Chaos is complex, unpredictable behavior in deterministic systems with sensitivity to initial conditions, topological mixing, and dense periodic orbits, appearing random . However, for chaotic scenarios involving memory, anomalous diffusion, multifractal behavior, and noise ; fractional systems, described by fractional-order differential equations, are preferred . Fractional calculus extends traditional calculus by allowing derivatives and integrals of non-integer orders . Fractional systems have shown to accurately model various phenomena such as anomalous diffusion viscoelasticity, and fractal behavior. Additionally, exploring fractional-order chaotic systems has unveiled hidden attractors, prompting studies of systems with no or stable equilibrium points displaying concealed chaotic attractors. Various fractional order chaotic maps have already been studied in the literature, such as the logistic map, sine map, Ikeda map, and others.
Previously, both commensurate (fractional orders are equal) and incommensurate (fractional orders are not equal) cases have been studied by using a fourth-order RungeKutta algorithm and the algorithm of Petrá. However, this study focused on commensurate fractional orders to use the Grünwald-Letnikov characterization of fractional derivatives. In our work, we assume that accurately modeling the complex dynamics of the human gait can be achieved using fractional order chaotic systems that incorporate memory effects and long-range dependence. Indeed, human gait exhibits oscillatory behavior. Apart from considering joint mechanics, the oscillating motion of the body’s center of mass (CoM) and the associated ground reaction forces (GRFs) are connected to a form of overall stiffness. Fractional derivatives, like Caputo or Grnwald-Letnikov characterizations, capture the system’s intricate variability by considering memory and non-local interactions . Adjusting the fractional derivative range controls the influence (bigger, or smaller) of past states. The relationship between fractional derivatives and chaoticity in gait analysis is complex due to physiological, neural, and environmental factors. Among the chaotic systems, the Rössler oscillator, first discovered by Otto Rössler, stands as a canonical example, enthralling the scientific and mathematical communities with its enigmatic properties. The Rössler oscillator, known for its complex dynamics, is a valuable model for studying oscillations, chaos, and nonlinear dynamics. Fractional versions add complexity and insight into chaotic behavior. Extending knowledge from fractional-order chaotic oscillators to human gait analysis, a novel Particularly-Shaped Adaptive Oscillator (PSAO) in exoskeletons synchronizes with users’ gait using joint angles, avoiding foot sensors. PSAO, tested in neuromuscular walking simulations, lowers metabolic walking costs for specific assistance modes and influences the walk ratio, which is crucial for minimizing metabolic costs during walking. This research aids in more efficient and synchronized exoskeleton-assisted walking.
In conclusion, fractional order chaotic systems offer a powerful framework for modeling complex dynamics, with potential applications in various fields, including the human gait system.