulation of a control parameter. Their performances are improved with respect to: (i) robustness, (ii) rate of convergences, (iii) reduction of waiting time, (iv) reduction of noise sensitivity. These features are analytically investigated, the achievements are rigorously proved and supported by numerical simulations. The proposed methods result successful for stabilizing unstable periodic orbits in some classical discrete maps like 1-D logistic and standard 2-D Hénon, but also in the hyperchaotic generalized n-D Hénon-like maps.

Locating and Stabilizing Unstable Periodic Orbits Embedded in the Horseshoe Map

Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.