Local and Global Stability of Certain Mixed Monotone Fractional Second Order Difference Equation with Quadratic Terms

This paper investigates the local and global character of the unique positive equilibrium of a mixed monotone fractional second-order difference equation with quadratic terms. The corresponding associated map of the equation decreases in the first variable, and it can be either decreasing or increasing in the second variable depending on the corresponding parametric values. We use the theory of monotone maps to study global dynamics. For local stability, we use the center manifold theory in the case of the non-hyperbolic equilibrium point. We show that the observed equation exhibits three types of global behavior characterized by the existence of the unique positive equilibrium, which can be locally stable, non-hyperbolic when there also exist infinitely many non-hyperbolic and stable minimal period-two solutions, and a saddle. Numerical simulations are carried out to better illustrate the results.

Stability of Non-Hyperbolic Equilibrium Point for Polynomial System of Differential Equations

In this paper, a polynomial system of plane differential equations is observed. The stability of the non-hyperbolic equilibrium point was analyzed using normal forms. The nonlinear part of the differential equation system is simplified to the maximum. Two nonlinear transformations were used to simplify first the quadratic and then the cubic part of the system of equations.

Nonlinear Compartment Models with Time-Dependent Parameters

A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C0-topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models.

Dynamical Analysis of a Novel 4-D Hyper-Chaotic System With One Non-Hyperbolic Equilibrium Point and Application in Secure Communication

In this article, a novel hyper-chaotic system has been introduced and its dynamical properties (i.e., phase plots, time series, lyapunov exponents, bifurcation diagrams, equilibrium points, Poincare sections, etc.) have been studied. Also, the novel chaotic systems have been synchronized using novel synchronization technique multi-switching compound difference synchronization and its application have been shown in the field of secure communication. Numerical simulations have been undertaken to validate the efficacy of the synchronization in secure communication.

Complex Dynamical Behaviors in a 3D Simple Chaotic Flow with 3D Stable or 3D Unstable Manifolds of a Single Equilibrium

This paper shows some examples of chaotic systems for the six types of only one hyperbolic equilibrium in changed chameleon-like chaotic system. Two of the six cases have hidden attractors. By adjusting the parameters in the system and controlling the stability of only one equilibrium, we can further obtain chaos with four kinds of conditions: (1) index-0 node; (2) index-3 node; (3) index-0 node foci; (4) index-3 node foci. Based on the method of focus quantities, we study three limit cycles (the outmost and inner cycles are stable, and the intermediate cycle is unstable) bifurcating from an isolated Hopf equilibrium. In addition, one periodic solution can be obtained from a nonisolated zero-Hopf equilibrium. The system may help us in better understanding, revealing an intrinsic relationship of the global dynamical behaviors with the stability of equilibrium point, especially hidden chaotic attractors.

Relationship Between Insulin Sensitivity and β-Cell Secretion in Nondiabetic Subjects with Rheumatoid Arthritis

Objective.In nondiabetic healthy individuals, insulin secretion and sensitivity are linked by a negative feedback loop characterized by a hyperbolic function. We aimed to study the association of traditional insulin resistance (IR) factors with insulin secretion and sensitivity, and to determine whether the hyperbolic equilibrium of this relation is preserved in patients with rheumatoid arthritis (RA).Methods.This was a cross-sectional study encompassing 361 nondiabetic individuals: 151 with RA and 210 controls. Insulin, C-peptide, and IR indices by homeostatic model (HOMA2) were assessed. A multivariable analysis was performed to evaluate the differences in the correlation of traditional IR-related factors with glucose homeostasis molecules, as well as IR indices between patients and controls. Nonlinear regression analysis was used to assess the hyperbolic relation of insulin sensitivity and secretion.Results.HOMA2-IR indices were higher in patients with RA than controls. Hepatic insulin extraction, as assessed by the insulin:C-peptide molar ratio, was lower in patients with RA after multivariable analysis (0.08 ± 0.02 vs 0.14 ± 0.07, p < 0.001). Traditional IR-related factors showed significantly lower adjusted correlation coefficients with IR indices in patients with RA. The association between insulin sensitivity and secretion showed a different hyperbolic relation in patients with RA: the variability explained by the curve was lower in RA (nonlinear r2= 0.845 vs r2= 0.928, p = 0.001) and β coefficients (−0.74, 95% CI −0.77 to −0.70 vs −1.09, 95% CI −1.17 to −1.02, ng/ml, p < 0.001) were different in RA.Conclusion.The traditional factors associated with IR in healthy individuals are less related to IR in patients with RA. Insulin sensitivity and secretion yield a different hyperbolic equilibrium in RA.

A New 7D Hyperchaotic System with Five Positive Lyapunov Exponents Coined

This paper reports the finding of a new seven-dimensional (7D) autonomous hyperchaotic system, which is obtained by coupling a 1D linear system and a 6D hyperchaotic system that is constructed by adding two linear feedback controllers and a nonlinear feedback controller to the Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest is that it has a hyperchaotic attractor with five positive Lyapunov exponents and a unique equilibrium in a large range of parameters. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, power spectrum and Poincaré projections verifies the existence of hyperchaotic and chaotic attractors. Moreover, stability of the hyperbolic equilibrium is analyzed and a complete mathematical characterization for 7D Hopf bifurcation is given. Finally, circuit experiment implements the hyperchaotic attractor of the 7D system, showing very good agreement with the simulation results.