Narrow Results

Electroencephalograph (EEG) data for normal individuals with eyes-closed and under stimuli is analyzed. The stimuli consisted of photo, audio, motor and mental activity. We use several measures from nonlinear dynamics to analyze and characterize the data. We find that the dynamics of the EEG signal is deterministic and chaotic but it is not a low dimensional chaotic system. The evoked responses lead to a redistribution of strengths relative to eyes-closed data. Basically, strength in α waves decreases whereas that in β wave increases. We also carried out simulations separately and in combination for δ, θ, α and β waves to understand the data. From the simulation results, it appears that the characteristics of EEG data are consequences of filtering the data with a relatively small range of frequency (0.5–32 Hz). In view of this, we believe that calculation of known nonlinear measures is not likely to be very useful for studying the dynamics of EEG data. We have also successfully modeled the EEG time series using the concept of state space reconstruction in the framework of artificial neural network. It gives us confidence that one would be able to understand, in a more basic way, how collectivity in EEG signal arises.

We show that a nonlinear dynamical system in Poincaré–Dulac normal form (in ℝⁿ) can be seen as a constrained linear system; the constraints are given by the resonance conditions satisfied by the spectrum of (the linear part of) the system and identify a naturally invariant manifold for the flow of the "parent" linear system. The parent system is finite dimensional if the spectrum satisfies only a finite number of resonance conditions, as implied e.g. by the Poincaré condition. In this case our result can be used to integrate resonant normal forms, and sheds light on the geometry behind the classical integration method of Horn, Lyapounov and Dulac.

The majority of experimental and theoretical friction studies considers translational motion, where all coarse-grained interface elements are displaced alike. An additional rotation of the slider introduces a well controlled displacement inhomogeneity across the interface. The friction response now consists of a force and a torque that are generally coupled. Recent results for a cylindrical slider are reviewed and applied to rigid objects with rotational symmetry about an axis. It is predicted that the difference between static and dynamic friction force can be suppressed, if a certain torque is applied. Moreover, we study the dynamics of the transition from sticking to sliding.

Rapid progress of experimental biology has provided a huge flow of quantitative data, which can be analyzed and understood only through the application of advanced techniques recently developed in theoretical sciences. On the other hand, synthetic biology enabled us to engineer biological models with reduced complexity. In this review we discuss that a multidisciplinary approach between this sciences can lead to deeper understanding of the underlying mechanisms behind complex processes in biology. Following the mini symposia "Noise and oscillations in biological systems" on Physcon 2011 we have collected different research examples from theoretical modeling, experimental and synthetic biology.

Energy potentials with double-well structures are typical in atoms and molecules systems. A manipulation scheme using Half Cycles Pulses (HCPs) is proposed to transfer a Gaussian wave packet between the two wells. On the basis of quantum mechanical simulations, the time evolution and the energy distribution of the wave packet are evaluated. The effect of time parameters, amplitude, and number of HCPs on spatial and energy distribution of the final state and transfer efficiency are investigated. After a carefully tailored HCPs sequence is applied to the initial wave packet localized in one well, the final state is a wave packet localized in the other well and populated at the lower energy levels with narrower distribution. The present scheme could be used to control molecular reactions and to prepare atoms with large dipole moments.

We present a general growth model based on nonextensive statistical physics. We show that the most common unidimensional growth laws such as power law, exponential, logistic, Richards, Von Bertalanffy, Gompertz can be obtained. This model belongs to a particular case reported in (Physica A 369, 645 (2006)). The new evolution equation resembles the “universality” revealed by West for ontogenetic growth (Nature 413, 628 (2001)). We show that for early times the model follows a power law growth as $N(t)\approx t^D$, where the exponent $D\equiv \frac{1}{1-q}$ classifies different types of growth. Several examples are given and discussed.

In this paper, a class of uncertain nonlinear systems is investigated and a sliding mode control (SMC) design is presented. The method is proposed for uncertain systems with model uncertainties, nonlinear dynamics and external disturbances. Using nominal system and related bounds of uncertainties, a chattering alleviating scheme is also proposed, which can ensure the robust SMC law. The performance and the significance of the controlled system are investigated under variation in system parameters and also in presence of an external disturbance. The simulation results indicate that performance of the proposed controller is effective compared to the existing controllers.

This paper pays attention to the piecewise continuous time modeling of a hydro-turbine governing system in the process of shutdown transient. As a pioneering work, the piecewise nonlinear dynamic transfer coefficients are introduced into the system. Moreover, for different hydrographs of shutdown transient process, the effect of different shutdown laws on the quality of shutdown transient process has been analyzed by numerical simulations. The nonlinear dynamic behaviors of the hydro-turbine governing system are illustrated by bifurcation diagrams, Poincaré maps, time waveforms and phase orbits. More importantly, these methods and analytic results are useful for the safe and stable operation of a hydropower station in the process of shutdown transient.

In this paper, we investigate the Bloch bands and develop a linear response theory for nonlinear systems, where the interplay between topological parameters and nonlinearity leads to new band structures. The nonlinear system under consideration is described by the Qi–Wu–Zhang model with Kerr-type nonlinearity, which can be treated as a nonlinear version of Chern insulator. We explore the eigenenergies of the Hamiltonian and discuss its Bloch band structures as well as the condition of gap closing. A cone structure in the ground Bloch band and tubed structure in the excited Bloch band is found. We also numerically calculate the linear response of the nonlinear Chern insulator to external fields, finding that these new band structures break the condition of adiabatic evolution and make the linear response not quantized. This feature of response can be understood by examining the dynamics of the nonlinear system.

In this study, the analytical elucidation for a generalized rotational harmonic system that possesses coherence and periodically excited force is reported. The methods of multiple scales within the interferometry are applied to evaluate the proposed problem and certain distinguishable cases for the rotational oscillators including normal harmonic oscillators without damped rotating are explored and discussed in detail. The distinctive computations for all mentioned chaotic cases about source peculiarities are deduced in detail. The acquired results are demonstrated in concrete graphical and numerical examples. Also, the coherence and the corresponding chaotic characteristics are discussed to probe the system intrinsic configurations. We can differentiate between correlations that result from particular multi-particle formation dynamics and even those caused by the influences of quantum symmetrization. We specifically demonstrate periodic flows and the interferences within the symmetrization for the partially chaotic systems obtained with the smashing of particles that is significant compared to the particle mass m. The partially chaotic system exhibits the coherence components which suppress the correlation intercept significantly and thus the current technique measures the degree of coherence precisely. The contemplated methodology can be applied to evaluating and analyzing many strong nonlinear oscillatory equations. Such an innovative approach can compute the problems of celestial mechanics and chemical reactions in engineering and medical fields.

This work investigates the nonlinear differential equations which have emerged as a substantial concentration of research within a multifariousness of nonlinear disciplines of science and the model computations elucidate the chaotic-coherence radiations with their corresponding convection parameters. The solution for droplets, temperature and chaotic properties of the systems is drawn using the nonlinear dynamics procedure and thus the graphical interpretation is carried out for the system controlling parameters. The significance of pertinent chaotic systems with correlations and their parameters is visualized graphically through the hybrid granular model. In particular, the chaotic-coherent profile is described with the statistical analysis for various system peculiarities and certain nonlinear equations explored in the current methodology through the feasible interferences transformation approach. The discussion part provides a detailed explanation of the schematic methods and certain conclusion observations based on the results of the present research. The findings of this research probe the new specific perceptiveness which commensurable to the system performance of incoherent material with peculiar aspects and such contemplation divulges the remunerative significance in engineering fields.

Ferroresonance is one of the most harmful and longest known power quality disturbances in the history of AC power systems. The ability of predicting transient and steady-state ferroresonance simulations mainly depends on the accuracy of the power transformer model. Most existing voltage transformer models apply single-valued nonlinear functions to represent the core nonlinearities. This study, based on our previous work, proposes a newly improved and accurate transformer iron core hysteresis model for ferroresonance simulation by extension of the classical arctangent model. To verify the proposed model’s accuracy and superiority, three different ferroresonant voltage and current waveform simulations were performed using both the proposed model and renowned EMTP Type-96 model under the same system parameters. In addition, simulation results were compared with the corresponding experimental measurements. The results indicate that the proposed model is easily implemented using numerical modeling method with good stability and convergence, and is sufficiently accurate for both transient and steady-state periodic ferroresonance analysis.

Over the past decade the study of fluidic droplets bouncing and skipping (or “walking”) on a vibrating fluid bath has gone from an interesting experiment to a vibrant research field. The field exhibits challenging fluids problems, potential connections with quantum mechanics, and complex nonlinear dynamics. We detail advancements in the field of walking droplets through the lens of Dynamical Systems Theory, and outline questions that can be answered using dynamical systems analysis. The paper begins by discussing the history of the fluidic experiments and their resemblance to quantum experiments. With this physics backdrop, we paint a portrait of the complex nonlinear dynamics present in physical models of various walking droplet systems. Naturally, these investigations lead to even more questions, and some unsolved problems that are bound to benefit from rigorous Dynamical Systems Analysis are outlined.

The conformable derivative and adequate fractional complex transform are implemented to discuss the fractional higher-dimensional Ito equation analytically. The Jacobi elliptic function method and Riccati equation mapping method are successfully used for this purpose. New exact solutions in terms of linear, rational, periodic and hyperbolic functions for the wave amplitude are derived. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. Numerical simulations of some obtained solutions with special choices of free constants and various fractional orders are displayed.

In numerous real-world applications, transverse vibrations of beams are nonlinear in nature. It is a task to solve nonlinear beam systems due to their substantial dependence on the 4 variables of the system and the boundary conditions. To comprehend the nonlinear vibration characteristics, it is essential to do a precise parametric analysis. This research demonstrates an approximation solution for odd and even nonlinear transverse vibrating beams using the Laplace-based variation iteration method, and the formulation of the beams depends on the Galerkin approximation. For the solution of the nonlinear differential equation, this method is efficient as compared to the existing methods in the literature because the solutions exactly match with the numerical solutions. The Laplace-based variation iteration method has been used for the first time to obtain the solution to this important problem. To demonstrate the applicability and precision of the Laplace-based iteration method, several initial conditions are applied to the governing equation for nonlinearly vibrating transverse beams. The natural frequencies and periodic response curves are computed using Laplace-based VIM and compared with the Runge–Kutta RK4 method. In contrast to the RK4, the results demonstrate that the proposed method yields excellent consensus. The Lagrange multiplier is widely regarded as one of the most essential concepts in variational theory. The result obtained are displayed in the table form.

Highlights

The highlights of the solution of the Euler–Bernoulli beam equation with quintic nonlinearity using Lagrange multiplier are:

• 1.Introducing the constraint of the boundary conditions into the equation using Lagrange multipliers.
• 2.Formulating the equations for the Lagrange multipliers and the deflection of the beam.
• 3.Solving the resulting system of algebraic equations using numerical methods.
• 4.Obtaining the deflection of the beam as a function of its length and the applied load.
• 5.Analyzing the behavior of the beam under different loads and boundary conditions.

In this paper, the variational iterative method (VIM) with the Laplace transform is utilized to solve the nonlinear problems of a simple pendulum and mass spring oscillator, which corresponds to the Duffing equation. Finding the Lagrange multiplier (LM) is a significant phase in the VIM, and variational theory is frequently employed for this purpose. This paper demonstrates how the Laplace transform can be utilized to locate the LM in a more efficient manner. The frequency obtained by Laplace-based VIM is the same as that defined in the already existing methods in the literature in order to ensure the clarity of the results. Numerous analytical techniques can be used to solve the Duffing equation, but we are the first to do it using a Laplace-based VIM and a distinctive LM. The fundamental results of my paper are that LM is also the same in the Elzaki transformation. In the vast majority of instances, Laplace-based VIM only requires one iteration to arrive at an answer with high precision and linearization, discretization or intensive computational work is required for this purpose. Comparing analytical results of VIM by Laplace transform to the built-in Simulink command in MATLAB which gives us the surety about the method’s applicability for solving nonlinear problems. Future work on the basic pendulum may examine the effects of nonlinearities and damping on its motion and the application of advanced control mechanisms to regulate its behavior. Future research on mass spring oscillators could examine the system’s response to random or harmonic input. The mass spring oscillator could also be used in vibration isolation to minimize vibrations from one building to another.

This paper delves into the exploration of directional recursion operators within the realm of regular space curves modeled by Heisenberg systems. The central objective is to introduce a myriad of recursive flows, encompassing ferromagnetic and antiferromagnetic solutions, alongside a family of general normalization operators in the normal and binormal directions. The study employs the extended compatible and inextensible flow model of curves to examine the evolution models, providing a comprehensive understanding of their dynamics. A significant aspect of the investigation involves elucidating the evolution model in terms of anholonomy shapes and their density. The directional recursive operator, a focus of this study, demonstrates distinct results compared to traditional approaches. The reliability and applicability of the obtained results extend to the examination of various linear and nonlinear continuous dynamical systems.

This paper introduces a groundbreaking method, Homotopy-based Fourier transform, integrating Fourier transform and Homotopy perturbation for refined nonlinear problem-solving. The modification enhances solution technique efficiency, notably accelerating convergence, particularly in solving the Korteweg–de Vries equation. Demonstrating versatility, the method effectively addresses ordinary and partial differential equations, showcasing its applicability across diverse mathematical scenarios. Moreover, the approach is extended to nonlinear dynamical systems, illustrating its robustness in handling complex dynamic behaviors. This method proves especially suitable for highly nonlinear differential equations, offering an efficient and effective tool for scientists and engineers dealing with intricate mathematical models. By significantly improving convergence rates, the Homotopy-based Fourier transform stands out as a valuable asset in unraveling the complexities of nonlinear systems across various scientific and engineering applications.

This review paper describes different lumped circuitry realizations of the chaotic dynamical systems having equilibrium degeneration into a plane object with topological dimension of the equilibrium structure equals one. This property has limited amount (but still increasing, especially recently) of third-order autonomous deterministic dynamical systems. Mathematical models are generalized into classes to design analog networks as universal as possible, capable of modeling the rich scale of associated dynamics including the so-called chaos. Reference state trajectories for the chaotic attractors are generated via numerical analysis. Since used active devices can be precisely approximated by using third-level frequency dependent model, it is believed that computer simulations are close enough to capture real behavior. These simulations are included to demonstrate the existence of chaotic motion.

Features of transition from regular types of oscillations to chaos in dynamic systems with finite and infinite dimensionality of phase space have been discussed. It has been found that for some types of nonlinearity, transition to the chaotic motion in these systems occurs according to the identical autoparametric scenario. The sequence of bifurcation phenomena looks as follows: equilibrium state ⇒ limit cycle ⇒ semitorus ⇒ strange attractor. On the basis of the results of numerical simulation a conclusion was made about the typical nature of such a scenario. The results of numerical calculations are confirmed by results of physical experiments carried out on the base of radiophysical self-oscillatory systems.