Synchronization of the self-excited pendula suspended on the vertically displacing beam

We consider the synchronization of a number of self-excited identical pendula hanging on the same beam which can move vertically. We identify different synchronous configurations and investigate their stability. An approximate analytical analysis of the energy balance allows to derive the synchronization conditions, phase difference between pendula and explains the observed types of synchronizations.     

Synchronization of pendula rotating in different directions

We study the synchronization of a number of pendula mounted on a horizontal beam which can roll on the parallel surface. Under the driving moment, the pendula rotate in different directions: one of them rotates counterclockwise, the rest rotate clockwise. It has been shown that after a transient different types of phase synchronization between pendula can be observed, despite opposite directions of rotations.     

Arnol′d Diffusion in a Pendulum Lattice

The main model studied in this paper is a lattice of pendula with a nearest‐neighbor coupling. If the coupling is weak, then the system is near‐integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain localized type, the neighboring pendula can exchange energy. In fact, the energy can be transferred between the pendula in any prescribed way. © 2014 Wiley Periodicals, Inc.     

Synchronous motion of two vertically excited planar elastic pendula

The dynamics of two planar elastic pendula mounted on the horizontally excited platform have been studied. We give evidence that the pendula can exhibit synchronous oscillatory and rotation motion and show that stable in-phase and anti-phase synchronous states always co-exist. The complete bifurcational scenario leading from synchronous to asynchronous motion is shown. We argue that our results are robust as they exist in the wide range of the system parameters.     

Theoretical aspects of synchronization in transverse galloping aeroelastic instability

We investigate, for the first time to the best of our knowledge, theoretical aspects of synchronization in transverse galloping aeroelastic instability. The current study is a generalization of previous studies that considered the dynamics of a single-cylinder, and therefore, precluded the option to study synchronization. Here, we consider both the deterministic and stochastic dynamics of a system comprising two weakly coupled cylinders, which are attached to the ground with linear springs and dashpots, and are immersed in a high velocity airstream. We derive the conditions for the instability threshold. We give a detailed and simplified procedure to compute the amplitudes, phase differences, and frequencies of the synchronized solutions. We calculate quantitative measures of the amplitude and phase noises, including an explicit calculation of the phase noise reduction due to synchronization, which can enhance the performance of transverse galloping-based energy harvesters. Furthermore, we provide simple mappings for the amplitudes and phase difference dynamics, which we show to be highly useful for understanding both the deterministic and the stochastic dynamics of the amplitudes and the phase difference dynamics from geometric point of view.     

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.     

Non-integrability of flail triple pendulum

We consider a special type of triple pendulum with two pendula attached to end mass of another one. Although we consider this system in the absence of the gravity, a quick analysis of of Poincaré cross sections shows that it is not integrable. We give an analytic proof of this fact analysing properties the of differential Galois group of variational equations along certain particular solutions of the system.     

Reduced-order synchronization of uncertain chaotic systems via adaptive control

We consider the coupling of two uncertain dynamical systems with different orders using an adaptive feedback linearization controller to achieve reduced-order synchronization between the two systems. Reduced-order synchronization is the problem of synchronization of a slave system with projection of a master system. The synchronization scheme is an exponential linearizing-like controller and a state/uncertainty estimator. As an illustrative example, we show that the dynamical evolution of a second-order driven oscillator can be synchronized with the canonical projection of a fourth-order chaotic system. Simulation results indicated that the proposed control scheme can significantly improve the synchronousness performance. These promising results justify the usefulness of the proposed output feedback controller in the application of secure communication.     

Observer-based chaotic synchronization in the presence of unknown inputs

This paper deals with the problem of synchronization of chaotic dynamical systems. We consider a drive-response type of synchronization via a scalar transmitted signal. Unlike most works we consider the presence of some unknown inputs in the drive system and that no knowledge about their nature is available. A reduced-order observer-based response system is designed to synchronize with the missing states. We show that under some assumptions the synchronization is exponentially achieved. The efficiency of our method is confirmed by numerical simulations of two well-known chaotic systems: Chua’s circuit and Lur’e system.     

Synchronization slaved by partial-states in lattices of non-autonomous coupled Lorenz equation

In this paper, we consider the synchronization between any two solutions and any two components of solutions for lattices of partial-stated coupled non-autonomous identical Lorenz equation. We give the boundedness of solutions with an absorbing set. Then, we show that the synchronization of solutions can be slaved by the coefficient in x or y-component. The eigenvalues of coupled operators play an essential role.     

Efficient synchronization of structurally adaptive coupled Hindmarsh–Rose neurons

The use of spikes to carry information between brain areas implies complete or partial synchronization of the neurons involved. The degree of synchronization reached by two coupled systems and the energy cost of maintaining their synchronized behavior is highly dependent on the nature of the systems. For non-identical systems the maintenance of a synchronized regime is energetically a costly process. In this work, we study conditions under which two non-identical electrically coupled neurons can reach an efficient regime of synchronization at low energy cost. We show that the energy consumption required to keep the synchronized regime can be spontaneously reduced if the receiving neuron has adaptive mechanisms able to bring its biological parameters closer in value to the corresponding ones in the sending neuron.     

On the synchronization of different chaotic oscillators

The synchronization of two different chaotic oscillators is studied, based on an open-loop control – the entrainment control. We consider two types of synchronization: complete synchronization and effectively complete synchronization. The sufficient conditions that two different systems can be synchronized by this method is discussed. Furthermore, a hierarchical idea to synchronize multiple chaotic subsystems is proposed.     

On a synchronization queue with two finite buffers

In this paper, we consider a synchronization queue (or synchronization node) consisting of two buffers with finite capacities. One stream of tokens arriving at the system forms a Poisson process and the other forms a PH-renewal process. The tokens are held in the buffers until one is available from each flow, and then a group-token is instantaneously released as a synchronized departure. We show that the output stream of a synchronization queue is a Markov renewal process, and that the time between consecutive departures has a phase type distribution. Thus, we obtain the throughput of this synchronization queue and the loss probabilities of each type of tokens. Moreover, we consider an extended synchronization model with two Poisson streams where a departing group-token consists of several tokens in each buffer. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bifurcation of slow motions in a stiff spring pendulum

We consider free oscillations of a double pendulum, where one of the pendula is modelled as a very stiff spring. Contrary to a single spring pendulum numerical simulations show an unexpected large influence of the fast longitudinal oscillations on the slow pendulum oscillations even for extremely large values of the stiffness. The transition from the regular motion, which is governed by the dynamics of a rigid double pendulum close to a periodic orbit, to the irregular motion with large contributions from the longitudinal oscillations occurs due to a subcritical symmetry breaking bifurcation of the periodic solution. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniform stability for the Timoshenko beam with tip load

In this paper we consider a hybrid elastic model consisting of a Timoshenko beam and a tip load at the free end of the beam. We show that uniform stabilization of the model which includes the rotary inertia of the tip load can be obtained when feedback boundary moment and force controls are applied at the point of contact between the beam and the tip load. However, in the presence of the load stabilization is “slower” and subject to a restriction on the boundary data at the free end of the beam.     

Non-Equilibrium States of a Photon Cavity Pumped by an Atomic Beam

We consider a beam of two-level randomly excited atoms that pass one-by-one through a one-mode cavity. We show that in the case of an ideal cavity, i.e. no leaking of photons from the cavity, the pumping by the beam leads to an unlimited increase in the photon number in the cavity. We derive an expression for the mean photon number for all times. Taking into account leaking of the cavity, we prove that the mean photon number in the cavity stabilizes in time. The limiting state of the cavity in this case exists and it is independent of the initial state. We calculate the characteristic functional of this non-quasi-free non-equilibrium state. We also calculate the total energy variation in both the ideal and the open cavities as well as the entropy production in the ideal cavity.     

New estimations for ultimate boundary and synchronization control for a disk dynamo system

We consider globally exponentially attractive sets and synchronization control for a disk dynamo system. First, based on generalized Lyapunov function theory and the extremum principle of function, we derive some new 4D ellipsoid estimations and a polydisk domain estimation of the globally exponentially attractive set of a 4D disk dynamo system without existence assumptions. Our results improve existing results on the globally exponentially attractive set as special cases and can lead to a series of new estimations. Second, we propose linear feedback control with a single input or two inputs to realize globally exponential synchronization of two 4D disk dynamo systems using inequality techniques. Some new sufficient algebraic criteria for the globally exponential synchronization of two 4D disk dynamo systems are obtained analytically. The controllers designed here have a simple structure and less conservation. Finally, numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

Noise and coupling induced synchronization in a network of chaotic neurons

The synchronization in four forced FitzHugh–Nagumo (FHN) systems is studied, both experimentally and by numerical simulations of a model. We show that synchronization may be achieved either by coupling of systems through bidirectional diffusive interactions, by introducing a common noise to all systems or by combining both ingredients, noise and coupling together. Here we consider white and colored noises, showing that the colored noise is more efficient in synchronizing the systems respect to white noise. Moreover, a small addition of common noise allows the synchronization to occur at smaller values of the coupling strength. When the diffusive coupling in the absence of noise is considered, the system undergoes the transition to subthreshold oscillations, giving a spike suppression regime. We show that noise destroys the appearance of this dynamical regime induced by coupling.     

Exponential stability for the Timoshenko system with two weak dampings

In this paper we consider a linear system of Timoshenko type beam equations with frictional dissipative terms. We show the exponential decay of the solution by using a method developed by Z. Liu and S. Zheng and their collaborators in past years. This method is very different from some others in the literature, such as the traditional energy method. It is our hope that the reader will find the method presented in this work is powerful and simple.     

Algebraic connectivity of directed graphs

We consider a generalization of Fiedler's notion of algebraic connectivity to directed graphs. We show that several properties of Fiedler's definition remain valid for directed graphs and present properties peculiar to directed graphs. We prove inequalities relating the algebraic connectivity to quantities such as the bisection width, maximum directed cut and the isoperimetric number. Finally, we illustrate an application to the synchronization in networks of coupled chaotic systems.