Nonlinear Dissipative Systems Research Articles

Shrinking shrimp-shaped domains and multistability in the dissipative asymmetric kicked rotor map.

An interesting feature in dissipative nonlinear systems is the emergence of characteristic domains in parameter space that exhibit periodic temporal evolution, known as shrimp-shaped domains. We investigate the parameter space of the dissipative asymmetric kicked rotor map and show that, in the regime of strong dissipation, the shrimp-shaped domains repeat themselves as the nonlinearity parameter increases while maintaining the same period. We analyze the dependence of the length of each periodic domain with the nonlinearity parameter, revealing that it follows a power law with the same exponent regardless of the dissipation parameter. Additionally, we find that the distance between adjacent shrimp-shaped domains is scaling invariant with respect to the dissipation parameter. Furthermore, we show that for weaker dissipation, a multistable scenario emerges within the periodic domains. We find that as the dissipation gets weaker, the ratio of multistable parameters for each periodic domain increases, and the area of the periodic basin decreases as the nonlinearity parameter increases.

Global large solutions for the nonlinear dissipative system modeling electro-hydrodynamics

In this paper, we are concerned with global existence of large solutions for a dissipative model arising from electro-hydrodynamics, which is the nonlinear nonlocal system coupled by the Poisson–Nernst–Planck equations and the incompressible Navier–Stokes equations through charge transport and external forcing terms. By introducing some proper weighted functions and fully using the algebraic structure of the system, we prove that, under some conditions imposed on the indices p, p1, q, r, α, there exist two positive constants c0, C0 such that if the initial data u0=(u0h,u03) and (v0, w0) satisfy ‖u0h‖Ḃp1,∞−1+3p1+‖u0h‖Ḃp1,∞−1+3p1α‖u03‖Ḃp1,∞−1+3p11−α+K0≤c0 with K0≔‖v0‖Ḃq,1−2+3q⁡expC0‖u0‖Ḃp,1−1+3p+C0‖w0‖Ḃr,1−2+3r+1expC0‖u0‖Ḃp,1−1+3p, then the system admits a unique global solution. Moreover, the global existence of large solution was also established in two dimensional case.

Stability of few-cycle light bullets in nonlinear metamaterials beyond the slowly varying envelope approximation

This letter explores stable few-cycle light bullets in nonlinear metamaterials beyond the slowly varying envelope approximation. Using a (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation, the letter explores solitonic behaviors of light bullets in materials with a negative refractive index under the few-cycle regime. The analysis involves a Lagrangian variational approach designed for dissipative nonlinear systems and utilizes direct numerical simulations to demonstrate the possibility of stable light bullets forming in metamaterials and corroborate the stability analysis. Based on the stability conditions for the system parameters using the Routh-Hurwitz criterion, optimal values of the cubic and quintic self-steepening effects promote stable propagation of light bullets, while an imbalance may lead to collapse. These findings suggest that combining few-cycle light bullets with metamaterial-based optics may enable new possibilities for ultrafast and nonlinear optics.

Modal analysis of non-conservative systems with friction-induced strong nonlinear damping by using response-controlled testing

In recent years, the response-controlled stepped-sine testing (RCT) method has made important contributions to the evolution of the single nonlinear mode (SNM) method from a postulate to a well-established theory with the accurate modal identification of various nonlinear mechanical systems ranging from simple benchmark beams to real missiles. Since the RCT method resides on the SNM theory which is based on Rosenberg’s periodic motion definition of nonlinear modes, it has been regarded as limited to weakly damped mechanical systems that mainly exhibit strong conservative nonlinearities. An important contribution of this current paper is extending the RCT method to non-conservative mechanical systems with complex nonlinear modes. The extension is first numerically validated on a strongly friction-damped cantilever beam and then experimentally validated on the actuation mechanism of a real control fin that exhibits strong softening-stiffening nonlinearity (up to 50% shift of natural frequency) due to friction/backlash and very high and nonlinear damping (up to 30% modal damping ratio) due to friction. Another important contribution of this paper is to demonstrate the applicability of the superposition principle in nonlinear dissipative systems with closely spaced modes at an acceptable accuracy if the nonlinear modes are similar in shape to their corresponding linear modes and the modal coupling terms in the equation of motion can be ignored. The assumption of similar nonlinear modes is reasonable for most practical applications that exhibit dissipative nonlinearities due to mechanical joints. Although the assumption of ignorable coupling terms does not reside on a solid theoretical framework, numerical studies on a friction-damped lumped system with closely spaced modes give promising results, pointing to the significant potential of the modal superposition in nonlinear systems.

Discovery of dissipative microwave photonic soliton molecules in dual-bandpass optoelectronic oscillator

Optoelectronic oscillators (OEOs), which have attracted extensive studies in the past decades, are high quality-factor optoelectronic feedback loops for generating various ultra-pure microwave signals. In essence, OEOs are also dissipative nonlinear systems with multiple timescale characteristics and abundant nonlinearities, which open the possibilities for exploring localized dissipative solitary waves. In this paper, we demonstrate a new-class temporal dissipative soliton, i.e., dissipative microwave photonic soliton molecule (DMPSM), in a dual-bandpass OEO. Both the numerical simulation and experiment are conducted to reveal the physical mechanism of DMPSM generation and to evaluate the characteristics of the generated DMPSM sequences. Unlike optical soliton molecules in mode-locked lasers, the formation of DMPSMs arises from the combined action of multiple timescale coupling, nonlinear bistability, and time-delayed feedback in the OEO cavity, where the soliton interval and number in a DMPSM can be well-controlled through varying the multiple timescale variables in the OEO cavity, and the repetition frequency of the DMPSMs can be tuned through changing that of the initially injected perturbation signal. Meanwhile, the generated DMPSM sequence performs with high stability and excellent coherence, which shows enormous application potentials in pulse radar detection, dense microwave comb generation, and neuromorphology.

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.

Lasing in Non-Hermitian Flat Bands: Quantum Geometry, Coherence, and the Fate of Kardar-Parisi-Zhang Physics.

We show that lasing in flat-band lattices can be stabilized by means of the geometrical properties of the Bloch states, in settings where the single-particle dispersion is flat in both its real and imaginary parts. We illustrate a general projection method and compute the collective excitations, which display a diffusive behavior ruled by quantum geometry through a peculiar coefficient involving gain, losses and interactions, and entailing resilience against modulational instabilities. Then, we derive an equation of motion for the phase dynamics and identify a Kardar-Parisi-Zhang term of geometric origin. This term is shown to exactly cancel whenever the real and imaginary parts of the laser nonlinearity are proportional to each other, or when the uniform-pairing condition is satisfied. We confirm our results through numerical studies of the π-flux diamond chain. This Letter highlights the key role of Bloch geometric effects in nonlinear dissipative systems and KPZ physics, with direct implications for the design of laser arrays with enhanced coherence.

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.

Representing the dissipation of infinite-dimensional linear port-Hamiltonian systems

It is well known that linear and non-linear dissipative port-Hamiltonian systems in finite dimensions admit an energy balance, relating the energy increase in the system with the supplied energy and the dissipated energy. The integrand in the dissipation term is then a function of the state variable. In this note, we answer the question of when this is possible for linear port-Hamiltonian systems in infinite dimensions.

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.