Nonlinear Perturbations Research Articles

Optimal Precursors for Central Pacific El Niño Events in GFDL CM2p1

Abstract Compared to the canonical eastern Pacific El Niño, the understanding and ability to predict the central Pacific (CP)-type event still need further improvement. In this study, the principal component analysis–based particle swarm optimization algorithm (PPSO) is applied in Geophysical Fluid Dynamics Laboratory Climate Model version 2p1 (GFDL CM2p1) to obtain the optimal precursors (OPRs) for CP El Niño events, based on the conditional nonlinear optimal perturbation (CNOP) method. For this, three normal years with neither El Niño nor La Niña events, i.e., three cases, are chosen as the reference states. The obtained OPRs for these cases exhibit a consistent positive sea surface temperature (SST) perturbation distribution in the subtropical North Pacific (20°–40°N, 175°E–140°W), which is further proven to be crucial for the evolution of CP El Niño based on the Northern and Southern Hemisphere significance test results. Mechanically, these positive SST perturbations are enhanced and reach the equatorial Pacific via wind–evaporation–SST (WES) feedback to evolve into a CP El Niño at the end of the year. The nonlinear approach is adopted to investigate the predictability of CP El Niño events and can shed some light on future studies.

Test of Causal Nonlinear Quantum Mechanics by Ramsey Interferometry with a Trapped Ion.

Quantum mechanics requires the time evolution of the wave function to be linear. While this feature has been associated with the preservation of causality, a consistent causal nonlinear theory was recently developed. Interestingly, this theory is unavoidably sensitive to the full physical spread of the wave function, rendering existing experimental tests for nonlinearities inapplicable. Here, using well-controlled motional superpositions of a trapped ion, we set a stringent limit of 5.4×10^{-12} on the magnitude of the unitless scaling factor ε[over ˜]_{γ} for the predicted causal nonlinear perturbation.

Distributed fusion filtering for cyber-physical systems under Round-Robin protocol: a mixed H 2/H ∞ framework

ABSTRACT In this paper, the distributed mixed fusion filter design problem is investigated for a class of cyber-physical systems subject to non-ideal measurements under Round-Robin protocol. To save the limited communication resources, the Round-Robin protocol is employed to schedule the data transmissions from the sensors to the remote filters. The measurement is imperfect by taking into account the noises, the random saturations, nonlinearity perturbations and packet dropouts. The objective of this study is to propose a desired fusion filtering method that ensures the prescribed performance constraints on both the local and fusion filtering error dynamics. With the aid of Lyapunov function and stochastic analysis technique, a sufficient condition is established to guarantee the mixed / performance index for the local filtering error systems, and then the filter parameter matrices are derived by resorting to the solutions to some matrix inequalities. Subsequently, on the basis of the obtained local state estimates, the proper fusion parameters are acquired by solving a convex optimisation problem. Finally, a numerical example is provided to demonstrate the effectiveness of the designed fusion filter.

Extent of the Impact of Arctic Atmospheric Uncertainty on Extended‐Range Forecasting of Cold Events in East Asia

AbstractThe extended‐range predictability of three simulated extreme cold events in East Asia in the Community Atmosphere Model (version 4) control experiment, whose atmospheric circulation backgrounds are similar to two recent observational cases, is investigated. The results show that they have a predictability of four pentads. Then, we evaluate the extent of the forecast uncertainty in the 4th pentad caused by the initial atmospheric uncertainties in the Arctic, which are large due to sparse instrumental observations. The initial uncertainty leading to the largest forecast uncertainty is obtained by the conditional nonlinear optimal perturbation method, and referred to as the CNOP‐type initial uncertainty. The forecast of the cold surge in the 4th pentad fails after adding the CNOP‐type initial uncertainty at day 0. In comparison, the forecast uncertainties are much weaker when the initial conditions are perturbed by noises, and the weaker influence may be due to the noises' lack of spatial structure. In terms of how the CNOP‐type initial uncertainty develops, a baroclinic structure is seen in the uncertainties on the first 2 days, followed by a propagating Rossby wave feature from the 2nd pentad to the 4th pentad. Meanwhile, synoptic transient eddy feedback also plays an essential role. The results suggest that the CNOP‐type initial uncertainty has potential to identify sensitive areas for targeted observations; plus, it could serve as a member of ensemble initial perturbations, since it indicates the largest uncertainty growth.

Finite-time dissipative control design for one-sided Lipschitz nonlinear singular Caputo fractional order systems

ABSTRACT This paper presents an efficient analytical approach for solving the problem of dissipative control in one-sided Lipschitz singular Caputo fractional-order systems subject to nonlinear perturbations. The approach is based on mathematical transformations, and fractional calculus. A state feedback controller is designed to ensure that the closed-loop system is regular, impulse-free, and finite-time bounded with a dissipativity performance index. The results are established using tractable strict linear matrix inequalities (LMIs) without any equality constraint. Two numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Flows for the masses: A multi-fluid non-linear perturbation theory for massive neutrinos

Velocity dispersion of the massive neutrinos presents a daunting challenge for non-linear cosmological perturbation theory. We consider the neutrino population as a collection of non-linear fluids, each with uniform initial momentum, through an extension of the Time Renormalization Group perturbation theory. Employing recently-developed Fast Fourier Transform techniques, we accelerate our non-linear perturbation theory by more than two orders of magnitude, making it quick enough for practical use. After verifying that the neutrino mode-coupling integrals and power spectra converge, we show that our perturbation theory agrees with N-body neutrino simulations to within 10% for neutrino fractions Ω ν,0 h 2 ≤ 0.005 up to wave numbers of k = 1 h/Mpc, an accuracy consistent with ≤ 2.5% errors in the neutrino mass determination. Non-linear growth represents a > 10% correction to the neutrino power spectrum even for density fractions as low as Ω ν,0 h 2 = 0.001, demonstrating the limits of linear theory for accurate neutrino power spectrum predictions. Our code FlowsForTheMasses is avaliable online at github.com/upadhye/FlowsForTheMasses.

On the linearization of infinite‐dimensional random dynamical systems

AbstractWe present a new version of the Grobman–Hartman's linearization theorem for random dynamics. Our result holds for infinite‐dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on the nonlinear perturbations, we do not require for the linear part to be nonuniformly hyperbolic in the sense of Pesin but rather (besides requiring the existence of stable and unstable directions) allow for the existence of a third (central) direction on which we do not prescribe any behavior for the dynamics. Moreover, under some additional nonuniform growth condition, we prove that the conjugacies given by the linearization procedure are Hölder continuous when restricted to bounded subsets of the space.

Nonlinear optimal perturbations and formation mechanism of localized wave packet in channel flow

The subcritical transition in a two-dimensional channel flow is characterized by the emergence of localized wave packet (LWP). In this paper, the nonlinear nonmodal optimization method is used, and when the target time is long enough the minimal seed, the minimal-energy perturbation triggering the transition is found to be the localized wave packet tilted upstream with energy scaling as Re−3.8 for Re>2500. The formation of LWP from a minimal seed includes three stages. First, the disturbance kinetic energy of the minimal seed increases to a local maximum through a transient growth governed by the Orr mechanism, and second, large scale vortex structures are generated due to the nonlinear forcing mainly contributed by the wave components and causes a streamwise symmetry breaking, i.e., a long but weak upstream tail and a strong downstream front are formed. As a result, the perturbation structure approaches the unstable lower branch solution and is attracted and captured at the last stage by the stable upper branch solution of LWP in the phase space.

Dimension shifting based intelligent algorithm framework to solve conditional nonlinear optimal perturbation

Conditional Nonlinear Optimal Perturbation (CNOP) method is an effective way to study the predictability of oceanic and climatic events. A framework combining the Feature Extraction method and Intelligent Algorithm (FEIA) is frequently used to solve CNOP because it is gradient-free and scalable at high-dimensional scales. However, the fixed latent subspace of FEIA framework makes it challenging to achieve both the quality of solutions and the solving efficiency. To overcome this bottleneck, this paper proposes Dimension Shifting based Intelligent Algorithm (DSIA) framework to solve CNOP in the large-scale model. DSIA framework adopts the dimension shifting strategy, which dynamically shifts search particles in different low-dimensional spaces. To verify the feasibility of DISA framework, we take a Regional Ocean Modeling System (ROMS) model of double-gyre variation as an experimental case. In experiments, we figure out that the selection of feature extraction method and the shifting dimension set are two influential factors for the performance of DSIA framework. Besides, in comparative experiments, DSIA framework yields better objective function values and more valid CNOP than FEIA framework. Moreover, convergence experiments demonstrate DSIA framework can solve CNOP with an appropriate number of function evaluations and has better convergence performance. In conclusion, experimental results prove that DSIA improves both quality of CNOP and solving efficiency.

A periodic boundary value problem with switchings under nonlinear perturbations

We investigate the existence of solutions of weakly nonlinear periodic boundary value problems for systems of ordinary differential equations with switchings and the construction of these solutions. We consider the critical case where the equation for the generating constants of a weakly nonlinear periodic boundary-value problem with switchings does not turn into an identity. We improve the classification of critical and noncritical cases and construct an iterative algorithm for finding solutions of weakly nonlinear periodic boundary value problems with switchings in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary value problem for the mathematical model of nonisothermal chemical reactions. To check the accuracy of the proposed approximations, we evaluate discrepancies in the original equation.

Nonclassical perturbation approach to a nonlinear multiparameter eigenvalue problem arising in electromagnetics

The paper focuses on a problem of nonlinear electromagnetic wave propagation in a plane shielded waveguide filled with nonlinear medium. We look for nonlinear eigenvalues that correspond to propagation constants of the waveguide. One of the main feature of the problem is that it has nonperturbative solutions and therefore the corresponding linearized problem cannot be used in any perturbation approach to find such eigenvalues (moreover, it turns out that the linearized problem is solvable in exceptional cases only). To solve the main problem, we develop a nonlinear perturbation approach based on usage of a certain simpler nonlinear problem that also has nonlinearizable solutions. Using this approach, we prove existence of nonlinearizable solutions to the main problem. Numerical results are also presented. Briefly, we present electromagnetic background for the main problem.

Error Evolutions and Analyses on Joint Effects of SST and SL via Intermediate Coupled Models and Conditional Nonlinear Optimal Perturbation Method

A seasonal predictability barrier has long been noticed in ENSO forecasting with numerical models. Previous studies explored the impact of seasonal optimal initial perturbation evolutions in sea surface temperature anomalies (SSTA) on ENSO forecasting using the intermediate coupled model (ICM) via the conditional nonlinear optimal perturbation (CNOP) method. In this paper, we investigate the joint effects of SSTA and sea level anomalies (SLA) from the perspective of the optimal growth initial error (OGE). After determining the four seasonal OGEs about SSTA and SLA (i.e., SSTA-OGE, SLA-OGE and Joint-OGE), we first demonstrate the patterns, evolutions and the resulting spring predictability barrier (SPB) of the above OGEs. Then, we analyze the mechanism of OGE evolutions and SPB. Finally, we conduct observing system simulation experiments to determine the best (economic) observation network. Our experimental results indicate that the ENSO evolution error induced by SSTA-OGE and Joint-OGE presents season dependency, but SLA-OGE has no impact on ENSO evolution. Moreover, Joint-OGEs induce error evolutions and the SPB with more significant intensity than SSTA-OGEs and SLA-OGEs. From mechanism analyses, the evolutions of SSTA-OGEs are mainly dominated by Bjerknes feedback. Further, the evolution dynamics of Joint-OGEs primarily contain the continuous heating between the upper ocean combined with Bjerknes feedback and thermal diffusion in response to the discharge process. In addition, comprehensive and economical sensitive areas are identified through Joint-OGE, including the central-eastern equatorial Pacific and the western and north-eastern tropical Pacific boundary, which contribute to the ENSO prediction benefits reaching 58.31% on average.

Optimal Arctic sea ice concentration perturbation in triggering Ural blocking formation

Utilizing the Community Atmospheric Model version 4, the role of Arctic sea ice concentration (SIC) in triggering the Ural Blocking (UB) event formation on subseasonal timescale is investigated. Taking eight zonal flows as the reference states, the optimal Arctic SIC perturbations that trigger zonal flows into UB events on subseasonal timescale are obtained with the conditional nonlinear optimal perturbation approach. The numerical results show that the Arctic SIC decline in the Greenland, Barents and Okhotsk Seas can trigger zonal flows into UB events on a timescale of four pentads (20 days). Further diagnosis shows that the SIC decline in these regions locally warms the low troposphere via diabatic processes in the first pentad. Then, dynamic processes, such as temperature advection, modulate the temperature in the middle troposphere and weaken the meridional temperature gradient between the Arctic and mid-latitudes upstream of the Ural sector. The weakened meridional temperature gradient further decelerates the background zonal flow near the Ural sector and triggers UB formation in four pentads. This investigation highlights the role of Arctic SIC in the Greenland, Barents and Okhotsk Seas in triggering the formation of UB events on subseasonal timescale. The spatial characteristics of the optimal SIC perturbations may provide scientific support for SIC target observations in capturing the precursor of UB formation and improving UB prediction skills on subseasonal timescale.

Thermalization of the Ablowitz-Ladik lattice in the presence of non-integrable perturbations.

We investigate the statistical mechanics of the photonic Ablowitz-Ladik lattice, the integrable version of the discrete nonlinear Schrödinger equation. In this regard, we demonstrate that in the presence of perturbations, the complex response of this system can be accurately captured within the framework of optical thermodynamics. Along these lines, we shed light on the true relevance of chaos in the thermalization of the Ablowitz-Ladik system. Our results indicate that when linear and nonlinear perturbations are incorporated, this weakly nonlinear lattice will thermalize into a proper Rayleigh-Jeans distribution with a well-defined temperature and chemical potential, in spite of the fact that the underlying nonlinearity is non-local and hence does not have a multi-wave mixing representation. This result illustrates that in the supermode basis, a non-local and non-Hermitian nonlinearity can in fact properly thermalize this periodic array in the presence of two quasi-conserved quantities.

Exact formulations of relativistic electrodynamics and magnetohydrodynamics with helically coupled scalar field

We present the general relativistic electrodynamics and magnetohydrodynamics with a helically coupled scalar field. We consider three component system with the fluid, scalar field and electromagnetic fields with the helical coupling. We derive three exact formulations: the covariant formulation, the ADM formulation, and the fully nonlinear and exact perturbation formulation. We also derive the weak-gravity limit with fully relativistic fluid and fields. The latter two formulations are presented in the cosmological context.

The Hamiltonian for von Zeipel–Lidov–Kozai oscillations

ABSTRACTThe Hamiltonian used in classical analyses of von Zeipel–Lidov–Kozai or ZLK oscillations in hierarchical triple systems is based on the quadrupole potential from a distant body on a fixed orbit, averaged over the orbits of both the inner and the outer bodies (‘double averaging’). This approximation can be misleading, because the corresponding Hamiltonian conserves the component of angular momentum of the inner binary normal to the orbit of the outer binary, thereby restricting the volume of phase space that the system can access. This defect is usually remedied by including the effects of the octopole potential, or by allowing the outer orbit to respond to variations in the inner orbit. However, in a wide variety of astrophysical systems, non-linear perturbations are comparable to or greater than these effects. The long-term effects of non-linear perturbations are described by an additional Hamiltonian, which we call Brown’s Hamiltonian. At least three different forms of Brown’s Hamiltonian are found in the literature; we show that all three are related by a gauge freedom, although one is much simpler than the others. We argue that investigations of ZLK oscillations in triple systems should include Brown’s Hamiltonian.

Nonlinear perturbation of a high-order exceptional point: Skin discrete breathers and the hierarchical power-law scaling

We study the nonlinear perturbation of a high-order exceptional point (EP) of the order equal to the system site number L in a Hatano–Nelson model with unidirectional hopping and Kerr nonlinearity. Notably, we find a class of discrete breathers that aggregate to one boundary, here named as skin discrete breathers (SDBs). The nonlinear spectrum of these SDBs shows a hierarchical power-law scaling near the EP. Specifically, the response of nonlinear energy to the perturbation is given by Em ∝ Γαm , where αm = 3 m–1 is the power with m = 1,…, L labeling the nonlinear energy bands. This is in sharp contrast to the L-th root of a linear perturbation in general. These SDBs decay in a double-exponential manner, unlike the edge states or skin modes in linear systems, which decay exponentially. Furthermore, these SDBs can survive over the full range of nonlinearity strength and are continuously connected to the self-trapped states in the limit of large nonlinearity. They are also stable, as confirmed by a defined nonlinear fidelity of an adiabatic evolution from the stability analysis. As nonreciprocal nonlinear models may be experimentally realized in various platforms, such as the classical platform of optical waveguides, where Kerr nonlinearity is naturally present, and the quantum platform of optical lattices with Bose–Einstein condensates, our analytical results may inspire further exploration of the interplay between nonlinearity and non-Hermiticity, particularly on high-order EPs, and benchmark the relevant simulations.

Penetrative convection: heat transport with marginal stability assumption

This paper investigates heat transport in penetrative convection with a marginally stable temporal-horizontal-averaged field or background field. Assuming that the background field is steady and is stabilised by the nonlinear perturbation terms, we obtain an eigenvalue problem with an unknown background temperature $\tau$ by truncating the nonlinear terms. Using a piecewise profile for $\tau$ , we derived an analytical scaling law for heat transport in penetrative convection as $Ra\rightarrow \infty$ : $Nu=(1/8)(1-T_M)^{5/3}Ra^{1/3}$ ( $Nu$ is the Nusselt number; $Ra$ is the Rayleigh number and $T_M$ corresponds to the temperature at which the density is maximal). A conditional lower bound on $Nu$ , under the marginal stability assumption, is then derived from a variational problem. All the solutions to the full system should deliver a higher heat flux than the lower bound if they satisfy the marginal stability assumption. However, data from the present direct numerical simulations and previous optimal steady solutions by Ding & Wu (J. Fluid Mech., vol. 920, 2021, A48) exhibit smaller $Nu$ than the lower bound at large $Ra$ , indicating that these averaged fields are over-stabilised by the nonlinear terms. To incorporate a more physically plausible constraint to bound heat transport, an alternative approach, i.e. the quasilinear approach is invoked which delivers the highest heat transport and agrees well with Veronis's assumption, i.e. $Nu\sim Ra^{1/3}$ (Astrophys. J., vol. 137, 1963, p. 641). Interestingly, the background temperature $\tau$ yielded by the quasilinear approach can be non-unique when instability is subcritical.

The Impact of Nonlinear Stochastic Perturbation on Dynamical Behavior of an HIV Infection System

The Impact of Nonlinear Stochastic Perturbation on Dynamical Behavior of an HIV Infection System

Adaptive neural networks-based integral sliding mode control for T-S fuzzy model of delayed nonlinear systems

This paper presents a method for designing adaptive neural networks (NNs) based on integral sliding mode control (ISMC) for delayed Takagi-Sugeno fuzzy systems with external disturbances. To do this, a fuzzy sliding surface is constructed with the information of the delayed states, and an adaptive radial basis function (RBF) NNs-based fuzzy ISMC is designed. Here, the RBF NN function is utilized to approximate the nonlinear perturbation term. The resulting stability criteria are derived based on suitable Lyapunov functionals with the help of generalized free-weight matrix-based inequality. The resulting conditions ensure the asymptotic stability with H∞ attenuation level. Moreover, the designed sliding motion achieved the reachability condition under the adaptive RBF NNs based on the sliding mode controller. To end with, a design and comparative examples are given to show the success of the nominated control scheme and less conservatism of the derived sufficient conditions.