Nonlinear Dissipative Systems Research Articles

Partially “invisible” pulsation of asymmetric soliton molecules

Pulsation is a universal phenomenon in all disciplines of nonlinear science, which generally exhibit various nonlinear soliton dynamic modes in nonlinear optics. With the emergence of real-time spectral measurement technology, a new phenomenon of optical soliton pulsation, “invisible” soliton pulsation, has gradually attracted researchers’ attention. In this paper, by solving the coupled Ginzburg–Landau equation, we have discovered the partially “invisible” pulsation phenomenon of asymmetric soliton molecules (SMs) composed of two unequal-intensity pulses in a normal-dispersion Mamyshev oscillator for the first time. It is indicated that the fluctuation periods of each internal pulse in asymmetric SMs in terms of peak power, energy, and relative phase difference are all four roundtrips. However, the oscillation period of asymmetric SMs energy is two roundtrips, only half of the above period. Further research shows that the oscillation of the relative phase difference between two internal pulses in SMs is related to the variation of their intensity difference, which can also affect the position change of the spectral modulation peak. This work will enrich the research on “invisible” soliton pulsation and is of great significance for promoting the development of nonlinear dissipative systems.

Simulation of matrix product states to unveil the initial state dependency of non-Gaussian dynamics of Kerr nonlinearity

We simulate a free dissipative and coherent-driven Kerr nonlinear system using a time-evolving block decimation (TEBD) algorithm to study the impact of the initial state on the exact quantum dynamics of the system. The superposition of two coherent branches results in non-classical time dynamics. The Wigner state representation confirms that the system ends up saturating to two different branches, through evolving different trajectories, resulting in de-Gaussification throughout evolution. Furthermore, we also see that the time evolution suffers the residual effect of the initial state.

Transient breathing dynamics of dissipative soliton in an ultrafast laser

The occurrence of periodically oscillatory behaviors in various nonlinear dissipative systems is known as breathing dynamics, which holds potential applications in spectroscopy. Herein, we have employed both experimental and numerical methods to investigate the breathing dynamics of dissipative soliton induced by Q-switched modulation in a mode-locked fiber laser. The findings indicate that the dynamics of breathing significantly influences the buildup stage of both stationary and breathing dissipative solitons. The breathing rate, breathing duration and breathing period are quite sensitive to the pump power. Numerical simulations are employed to generate similar breathing dynamics within the context of a model that combines equations describing the population inversion in a mode-locked laser. These obtained results provide confirmation that the Q-switching modulation is responsible for the initiation of breathing oscillations, and present novel prospects for the development of different operational frameworks for ultrafast lasers.

Real-time observation of soliton pulsation and explosion in an Yb-doped fiber laser

Nonlinear dissipative systems support pulsating solutions and exhibit many interesting dynamic behaviors. Here, we report various soliton pulsations and explosions in an all-normal dispersion ytterbium-doped fiber laser by utilizing the dispersive Fourier transform technique. By the different soliton dynamics, these phenomena can be classified into single-soliton pulsation and explosion, dual-soliton synchronous and asynchronous pulsation, as well as dual-soliton asynchronous explosion. Solitons exhibit identical periodic fluctuations in the synchronous pulsation while showing the anti-phase behavior in asynchronous pulsation. The dual-soliton asynchronous pulsation might be related to the periodic modulation of the polarization state of solitons. As for the dual-soliton asynchronous explosion, it can be regarded as the asynchronously triggered transient solitons containing periodic explosion through the gain-mediated soliton interactions. These findings may provide new insights into complex dynamics in the field of ultrafast lasers.

Nonlinear dynamical systems: Time reversibility versus sensitivity to the initial conditions

Time reversal of vast classes of phenomena has direct implications with predictability, causality and the second principle of thermodynamics. We propose a new method of investigating reversibility in time series data through the instance of a paradigmatic dissipative nonlinear dynamical system, namely the logistic map xt+1=1−axt2. The method splits the original data into two series, one symmetrical and the other antisymmetrical with respect to time. A close relation is revealed between time reversibility and the sensitivity to the initial conditions. Indeed, depending on the initial condition and the size of the time series, time reversal can enable the recovery, within a small error bar, of past information when the Lyapunov exponent is non-positive, notably at the Feigenbaum point (edge of chaos), where weak chaos is known to exist. Past information is gradually lost for increasingly large Lyapunov exponent (strong chaos), notably at a=2 where it attains a large value.

Some Modified Equations of the Sine-Hilbert Type

Three modified sine-Hilbert (sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided. Based on these bilinear equations, some exact solutions to the three modified equations are derived.

Application of information technology to create the digital acoustic models of VVER

The IT used to create DAMPS is based on an interdisciplinary approach. DAMNR is presented as a self-oscillatory system, belonging to a special class of nonlinear dissipative systems capable of generating undamped oscillations. The use of DAMNR in the design and operation of NPP makes it possible to identify the sources of occurrence of ASW in the coolant, the conditions for their occurrence and frequency, and determine the influence of geometric parameters and the layout of coolant circuit equipment on the interaction of neutronic-physical, thermal–hydraulic and vibroacoustic processes. The use of DAMNR makes it possible to optimize engineering and design solutions when creating NPP by eliminating the conditions that cause the occurrence of unwanted self-oscillations of the coolant and vibroacoustic resonances in normal and emergency operating modes, as well as under conditions of shock impacts and seismic loads. Universal models and analysis methods were verified at Novovoronezh NPP.

Non-extensive (Tsallis) q-statistics and auroral glow

It is well-known that the auroral region of the magnetosphere–ionosphere interaction is an open, nonlinear dissipative system far from the equilibrium state. It is in this region that auroras are regularly observed, demonstrating not only a wide variety of dynamic forms but also a wide range of temporal and spatial scales. Due to the memory effects and fractal properties of auroral plasma, as well as strong correlations between its individual parts, the dynamics of the high-latitude system are primarily determined by long-range electromagnetic forces. Therefore, it is expected that non-additive and non-extensive thermodynamic principles may characterize their macroscopic behavior. In the presented study, it is shown that pulsating auroras exhibit non-extensive properties and can be described by q-statistics. The correlation of the parameter q with traditional indicators of nonlinear dynamics, such as the flatness coefficient, standard deviation, and scaling index, has been investigated. Small-scale changes are superimposed on large-scale processes characterized by a low correlation dimension, and this leads to an increase in the correlation dimension during geomagnetic activity, which approximately coincides with increased values of q. It is concluded that the q-statistics (Tsallis distribution) can be used to analyze optical phenomena in the auroral region.

Dynamic analysis of the effects of dissipative elements and modulational instability in a multicoupled nonlinear electrical transmission line with the propagation of new rogue waveforms

In this study, we consider a nonlinear multicoupled discrete electrical transmission line consisting of several modified Noguchi lines and analyze the dynamics of the effects of dissipative elements on modulated waves. This analysis shows that the dispersion element (C S ) and solution parameter (γ) strongly contribute to the increase in voltage amplitudes and to the modulation of these new rogue waveforms, unlike the dissipative element (G). Using a semi-discrete approximation, we demonstrate that the dynamics of modulated waves in such a dissipative electrical system can be governed by a system of nonlinear Schrödinger equations, the Manakov system, and system parameters. The phenomenon of modulational instability in this dissipative electrical system is studied, and areas of instability are shown. We found that the dissipative element of this system increased and decreased the areas of instability. Under the condition of this Manakov system, we determine the approximate modulated wave solutions that are then used for the dynamic analysis of the effects of dissipative elements when transmitting these new rogue waveforms through this dissipative electrical system. The effects of the parameters of this nonlinear dissipative electrical system, such as dispersive, dissipative, and solution parameters, in the dominant direction of propagation of these new rogue wave signals are presented. Based on these results, we observe that the effects of dissipative elements do exist in this nonlinear dissipative electrical system and that these dissipative elements would also impact the areas of modulational instability, which could gradually disappear in this electrical system.

General fractional classical mechanics: Action principle, Euler–Lagrange equations and Noether theorem

Standard action principle and first Noether theorem are generalized to take into account general form of nonlocality in time (memory) and to describe dissipative and non-Lagrangian nonlinear systems. General fractional calculus (GFC) of the Luchko form is used to take into account a wide class of nonlocalities time compared to the usual fractional calculus. Nonlocality is described by a pair of operator kernels belonging to the Luchko set. Non-holonomic variation equations of Sedov type is used to take into account equations of motion for wide class of dissipative and non-Lagrangian systems. In addition, within the framework of the proposed approach, equations of motion are considered not only with general fractional derivatives, but also for general fractional integrals. Example of application is considered.

Simulation of kerr nonlinearity: revealing initial state dependency

We simulate coherent driven free dissipative Kerr nonlinear system numerically using time-evolving block decimation (TEBD) algorithm and time propagation on the Heisenberg equation of motion using Euler’s method to study how the numerical results are analogous to classical bistability . The system evolves through different trajectories to stabilize different branches for different external drives and initial conditions. The Wigner state reprentation confirms the system to suffer a residual effect of initial state throughout the non-classical dynamical evolution and the metastable states of the system . Furthermore, we also see the numerically simulated spectral density remains significantly different from analytical counterparts when initial states do not lie to the same branch of the final state.

Feigenbaum scenario without parameters.

Typically, the period-doubling bifurcations exhibited by nonlinear dissipative systems are observed when varying systems' parameters. In contrast, the period-doubling bifurcations considered in the current research are induced by changing the initial conditions, whereas parameter values are fixed. Thus, the studied bifurcations can be classified as the period-doubling bifurcations without parameters. Moreover, we show a cascade of the period-doubling bifurcations without parameters, resulting in a transition to deterministic chaos. The explored effects are demonstrated by means of numerical modeling on an example of a modified Anishchenko-Astakhov self-oscillator where the ability to exhibit bifurcations without parameters is associated with the properties of a memristor. Finally, we compare the dynamics of the ideal-memristor-based oscillator with the behavior of a model taking into account the memristor forgetting effect.

Characteristics of a Low-Pressure Carbon-Dioxide Surface Dielectric Barrier Discharge Plasma Ion-Wind Thruster

This study investigates the discharge and dynamic characteristics of surface dielectric barrier discharge ion-wind thrusters made of polymethyl methacrylate, quartz, and polyvinylidene difluoride on a Martian surface filled with carbon 4 dioxide. Surface dielectric barrier discharge is a typical dissipative nonlinear spatiotemporal dynamical system. Space-charge accumulation and electron backflow are the main causes of gas multipulse asymmetric discharge. The heat generated by high-velocity particle bombardment and polarization loss in the filamentary discharge channel results in the partial breakdown of polymethyl methacrylate and polyvinylidene difluoride, reducing the electrical capacity of both dielectrics. The space-charge and shielding effects of the discharge plasma region are the main internal mechanisms influencing the thrust and thrust-to-power ratio. The maximum surface flow velocities of the three ion-wind thrusters are 22.1, 23.2, and , respectively. The maximum thrust reaches approximately 84.48, 97.48, and , respectively.The maximum thrust-to-power ratio is 15.18, 17.03, and , respectively. The change in the thrust and thrust-to-power ratio spans four orders of magnitude and can be adjusted in a broad range.

Identification and study of the characteristics of friction oscillations in the brake

Purpose.The task of researching modes of established frictional oscillations of the braking mechanism is to find a solution to the initial dynamic problem with friction that satisfies the periodicity conditions. At the same time, the period of motion of the dynamic system is not known in advance. This dynamic system is described by a non-linear dissipative non-autonomous system of differential equations. The methods. The developed technique of spectral analysis of the braking mechanism's oscillations is based on the assumption that its movements are periodic. If deterministic chaos occurs in the analyzed dynamic system, then the autocorrelation function of the time series of movements must have a finite carrier, that is, vanish outside a finite time interval. Findings. In the paper, the method of computational experiment is used to identify and study the characteristics of oscillatory processes in brake mechanisms. At the first stage of the computational experiment, a numerical solution of the considered dynamic problem with friction is carried out using a computational algorithm. As a result, the time series of block movements are calculated. At the second stage of the computational experiment, the obtained time series are studied. The originality. The paper uses phase diagrams in the "displacement-velocity" variables to analyze the process of establishing the oscillations of the brake mechanism and visual detection of attractors. When studying the dependence of amplitudes of displacements, velocities and accelerations of the dynamic system under consideration on changes in its parameters, the method of continuation by parameter was used with a stepwise change in the parameters of the system. Practical implementation. The developed mathematical model of vibrations of the braking mechanism and the computational algorithm for its numerical study are implemented in the form of a computer program for personal computers in the FORTRAN algorithmic language. Almost all available commercial compilers can be used to compile the program, including Compaq Visual Fortran 6.6 and Intel Visual Fortran 10, as well as non-commercial compilers distributed under the GNU license.

Partial stability of nonlinear dissipative feedback systems

Partially stable systems involve dynamical systems whose motion lie in a subspace of the state space resulting in system stability with respect to part of the system's states. In this paper, we develop partial stability theorems for nonlinear continuous-time and discrete-time dissipative feedback systems. Specifically, by invoking additional structural constraints on the forward loop and feedback loop system storage functions, we develop feedback interconnection partial stability results for dissipative nonlinear dynamical systems. Our results provide extensions of the positivity and small gain theorems for guaranteeing partial stability of feedback interconnected systems.

Unveiling Laser Radiation of Multiple Optical Solitons by Nonlinear Fourier Transform

AbstractMode‐locked fiber lasers are applied in versatile scientific fields and exhibit a rich diversity of nonlinear dynamics. However, the nontrivial coexistence of solitons and embedded dispersive wave radiation prevents unveiling the real nonlinear dynamics in mode‐locked fiber lasers, such as multiple solitons interaction. Here, nonlinear Fourier transform (NFT) is applied as a signal processing tool to reveal nonequilibrium multiple soliton dynamics. It is feasible to isolate solitons from the continuous wave background in a fiber laser. The real‐time coherent homodyne detection methodology is used to measure the full‐field dynamic evolution of multiple solitons, including multiple solitons buildup and sequentially nonequilibrium evolution with complex splitting, drifting, and collision processes. With the approach of inverse NFT, the corresponding various pure solitons buildup and collision are reconstructed. The eigenvalue probability distributions are used to classify different lasing regimes. Moreover, the controllable multiple solitons drifting is achieved and characterized by using all‐optical methods. Experimental results suggest that NFT can be used to identify localized soliton nonequilibrium evolution excluding dispersive wave radiation influence, which provides a new window into the physics of the underlying laser dynamics and uncovers real soliton interaction in dissipative nonlinear systems.

Dynamics of soliton explosions in a polarization-multiplexed ultrafast fiber laser

Soliton explosions are among the most striking soliton phenomena in nonlinear dissipative systems. In this regime, quasi-stable solitons suffer abrupt structural collapses and then recover to their original state. Here, we report on the first experimental observation of soliton explosions triggered by vector dissipative soliton collisions in a polarization-multiplexed ultrafast fiber laser. The fiber laser simultaneously generates two orthogonally polarized pulse trains with different repetition rates, which enables periodic soliton collisions in the cavity. Based on the dispersive Fourier transform technique, the spectral dynamics of collision-induced soliton explosions are identified in real time. The corresponding time-domain measurements demonstrate opposite temporal shifts in the two output pulse trains. Numerical simulations validate the experimental observations and give more insights into the physical mechanism for soliton explosions. Our findings can shed new light on the dynamics of soliton explosions.

Power-law Distribution of Solar Cycle–modulated Coronal Jets

Power-law distributions have been studied as a significant characteristic of nonlinear dissipative systems. Since discovering the power-law distribution of solar flares that was later extended to nanoflares and stellar flares, it has been widely accepted that different scales of flares share the same physical process. Here we present the newly developed semiautomated jet identification algorithm and its application for detecting more than 1200 off-limb solar jets during Solar Cycle 24. Power-law distributions have been revealed between the intensity/energy and frequency of these events, with indices found to be analogous to those for flares and coronal mass ejections (CMEs). These jets are also found to be spatially and temporally modulated by the solar cycle, forming a butterfly diagram in their latitudinal-temporal evolution, experiencing quasi-annual oscillations in their analyzed properties, and very likely gathering in certain active longitudinal belts. Our results show that coronal jets display the same nonlinear behavior as that observed in flares and CMEs, in solar and stellar atmospheres, strongly suggesting that they result from the same nonlinear statistics of scale-free processes as their counterparts in different scales of eruptive events. Although these jets, like flares and other large-scale dynamic phenomena, are found to be significantly modulated by the solar cycle, their corresponding power-law indices still remain similar.

Dichromatic "Breather Molecules" in a Mode-Locked Fiber Laser.

Bound states of solitons ("molecules") occur in various settings, playing an important role in the operation of fiber lasers, optical emulation, encoding, and communications. Soliton interactions are generally related to breathing dynamics in nonlinear dissipative systems, and maintain potential applications in spectroscopy. In the present work, dichromatic breather molecules (DBMs) are created in a synchronized mode-locked fiber laser. Real-time delay-shifting interference spectra are measured to display the temporal evolution of the DBMs, that cannot be observed by means of the usual real-time spectroscopy. As a result, robust out-of-phase vibrations are found as a typical intrinsic mode of DBMs. The same bound states are produced numerically in the framework of a model combining equations for the population inversion in the mode-locked laser and cross-phase-modulation-coupled complex Ginzburg-Landau equations for amplitudes of the optical fields in the fiber segments of the laser cavity. The results demonstrate that the Q-switching instability induces the onset of breathing oscillations. The findings offer new possibilities for the design of various regimes of the operation of ultrafast lasers.

Isomeric dynamics of multi-soliton molecules in passively mode-locked fiber lasers

The self-assembly of optical solitons propagating in nonlinear dissipative systems spreads the concept of soliton molecules. Assisted with the real-time spectral interferometry, plentiful internal dynamics has been probed within the multi-pulse patterns, emphasizing the striking analogies with the matter molecules. Therefrom, these particle-like behaviors would yield more intriguing landscapes toward the extended degrees of freedom considering increased constituents. Here, we transfer the concept of “isomer” to the experimental investigation on the unexplored isomeric dynamics of soliton molecules in parallel. Particularly, two isomers for soliton triplets and four isomers for soliton quadruplets are captured under different self-assembled forms, within each of which the binding separations and relative phases of the constituents are governed by mutual soliton interactions. With the diverse separation-phase evolving trajectories mapped in the interaction plane, detailed insights into the temporal distribution and the transient dynamics are displayed with respect to a panorama of the isomeric dynamics. The perspective of optical isomers shed new light on the analogy with matter molecules, and the underlying isomeric dynamics may stimulate the artificial manipulation of various soliton molecules for ultrafast applications.