Nonlinear Dynamic Mode Research Articles

Study of nonlinear dynamic modes of native DNA via mathematical modeling methods

One of the main problems in the nonlinear dynamics of DNA is the identification of physical mechanisms responsible for the mechanical features of its behaviour. The formation and interaction of open states in the DNA structure may be one such mechanism. In this paper we propose a mathematical model of native DNA, which allows us to describe its thermodynamic and kinetic properties taking into account the collective behaviour of the ensemble of open states. The concept of a microscopic open state in DNA and its associated parameter, the displacement vector of nitrogenous bases, are introduced. By averaging the base displacement vectors over the ensemble of states, the thermodynamic variable is determined. According to the statistical model of DNA in the self-consistent field approximation, the structural parameter of the system thermalization is derived. This parameter reflects the statistical self-similarity in the behaviour of the ensemble of open states. Regularities of "criticality" for different ranges of the structural parameter are established and phenomenological representations of the free energy in the framework of the Ginzburg-Landau approach are proposed. Numerical modelling of the dynamics of native DNA is carried out for different ranges of the structural parameter, which has allowed us to establish the types of self-similar solutions and the corresponding open-state collective modes. It is shown that the latter have the nature of finite-amplitude fluctuations in the form of breahters and autosoliton modes, as well as blow-up dissipative structures. The sensitivity of the model to initial conditions is investigated, and the influence of nonlinearity inherent in the model on the modelling results is established. The limitations and prospects of using the approach proposed in this work for mathematical modelling of native DNA are discussed.

Nonlinear dynamic mode decomposition from time-resolving snapshots based on deep convolutional autoencoder

A novel nonlinear decomposition method of flow field is proposed based on the dynamic mode decomposition (DMD) and nonlinear mode decomposition autoencoder. The flow fields are indexed in time order and then input to the nonlinear neural networks to learn the connected observables and decomposed fields. The reconstructions of input fields are assumed to be the summation of the nonlinear decomposed fields. The nonlinear decoders are regarded as the nonlinear modes, which concentrate most of the total energy of the flow field data. The resulting nonlinear dynamic mode decomposition autoencoder reports a series of orderly low-dimensional representations and decomposition fields. Besides, the proposed method can be used for dynamic modeling and returns more stable and accurate predictions with a few number of low-dimensional representations. The present method is tested with the benchmark case, flow around a circular cylinder at Reynolds number Re = 100. The results in this example indicate that the proposed method achieves higher reconstruction accuracy using fewer modes, while retaining similar temporal dynamics and mode information as proper orthogonal decomposition and DMD.

Plane vibrational modes and localized nonlinear excitations in carbon nanotube bundle

Bulk nanomaterials made of carbon nanotubes (CNTs) possess unique physical and mechanical properties that are used in a number of applications, for example, for creating mass or force sensors, protection against shock and vibration. One of the specific implementations of such materials is a CNT bundle in which aligned CNTs are held together due to van der Waals interactions. Linear and nonlinear dynamic excitations in such materials are still poorly understood. Here we study plane waves and spatially localized nonlinear dynamic modes in a CNT bundle under plane strain conditions, when the CNT cross sections completely determine the deformed state of the bundle. In particular, we analyze vibration spectrum of single CNT, dispersion curves of phonons propagating along the close packed direction of CNT bundle, compressive solitons and discrete breathers in the form of single CNT oscillating at large amplitude in the CNT bundle. It is shown that compressive solitons can propagate long distances in the bundles of CNTs having radius no more than about 0.8 nm. Discrete breathers can exist in bundles of CNTs of any radius. The results obtained revealed new channels of energy localization and transport in CNT bundles, which is important for their use in various technologies.

Tracking Control of Hypersonic Vehicles with Input Saturation Based on Fast Terminal Sliding Mode

In this paper, the track problem of hypersonic vehicle is studied and analyzed with external disturbances, parameter uncertainty and input saturation. First, to make the design of controller convenient, the input and output linearized high-order model of hypersonic vehicle is transformed into a multi-variable second-order system model by introducing auxiliary error variables. To ensure the sliding mode manifold is practical finite time stable, an adaptive fast nonsingular terminal sliding mode controller is designed for hypersonic vehicle with unknown upper bound of disturbance. An adaptive anti-saturation fast nonsingular terminal sliding mode controller is designed by introducing hyperbolic tangent function an auxiliary system, which further solves the input saturation problem. Not only the requirement for actuator physical restraint is satisfied, but also the finite time stability of system sliding mode manifold is guaranteed at the same time. The vigorous proof is given using Lyapunov theory for the designed controllers. The numerical simulations are also conducted with the longitudinal nonlinear dynamic mode of hypersonic vehicle. The results demonstrate the effectiveness of the two designed controllers.

Method for reconstructing nonlinear modes with adaptive structure from multidimensional data.

We present a detailed description of a new approach for the extraction of principal nonlinear dynamical modes (NDMs) from high-dimensional data. The method of NDMs allows the joint reconstruction of hidden scalar time series underlying the observational variability together with a transformation mapping these time series to the physical space. Special Bayesian prior restrictions on the solution properties provide an efficient recognition of spatial patterns evolving in time and characterized by clearly separated time scales. In particular, we focus on adaptive properties of the NDMs and demonstrate for model examples of different complexities that, depending on the data properties, the obtained NDMs may have either substantially nonlinear or linear structures. It is shown that even linear NDMs give us more information about the internal system dynamics than the traditional empirical orthogonal function decomposition. The performance of the method is demonstrated on two examples. First, this approach is successfully tested on a low-dimensional problem to decode a chaotic signal from nonlinearly entangled time series with noise. Then, it is applied to the analysis of 250-year preindustrial control run of the INMCM4.0 global climate model. There, a set of principal modes of different nonlinearities is found capturing the internal model variability on the time scales from annual to multidecadal.

Observation of droplet soliton drift resonances in a spin-transfer-torque nanocontact to a ferromagnetic thin film

Magnetic droplet solitons are non-linear dynamical modes that can be excited in a thin film with perpendicular magnetic anisotropy with a spin-transfer-torque. Although droplet solitons have been proved to be stable with a hysteretic response to applied currents and magnetic fields at low temperature, measurements at room temperature indicate less stability and reduced hysteresis width. Here, we report evidence of droplet soliton drift instabilities, leading to drift resonances, at room temperature that explains their lower stability. Micromagnetic simulations show that the drift instability is produced by an effective field asymmetry in the nanocontact region that can have different origins.

Principal nonlinear dynamical modes of climate variability.

We suggest a new nonlinear expansion of space-distributed observational time series. The expansion allows constructing principal nonlinear manifolds holding essential part of observed variability. It yields low-dimensional hidden time series interpreted as internal modes driving observed multivariate dynamics as well as their mapping to a geographic grid. Bayesian optimality is used for selecting relevant structure of nonlinear transformation, including both the number of principal modes and degree of nonlinearity. Furthermore, the optimal characteristic time scale of the reconstructed modes is also found. The technique is applied to monthly sea surface temperature (SST) time series having a duration of 33 years and covering the globe. Three dominant nonlinear modes were extracted from the time series: the first efficiently separates the annual cycle, the second is responsible for ENSO variability, and combinations of the second and the third modes explain substantial parts of Pacific and Atlantic dynamics. A relation of the obtained modes to decadal natural climate variability including current hiatus in global warming is exhibited and discussed.

Confined Dissipative Droplet Solitons in Spin-Valve Nanowires with Perpendicular Magnetic Anisotropy

Magnetic dissipative droplets are localized, strongly nonlinear dynamical modes excited in nanocontact spin valves with perpendicular magnetic anisotropy. These modes find potential application in nanoscale structures for magnetic storage and computation, but dissipative droplet studies have so far been limited to extended thin films. Here, numerical and asymptotic analyses are used to demonstrate the existence and properties of novel solitons in confined structures. As a nanowire's width is decreased with a nanocontact of fixed size at its center, the observed modes undergo transitions from a fully localized two-dimensional droplet into a two-dimensional droplet edge mode and then a pulsating one-dimensional droplet. These solitons are interpreted as dissipative versions of classical, conservative solitons, allowing for an analytical description of the modes and the mechanisms of bifurcation. The presented results open up new possibilities for the study of low-dimensional solitons and droplet applications in nanostructures.

Regular and chaotic dynamics of a chain of magnetic dipoles with moments of inertia

The nonlinear dynamic modes of a chain of coupled spherical bodies having dipole magnetic moments that are excited by a homogeneous ac magnetic field are studied using numerical analysis. Bifurcation diagrams are constructed and used to find conditions for the presence of several types of regular, chaotic, and quasi-periodic oscillations. The effect of the coupling of dipoles on the excited dynamics of the system is revealed. The specific features of the Poincare time sections are considered for the cases of synchronous chaos with antiphase synchronization and asynchronous chaos. The spectrum of Lyapunov exponents is calculated for the dynamic modes of an individual dipole.

Nonlinear dynamics of the dual-wavelength CO2 laser

Theoretical studies have been made of the nonlinear-dynamic modes of operation of a dual-wavelength CO2 laser with continuous pumping by electric discharge and loss modulation in one of the lasing channels. It is shown that by varying the modulation depth and frequency, it is possible to control, over a fairly wide range, the time and energy parameters of lasing in both channels, going from regular to chaotic modes of operation.

Multi-basin dynamics of a protein in a crystal environment

The dynamics of the small protein crambin is studied in the crystal environment by means of a 5.1 nanoseconds molecular dynamics (MD) simulation. The resulting trajectory is analyzed in terms of a small set of nonlinear dynamical modes that best describe the molecule's fluctuations. These modes are nonlinear in the sence that they describe a trajectory exhibiting multiple transitions among local minima at various timescales. Nonlinear modes are responsible for most of the protein atomic fluctuations. An ultrametric hierarchy of sampled local minima describes the protein trajectory when structures are classified in terms of their interconfigurational mean squared distance. Transitions among minima involve small changes in the relative atomic positions of many atoms in the protein. The character of the MD trajectory fits within the framework of rugged energy landscape dynamics. This MD simulation clarifies the unique statistical features of the barriers between minima in the energy-like configurational landscape. Longer timescale dynamics seem to sample transitions between minima separated by relatively higher barriers. The MD trajectory of the system in configurational space can be described in terms of diffusion of a particle in real space with a waiting time distribution due to partial trapping in shallow minima. A description of the dynamics in terms of an open Newtonian system (the protein) coupled to a stochastic system (the solvent and fast quasiharmonic modes of the protein) reveals that the system loses memory of its configurational space within a few picoseconds. The diffusion of the protein in configurational space is anomalous in the sense that the mean square displacement increases sublinearly with time, i.e., as a power law with an exponent that is smaller than unity.

A New Identification Method for Serial Manipulator Arms

Developement of the methods to identify manipulator dynamics precisely is essential for the advanced control and design of the manipulator systems. Although dynamic models of manipulators are highly non linear, the form of nonlinear terms in the models are determined by the structures of the manipulators and all the coefficients are derived from the kinematic parameters of the manipulator components. A new method to identify these coefficients and to construct nonlinear dynamic mode is for serial manipulators with arbitrary number of links is proposed. The method consists of three types of simple enough tests performed without decomposing the manipulator into parts. The application to a real industrial robot arm shows that this method is very powerful and practical.