This paper introduces a modern version of the classical Huygens' experiment on synchronization of pendulum clocks. The version presented here consists of two monumental pendulum clocks--ad hoc ...designed and fabricated--which are coupled through a wooden structure. It is demonstrated that the coupled clocks exhibit 'sympathetic' motion, i.e. the pendula of the clocks oscillate in consonance and in the same direction. Interestingly, when the clocks are synchronized, the common oscillation frequency decreases, i.e. the clocks become slow and inaccurate. In order to rigorously explain these findings, a mathematical model for the coupled clocks is obtained by using well-established physical and mechanical laws and likewise, a theoretical analysis is conducted. Ultimately, the sympathy of two monumental pendulum clocks, interacting via a flexible coupling structure, is experimentally, numerically, and analytically demonstrated.
A seemingly ubiquitous irrational number often appearing in nature and in man-made things like structures, paintings, and physical systems, is the golden number. Here, we show that this astonishing ...number appears in the periodic solutions of an underactuated mass-spring oscillator driven by a nonlinear self-excitation. Specifically, by using the two-time scale perturbation method, it is analytically demonstrated that the golden number appears in the ratio of amplitudes, as well as in the oscillation frequency of the periodic solution, which is referred to as golden solution and, by applying the Poincaré method, it is demonstrated that this solution is asymptotically stable. Additionally, the analytic results are illustrated by means of numerical simulations and also, an experimental study is conducted.
This paper focuses on the application of the Poincaré method of ‘small parameter’ for the study of coupled dynamical systems. Specifically, our attempt here is to show that, by using the Poincaré ...method, it is possible to derive conditions for the onset of synchronization in coupled (oscillatory) systems. A case of study is presented, in which conditions for the existence and stability of synchronous solutions, occurring in two nonlinear oscillators interacting via delayed dynamic coupling, are derived. Ultimately, it is demonstrated that the Poincaré method is indeed an effective tool for analyzing the synchronous behavior observed in coupled dynamical systems.
•A mathematical framework for studying synchronization is presented.•The framework is based on the Poincaré method of small parameter.•The proposed methodology is applicable to weakly nonlinear coupled systems.•For the sake of illustration a particular example is analyzed.
Crouch gait is one of the most common gait abnormalities; it is usually caused by cerebral palsy. There are few works related to the modeling of crouch gait kinematics, crouch gait analysis, and ...visualization in both the workspace and joint space. In this work, we present a quaternion-based method to solve the forward kinematics of the position of the lower limbs during walking. For this purpose, we propose a modified eight-DoF human skeletal model. Using this model, we present a geometric method to calculate the gait inverse kinematics. Both methods are applied for gait analysis over normal, mild, and severe crouch gaits, respectively. A metric-based comparison of workspace and joint space for the three gaits for a gait cycle is conducted. In addition, gait visualization is performed using Autodesk Maya for the three anatomical planes. The obtained results allow us to determine the capabilities of the proposed methods to assess the performance of crouch gaits, using a normal pattern as a reference. Both forward and inverse kinematic methods could ultimately be applied in rehabilitation settings for the diagnosis and treatment of diseases derived from crouch gaits or other types of gait abnormalities.
This paper presents a First-Order Recurrent Neural Network activated by a wavelet function, in particular a Morlet wavelet, with a fixed set of parameters and capable of identifying multiple chaotic ...systems. By maintaining a fixed structure for the neural network and using the same activation function, the network can successfully identify the three state variables of several different chaotic systems, including the Chua, PWL-Rössler, Anishchenko–Astakhov, Álvarez-Curiel, Aizawa, and Rucklidge models. The performance of this approach was validated by numerical simulations in which the accuracy of the state estimation was evaluated using the Mean Square Error (MSE) and the coefficient of determination (r2), which indicates how well the neural network identifies the behavior of the individual oscillators. In contrast to the methods found in the literature, where a neural network is optimized to identify a single system and its application to another model requires recalibration of the neural algorithm parameters, the proposed model uses a fixed set of parameters to efficiently identify seven chaotic systems. These results build on previously published work by the authors and advance the development of robust and generic neural network structures for the identification of multiple chaotic oscillators.
In this paper, we study the synchronization of two Hindmarsh-Rose neuronal models interacting to each other through a dynamic coupling. The design of the dynamic interconnection is inspired in the ...so-called Huygens’ coupling, which in its simplest form is modeled by a second order linear system. In the analysis, it is assumed that only one state variable is available for measurement and the stability of the synchronous behavior is investigated by using the master stability function approach, in combination with the largest transverse Lyapunov exponent. Ultimately, the proposed synchronization scheme is experimentally validated by using electronic circuits, which emulate the dynamics of the Hindmarsh-Rose neuronal model.
This paper investigates the onset of synchronization in a network of linear non-minimum phase oscillators. We provide a prototype example which illustrates that full synchronization in such ...oscillator network under diffusive coupling may appear for only limited coupling strength, but, when the coupling is extended to a linear dynamic one, the synchronization appears for sufficiently large coupling. Also, the example at hand shows that if the coupling strength is fixed, there exists a limitation on the maximum number of nodes that can be synchronized when using diffusive couplings. In contrast, dynamic coupling allows removing such limitation. The obtained theoretical results are illustrated by computer simulations.
We present a method for estimating the dynamical noise level of a ‘short’ time series even if the dynamical system is unknown. The proposed method estimates the level of dynamical noise by ...calculating the fractal dimensions of the time series. Additionally, the method is applied to EEG data to demonstrate its possible effectiveness as an indicator of temporal changes in the level of dynamical noise.
•A dynamical noise level estimator for time series is proposed.•The estimator does not need any information about the dynamics generating the time series.•The estimator is based on a novel definition of time series dimension (TSD).•It is demonstrated that there exists a monotonic relationship between the•TSD and the level of dynamical noise.•We apply the proposed method to human electroencephalographic data.
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