Narrow Results

The aim of this paper is to study the stability of the solutions of some uncertain delay (integro) differential equations under different conditions. In this regard, applying the Lyapunov direct method and fixed point technique, some sufficient conditions will be presented for the stability of a linear scalar uncertain delay differential equation. In some other cases that the conditions make it difficult to define a Lyapunov function, the stability problem will be studied only by fixed point method.

Medical cyber-physical system (MCPS) is usually used in hospitals to provide continuous services (e.g., glycemic monitor and alarm) for patients. In this paper, we introduce cloud computing to integrate multiple MCPSs of hospitals to construct medical cyber-physical cloud system (MCPCS). Such complex MCPCS need stability analysis to guarantee collaboration among multiple MCPSs. We treat the MCPCS as a complex network, where each MCPS is considered as a node. Then, we introduce the Lyapunov stability theory for synchronization stability analysis of MCPCS by considering multi-closed-loops. For different network types, the dynamic equations of MCPSs were improved through introducing feedback in the multi-closed-loops. For the problem that different networks cause different delays, we select three networks to solve sufficient conditions for the synchronization stability of MCPCS. Experiments are performed to confirm the efficiency of sufficient conditions and the synchronization stability of MCPCS.

In this paper, the implementation of the Fuzzy Sliding Mode Controller (FSMC) in the inner loop of the cascaded control structure of the DC–DC Boost Converter is presented. On account of nonlinearity and nonminimum phase nature, switched-mode DC–DC converters show a poor response in their dynamic characteristics. In most of the works, the inner loop is served by SMC/FLC, and the outer loop by PI. In this study, the proposed FSMC, which is the combination of SMC and FLC is recommended in the inner current loop which reduces the chattering phenomena and improves the robustness against uncertainties, disturbances and varying circuit parameters with the reaching law. The Lyapunov approach is considered to study the stability of the proposed controller. A comparative analysis is made with the results obtained from the proposed FSM controller, Fuzzy and SMC control. The effectiveness of the FSMC controller is validated by observing its system performance.

We call a chaotic dynamical system pseudo-deterministic when it does not produce numerical, or pseudo-trajectories that stay close, or shadow chaotic true trajectories, even though the model equations are strictly deterministic. In this case, single chaotic trajectories may not be meaningful, and only statistical predictions, at best, could be drawn on the model, like in a stochastic system. The dynamical reason for this behavior is nonhyperbolicity characterized either by tangencies of stable and unstable manifolds or by the presence of periodic orbits embedded in a chaotic invariant set with a different number of unstable directions. We emphasize herewith the latter by studying a low-dimensional discrete-time model in which the phenomenon appears due to a saddle-repeller bifurcation. We also investigate the behavior of the finite-time Lyapunov exponents for the system, which quantifies this type of nonhyperbolicity as a system parameter evolves past a critical value. We argue that the effect of unstable dimension variability is more intense when the invariant chaotic set of the system loses transversal stability through a blowout bifurcation.

Melnikov's method is used to analytically predict the onset of chaotic instability in a rotating body with internal energy dissipation. The model has been found to exhibit chaotic instability when a harmonic disturbance torque is applied to the system for a range of forcing amplitude and frequency. Such a model may be considered to be representative of the dynamical behavior of a number of physical systems such as a spinning spacecraft. In spacecraft, disturbance torques may arise under malfunction of the control system, from an unbalanced rotor, from vibrations in appendages or from orbital variations. Chaotic instabilities arising from such disturbances could introduce uncertainties and irregularities into the motion of the multibody system and consequently could have disastrous effects on its intended operation. A comprehensive stability analysis is performed and regions of nonlinear behavior are identified. Subsequently, the closed form analytical solution for the unperturbed system is obtained in order to identify homoclinic orbits. Melnikov's method is then applied on the system once transformed into Hamiltonian form. The resulting analytical criterion for the onset of chaotic instability is obtained in terms of critical system parameters. The sufficient criterion is shown to be a useful predictor of the phenomenon via comparisons with numerical results. Finally, for the purposes of providing a complete, self-contained investigation of this fundamental system, the control of chaotic instability is demonstated using Lyapunov's method.

Detecting the state of the Duffing oscillator, a type of well-known chaotic oscillator, deeply affects the accuracy of its application. Considering this, the present paper introduced a novel method for detecting the state of the Duffing oscillator. Binary outputs, simple calculation, high precision and fast response time were the main advantages of the phase space trajectory autocorrelation. Also, this study explained the largest Lyapunov exponent as well as a number of other methods commonly employed in detecting the state of the Duffing oscillator. The precision and effectiveness of the method introduced was compared with other well-known state detection methods such as the 0-1 test and the largest Lyapunov exponent.

Definitions of Hausdorff–Lebesgue measure and dimension are introduced. Combination of Hausdorff and Lebesgue ideas are used. Methods for upper and lower estimations of attractor dimensions are developed.

Fractal geometry can adequately represent many complex and irregular objects in nature. The fractal dimension is typically computed by the box-counting procedure. Here I compute the box-counting and the Kaplan-Yorke dimensions of the 14-dimensional models of the Drosophila circadian clock. Clockwork Orange (CWO) is transcriptional repressor of direct target genes that appears to play a key role in controlling the dynamics of the clock. The findings identify these models as strange attractors and highlight the complexity of the time-keeping actions of CWO in light-day cycles. These fractals are high-dimensional counterexamples of the Kaplan-Yorke conjecture that uses the spectrum of the Lyapunov exponents.

In potential games, the best-reply dynamics results in the existence of a cost function such that each player's best-reply set equals the set of minimizers of the potential given by the opponents' strategies. The study of sequential best-reply dynamics dates back to Cournot and, an equilibrium point which is stable under the game's best-reply dynamics is commonly said to be Cournot stable. However, it is exactly the best-reply behavior that we obtain using the Lyapunov notion of stability in game theory. In addition, Lyapunov theory presents several advantages. In this paper, we show that the stability conditions and the equilibrium point properties of Cournot and Lyapunov meet in potential games.

We propose a methodology to obtain the amplitude of a nonlinear differential equation that may not satisfy Lyapunov’s global stability criterion. This theory is applied to the MEMS resonator which has a high-quality factor. The derivative of the Lyapunov function approximated for a finite time and an optimization problem was formulated. The local optima were obtained using the Karush–Kuhn–Tucker conditions, for which the amplitude was analytically formulated. The obtained amplitude, when compared with that by the numerical method, showed the validity of the analytical approximation for a useful range of the nonlinearity, but accurate only at an excitation frequency $\mathrm{\Omega }=0.913$. This methodology will be useful to approximate the damping in a system if one obtains the amplitude from the experimental data near this excitation frequency.

A new viral infection model with cell-to-cell infection and periodic antiretroviral therapy was proposed and analyzed. With the help of defined basic reproduction number ${\Re }_0$, we proved that if ${\Re }_0<1$, then the infection-free steady state is globally attractive. If ${\Re }_0>1$, then the infection is uniformly persistent. Furthermore, the globally asymptotical stability of infection-free steady state has been established for the critical case of ${\Re }_0=1$. Also, the global asymptotic stability of the infection steady state for a special case of the model has been established by applying the method of Lyapunov. Numerical simulations show that increasing the drug efficacy for blocking virus-to-cell infection, cell-to-cell infection and producing non-infectious virus contributes to weakening the severity of viral infection.