Study of Chaotic and Regular Modes of the Fractional Dynamic System of Selkov

The paper investigates the dynamic modes of the Sel’kov fractional self-oscillating system in order to simulate the interaction of cracks. The spectra of the maximum Lyapunov exponents, constructed depending on the parameters of the dynamic system, are used as a research tool. The maximum Lyapunov exponents were constructed according to the Benettin-Wolf algorithm. It is shown that the existence of chaotic regimes is possible. In particular, the spectrum of the maximum Lyapunov exponents of the order of the fractional derivative contains positive values, which indicates the presence of a chaotic regime. Phase trajectories were also constructed to confirm these results. It was also confirmed that the orders of fractional derivatives are responsible for dissipation in the system under consideration.

Pipeline FPGA-Based Implementations of ANNs for the Prediction of up to 600-Steps-Ahead of Chaotic Time Series

Chaotic time series prediction can be performed by applying different architectures of artificial neural networks (ANNs) that can be implemented on field-programmable gate arrays (FPGAs). However, the main challenges are the reduction of hardware resources to develop faster ANNs and the prediction capabilities for large horizons. In this manner, the contribution is devoted to introduce pipeline architectures in which some registers are placed between combinational blocks to divide the logic into shorter stages that can run with a faster clock. The cases of study are the multilayer perceptron (MLP), nonlinear autoregressive with exogenous input (NARX), and echo state network (ESN). In addition, another contribution is devoted to introduce the application of the decimation technique to extend the prediction horizon of the ANNs from 12 to 600-steps-ahead. The prediction capabilities of the MLP, NARX and ESN are compared by using eight chaotic time series with different maximum Lyapunov exponents. The pipeline FPGA-based implementations show that the ESN with a reservoir of at least 30 neurons guarantees a large prediction horizon of 600-steps-ahead. Another important advantage of the ESN is that its FPGA-based implementation can be performed by reusing one neuron, thus requiring the lowest quantity of hardware resources.

Codimension-2 Bifurcation Analysis and Control of a Discrete Mosquito Model with a Proportional Release Rate of Sterile Mosquitoes

This paper concerns a discrete wild and sterile mosquito model with a proportional release rate of sterile mosquitoes. It is shown that the discrete model undergoes codimension-2 bifurcations with 1 : 2, 1 : 3, and 1 : 4 strong resonances by applying the bifurcation theory. Some numerical simulations, including codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams, and phase portraits, are also presented to illustrate the validity of theoretical results and display the complex dynamical behaviors. Moreover, two control strategies are applied to the model.

Bifurcation Analysis of a Discrete-Time Two-Species Model

We study the local dynamics and bifurcation analysis of a discrete-time modified Nicholson–Bailey model in the closed first quadrant R+2. It is proved that model has two boundary equilibria: O0,0,Aζ1−1/ζ2,0, and a unique positive equilibrium Brer/er−1,r under certain parametric conditions. We study the local dynamics along their topological types by imposing method of Linearization. It is proved that fold bifurcation occurs about the boundary equilibria: O0,0,Aζ1−1/ζ2,0. It is also proved that model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium Brer/er−1,r and meanwhile stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasi-periodic oscillations between host and parasitoid populations. Some simulations are presented to verify theoretical results. Finally, bifurcation diagrams and corresponding maximum Lyapunov exponents are presented for the under consideration model.

QUANTIFYING SENSITIVE DEPENDENCE OF INITIAL CONDITION USING STRUCTURAL EQUATION MODELING

Human systems display sensitive dependence of initial condition. That is, even though two individuals may be similar in most regards, small differences between these individuals may have far reaching consequences later in life. In dynamical systems analysis, this sort of behavior is quantified with maximum Lyapunov exponents. These exponents quantify the degree to which small differences in initial condition between two systems affect trajectories of these systems later in time. Current methods for estimating maximum Lyapunov exponents are sensitive to noise and this sensitivity leads to estimation errors when researchers attempt to estimate these exponents on data obtained from human participants. Additionally, most current methods only allow for maximum Lyapunov exponent estimation using univariate time series. In this presentation, we present a method for using structural equation modeling for estimating latent maximum Lyapunov exponents from noisy multivariate time series and discuss applications of this method for analyzing human generated data.

The study of chaotic and regular regimes of the fractal oscillators FitzHugh-Nagumo

In this paper we study the conditions for the existence of chaotic and regular oscillatory regimes of the hereditary oscillator FitzHugh-Nagumo (FHN), a mathematical model for the propagation of a nerve impulse in a membrane. To achieve this goal, using the non-local explicit finite-difference scheme and Wolf’s algorithm with the Gram-Schmidt orthogonalization procedure and the spectra of the maximum Lyapunov exponents were also constructed depending on the values of the control parameters of the model of the FHN. The results of the calculations showed that there are spectra of maximum Lyapunov exponents both with positive values and with negative values. The results of the calculations were also confirmed with the help of oscillograms and phase trajectories, which indicates the possibility of the existence of both chaotic attractors and limit cycles.

Research dynamic modes of stick-slip effect with the account of hereditarity

In study with the help of the spectrum of maximal Lyapunov exponents, dynamic regimes of the stick-slip effect were studied with allowance effect of hereditarity. Spectrum of the Lyapunov exponents were constructed using the Wolff algorithm with Gram-Schmidt orthogonalization depending on the values of the control parametersfriction and adhesion coefficients, as well as fractional index values, which determine the heredity of the dynamical system under consideration. The existence of an area of positive values of the maximum Lyapunov exponents is shown, which indicates the presence of chaotic regimes. Oscillograms and phase trajectories are constructed.

Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response

The dynamic behavior of a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response is examined. We algebraically show that the system undergoes a bifurcation (flip or Neimark-Sacker) in the interior of R+2. Numerical simulations are presented not only to validate analytical results but also to show chaotic behaviors which include bifurcations, phase portraits, period 2, 4, 6, 8, 10, and 20 orbits, invariant closed cycle, and attracting chaotic sets. Furthermore, we compute numerically maximum Lyapunov exponents and fractal dimension to justify the chaotic behaviors of the system. Finally, a strategy of feedback control is applied to stabilize chaos existing in the system.

Associations Between Step Duration Variability and Inertial Measurement Unit Derived Gait Characteristics

Inertial measurement units (IMU) provide a convenient tool for gait stability assessment. However, it is unclear how various gait characteristics relate to each other and whether gait characteristics can be obtained from resultant acceleration. Therefore, step duration variability was measured in treadmill walking from 39 young ambulant volunteers (age 24.2 [± 2.5] y; height 1.79 [± 0.09] m; mass 71.6 [± 12.0] kg) using motion capture. Accelerations and gyrations were simultaneously recorded with an IMU. Harmonic ratio, maximum Lyapunov exponents, and multiscale sample entropy (MSE) were calculated. Step duration variability was positively associated with MSE with coarseness levels = 3–6 (r = –.33 to –.42, P ≤ .045). Harmonic ratio and MSE with all coarseness levels were negatively associated (r = –.45 to –.57, P ≤ .004). The MSE with coarseness level = 2 was negatively associated with short-term maximum Lyapunov exponents (r = –.32, P = .047). The agreement between resultant and vertical acceleration derived gait characteristics was excellent (ICC = 0.97–0.99). In conclusion, MSE with varying coarseness levels was associated with the other gait characteristics evaluated in the study. Resultant and vertical acceleration derived results had excellent agreement, which suggests that resultant acceleration is a viable alternative to considering the acceleration dimensions independently.