Deriving main rhythms of the human cardiovascular system from the heartbeat time series and detecting their synchronization

We demonstrate a possibility of determining the instantaneous phases and instantaneous frequencies of the main rhythmic processes governing the cardiovascular dynamics in humans from heart rate variability data with the methods using bandpass filtration, empirical mode decomposition and wavelet transform. For the cases of spontaneous respiration and paced respiration with a fixed frequency we investigate synchronization between the rhythms of the cardiovascular system analyzing univariate data in the form of the heartbeat time series. It is shown that the main heart rhythm and the rhythm of slow regulation of blood pressure with fundamental frequency close to 0.1 Hz can be synchronized with respiration.     

Nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions

This paper deals with the nonlinear dynamics and synchronization of coupled electromechanical systems with multiple functions, described by an electrical Duffing oscillator magnetically coupled to linear mechanical oscillators. Firstly, the amplitudes of the sub- and super-harmonic oscillations for the resonant states are obtained and discussed using the multiple time scales method. The equations of motion are solved numerically using the Runge–Kutta algorithm. It is found that chaotic and periodic orbit coexist in the electromechanical system depending on the set of initial conditions. Secondly, the problem of synchronization dynamics of two coupled electromechanical systems both in the regular and chaotic states is also investigated, and estimation of the coupling coefficient under which synchronization and no-synchronization take place is made.     

Synchronization analysis through coupling mechanism in realistic neural models

Synchronization which relates to the system’s stability is important to many engineering and neural applications. In this paper, an attempt has been made to implement response synchronization using coupling mechanism for a class of nonlinear neural systems. We propose an OPCL (open-plus-closed-loop) coupling method to investigate the synchronization state of driver-response neural systems, and to understand how the behavior of these coupled systems depend on their inner dynamics. We have investigated a general method of coupling for generalized synchronization (GS) in 3D modified spiking and bursting Morris–Lecar (M-L) neural models. We have also presented the synchronized behavior of a network of four bursting Hindmarsh–Rose (H-R) neural oscillators using a bidirectional coupling mechanism. We can extend the coupling scheme to a network of N neural oscillators to reach the desired synchronous state. To make the investigations more promising, we consider another coupling method to a network of H-R oscillators using bidirectional ring type connections and present the effectiveness of the coupling scheme.     

Generalizations of the concept of marginal synchronization of chaos

Generalizations of the concept of marginal synchronization between chaotic systems, i.e. synchronization with zero largest conditional Lyapunov exponent, are considered. Generalized marginal synchronization in drive–response systems is defined, for which the function between points of attractors of different systems is given up to a constant. Auxiliary system approach is shown to be able to detect this synchronization. Marginal synchronization in mutually coupled systems which can be viewed as drive–response systems with the response system influencing the drive system dynamics is also considered, and an example from solid-state physics is analyzed. Stability of these kinds of synchronization against changes of system parameters and noise is investigated. In drive–response systems generalized marginal synchronization is shown to be rather sensitive to the changes of parameters and may disappear either due to the loss of stability of the response system, or as a result of the blowout bifurcation. Nonlinear coupling of the drive system to the response system can stabilize marginal synchronization.     

Energy–vector method in mechanical oscillations

The new conception of the space, and its construction are introduced. The space allows for a geometrical view on energy changes in mechanical vibrating systems. It posses all the possibilities of the system dynamics analysis that are used in the traditional phase space but it also shows an amount of the energy accumulated in the system, an energy changes, flow, dissipation, synchronization, an energy attractors. Thus it increases our knowledge and intuition on energy changes in vibrating systems. Dynamics of the chaotic system is studied using the energy space. New types of maps are introduced.     

A practical projective synchronization approach for uncertain chaotic systems with dead-zone input

In this paper, a practical projective synchronization problem of master–slave chaotic systems is investigated. More specifically, a fuzzy adaptive slave chaotic system subject to dead-zone nonlinearity in the input channel is proposed using only the measurable output of the master system thanks to a suitable observer. A practical projective synchronization between the master and slave systems is achieved by an adequate fuzzy adaptive control system. The underlying parameter adaptation design as well as stability analysis are carried out using a Lyapunov based approach. Unlike the previous works, in the design of the proposed synchronization scheme, we do not require to know the uncertainties function and that the dynamics of the original synchronization error are strictly positive real (SPR). In fact, herein, the uncertainties function is estimated by a fuzzy adaptive system and the dynamics of the original synchronization error are augmented by a low pass filter designed to satisfy the SPR condition. Simulation results are given to show the effectiveness of the proposed practical projective synchronization scheme.     

A Mathematical model for carbon dioxide exchange during mechanical ventilation with Tracheal Gas Insufflation (TGI)

A mathematical model is presented to study the effect of Tracheal Gas Insufflation (TGI) on the alveolar ventilation during passive, mechanical ventilation. Equations are derive that relate the physiologic dead space and alveolar ventilation to the physiologic parameters of the patient (compliance, inspiratory and expiratory resistances, anatomical dead space, cardiac output, and shunting), the ventilator settings (frequency, inspiratory time fraction, and pressure waveform), and the catheter settings (flow rate and flow duration). It is shown that optimal alveolar ventilation exists for certain frequencies of ventilation and these frequencies are dependent on the catheter flow.     

Synchronized states in a ring of four mutually coupled self-sustained electromechanical devices

In this paper, we study the dynamics of a ring of four mutually coupled identical self-sustained electromechanical devices both in their autonomous and nonautonomous chaotic states. The transition boundaries that can occur between instability and complete synchronization states when the coupling strength varies are derived. Numerical simulations are then performed to support the accuracy of the analytical approach.     

Adaptive synchronization of T–S fuzzy chaotic systems with unknown parameters

This paper presents a fuzzy model-based adaptive approach for synchronization of chaotic systems which consist of the drive and response systems. Takagi–Sugeno (T–S) fuzzy model is employed to represent the chaotic drive and response systems. Since the parameters of the drive system are assumed unknown, we design the response system that estimates the parameters of the drive system by adaptive strategy. The adaptive law is derived to estimate the unknown parameters and its stability is guaranteed by Lyapunov stability theory. In addition, the controller in the response system contains two parts: one part that can stabilize the synchronization error dynamics and the other part that estimates the unknown parameters. Numerical examples, including Duffing oscillator and Lorenz attractor, are given to demonstrate the validity of the proposed adaptive synchronization approach.     

On synchronization of a forced delay dynamical system via the Galerkin approximation

A forced scalar delay dynamical system is analyzed from the perspective of bifurcation and synchronization. In general first order differential equations do not exhibit chaos, but introduction of a delay feedback makes the system infinite dimensional and shows chaoticity. In order to study the dynamics of such a system, Galerkin projection technique is used to obtain a finite dimensional set of ordinary differential equations from the delay differential equation. We compare the results of simulation with those obtained from direct numerical simulation of the delay equation to ascertain the accuracy of the truncation process in the Galerkin approximation. We have considered two cases, one with five and the other with eight shape functions. Next we study two types of synchronization by considering coupling of the above derived equations with a forced dynamical system without delay. Our analysis shows that it is possible to have synchronization between two such systems. It has been shown that the chaotic system with delay feedback can drive the system without delay to achieve synchronization and the opposite case is also equally valid. This is confirmed by the evaluation of the conditional Lyapunov exponents of the systems.     

Generalized synchronization of strictly different systems: Partial-state synchrony

Generalized synchronization (GS) occurs when the states of one system, through a functional mapping are equal to states of another. Since for many physical systems only some state variables are observable, it seems convenient to extend the theoretical framework of synchronization to consider such situations. In this contribution, we investigate two variants of GS which appear between strictly different chaotic systems. We consider that for both the drive and response systems only one observable is available. For the case when both systems can be taken to a complete triangular form, a GS can be achieved where the functional mapping between drive and response is found directly from their Lie-algebra based transformations. Then, for systems that have dynamics associated to uncontrolled and unobservable states, called internal dynamics, where only a partial triangular form is possible via coordinate transformations, for this situation, a GS is achieved for which the coordinate transformations describe the functional mapping of only a few state variables. As such, we propose definitions for complete and partial-state GS. These particular forms of GS are illustrated with numerical simulations of well-known chaotic benchmark systems.     

Adaptive numerical components for PDE-based simulations

Numerical simulations based on nonlinear partial differential equations (PDEs) using Newton-based methods require the solution of large, sparse linear systems of equations at each nonlinear iteration. Typically in large-scale parallel simulations such linear systems are solved by using preconditioned Krylov methods. In many cases, especially in time-dependent problems, the attributes of the linear systems can change throughout the stimulation, potentially leading to varying times for solving the linear systems during different nonlinear iterations. We present an approach to characterizing the nonlinear and linear system solution and using the resulting application performance information to dynamically select linear solver methods, with the goal of reducing the total time to solution. We discuss the effect of these adaptive heuristics on fluid dynamics and radiation transport codes. We also introduce general component infrastructure to support dynamic algorithm selection and adaptation in applications involving the solution of nonlinear PDEs. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chaos control for Willamowski–Rössler model of chemical reactions

Real systems evolving towards complex state encounter chaotic behavior. This behavior is very important in chemical processes or in biological structures because it defines the direction of the evolution of the system. From this point of view, the capability of deliberate control of these phenomena has a great practical impact despite the fact that it is very difficult; this is the reason why theoretical models are useful in these situations. In order to obtain chaos control in chemical reactions, the analysis of the dynamics of Willamowski–Rössler system involving the synchronization of two Minimal Willamowski–Rössler (MWR) systems based on the adaptive feedback method of control is presented in this work. As opposed to previous studies where in order to obtain synchronization 3 controllers were used, implying from a practical point of view the control of the concentrations of three chemical species, in this study we showed that the use of just one is sufficient which in practice is important as controlling the concentration of a single chemical species would be much easier. We also showed that the transient time until synchronization depends on initial conditions of two systems, the strength and number of the controllers and we attempted to identify the best conditions for a practical synchronization.     

Synchronization and adaptive synchronization of nuclear spin generator system

This study addresses the synchronization and adaptive synchronization problems of nuclear spin generator (NSG) systems with unknown system parameters. We show that the NSG system can be synchronized by using drive-response systems. Adaptive control law is applied to achieve the state synchronization of two identical NSG systems. Lyapunov direct method of stability is used to prove the asymptotic stability of solutions for the error dynamical system. Numerical simulation is used to show the effectiveness of the proposed control schemes.     

分数阶Duffing混沌系统的动力性态及其由单一主动控制的混沌同步

随着物理与技术的深入研究,分数阶非线性系统的动力性态及其分数阶混沌系统的同步成为研究的焦点.研究了分数阶Duffing系统的动力性态包括混沌性质,并且由分数阶非线性稳定性准则得到了分数阶非自治系统的混沌同步.特别地,研究了由单一主动控制的分数阶Duffing系统的同步.相应的数值结果演示了方法的有效性.     

Linear and nonlinear mathematical models for noninvasive ventilation

Two mathematical models are proposed for passive, noninvasive ventilation. Both models take the form of coupled ordinary differential equations that describe the volume in a single compartment lung. One model is linear and the other nonlinear; both models are derived from basic pressure balances in the lung-ventilator system. These models are also compared to a physical model using a test lung. Both the physical and mathematical models exhibit instabilities that appear to have important clinical implications. The simulations from these models, and the forms of their governing equations, suggest that the presence of an airway leak proximal to the airway opening during pressure support noninvasive ventilation may render this mode of ventilation dynamically unstable. The mathematical models are extended to incorporate a special type of nonpassive ventilation where the total cycle times of the ventilator depend on the inspiratory phases of these cycles.     

Optimal synchronization of two different in‐commensurate fractional‐order chaotic systems with fractional cost function

This article investigates the optimal synchronization of two different fractional‐order chaotic systems with two kinds of cost function. We use calculus of variations for minimizing cost function subject to synchronization error dynamics. We introduce optimal control problem to solve fractional Euler–Lagrange equations. Optimal control signal and minimum time of synchronization are obtained by proposed method. Examples show the optimal synchronization of two different systems with two different cost functions. First, we use an ordinary integer cost function then we use a fractional‐order cost function and comparing the results. Finally, we suggest a cost function which has the optimal solution of this problem, and we can extend this solution to solve other synchronization problems. © 2016 Wiley Periodicals, Inc. Complexity 21: 401–416, 2016     

Antiphase synchronization of chaos by noncontinuous coupling: two impacting oscillators

We show that two identical chaotic oscillators can evolve in antiphase synchronization regime when noncontinuous coupling between them is introduced. As an example, we consider dynamics of two mechanical oscillators coupled by impacts.