Original Phase Space Research Articles

Deep-CNN based knowledge learning with Beluga Whale optimization using chaogram transformation using intelligent sensors for speech emotion recognition

In recent times, there has been a lot of focus placed on speech emotion recognition, often known as SER. SER is a fundamental approach to emotional human-machine interaction. Utilising DCNNs has resulted in significant advancements being made, notably with regard to the learning of high-level characteristics for SER. However, one of the primary challenges that arises when applying deep neural networks to SER is the problem of overfitting, which occurs commonly when too many parameters are chosen using insufficiently large datasets. The majority of the research that has been done to attempt to solve this problem has focused on translating the audio input into a picture and employing transfer learning methods. According to research, the patterns generated in this region include significant emotional aspects of the speaker. A new speech signal format called Chaogram was designed to project these patterns, which would result in three channels like RGB images, in order to give an input that works with VGG-based Deep-CNNs. The original phase space is then used to reconstruct the voice samples in a three-dimensional space. In the subsequent stage, the Chaogram photos' finer details were accentuated by image enhancing techniques. To learn Chaogram's high-level features and emotion classifications, the Visual Geometry Group (VGG) deep convolutional neural network (DCNN) is employed after being pre-trained on the massive ImageNet dataset using intelligent sensors. We next apply transfer learning to our datasets to further improve the provided model, which we combine with the proposed model. To enhance the hyper-parameter layout of architecture-determined CNNs, a new Deep-CNN with BWO (Beluga Whale Optimization) is introduced. In order to apply Deep–CNN–KL and to the field of SER, thispropose a new method of representing speech signals, which we call Chaograms, by mapping the signal onto a 2D image. On two publicly available emotion datasets, these findings demonstrate the ability of the proposed approach, which has the potential to significantly improve SER applications: EMO-DB and eNTERFACE05. Image enhancement methods that place more emphasis on such features may lead to more precise classification in the performance analysis. Classification accuracy can be considerably improved by adapting these images to work with the pre-trained Deep-CNN inputs.

Optimal control for sampling the transition path process and estimating rates

Many processes in nature such as conformal changes in biomolecules and clusters of interacting particles, genetic switches, mechanical or electromechanical oscillators with added noise, and many others are modeled using stochastic differential equations with small white noise. The study of rare transitions between metastable states in such systems is of great interest and importance. The direct simulation of rare transitions is difficult due to long waiting times. Transition path theory is a mathematical framework for the quantitative description of rare events. Its crucial component is the committor function, the solution to a boundary value problem for the backward Kolmogorov equation. The key fact exploited in this work is that the optimal controller constructed from the committor leads to the generation of transition trajectories exclusively. We prove this fact for a broad class of stochastic differential equations. Moreover, we demonstrate that the committor computed for a dimensionally reduced system and then lifted to the original phase space still allows us to construct an effective controller and estimate the transition rate with reasonable accuracy. Furthermore, we propose an all-the-way-through scheme for computing the committor via neural networks, sampling the transition trajectories, and estimating the transition rate without meshing the space. We apply the proposed methodology to four test problems: the overdamped Langevin dynamics with Mueller’s potential and the rugged Mueller potential in 10D, the noisy bistable Duffing oscillator, and Lennard-Jones-7 in 2D.

Improved compact representation method for Monte Carlo phase spaces

Phase space files used in particle Monte Carlo simulations can be large, and thereby awkward to distribute and slow to load into memory. To circumvent this, compact source models and phase space representations have been developed. In this study, the aim is to increase the versatility of a previously presented method, based on principal component analysis whitening, by evaluating several modifications to the method on simulated cone-beam computed tomography projection images. Scaling the phase space components before whitening was seen to give the largest improvement and allowed for accurate modelling of phase spaces with a shifted detector and/or a bowtie filter present, configurations where the original method struggled. The modified method reproduced results using the original phase space to within a few percent both in energy fluence on the detector and in simulated projection and scatter images of a patient pelvis, without obvious position-dependence of the errors. Hence, the modified method was deemed sufficiently accurate for most applications relying on cone-beam computed tomography simulations. The final aim is to use the more versatile method for a broader range of phase space-based applications such as linear accelerator beam simulations and room shielding calculations.

Modified gauge-unfixing formalism and gauge symmetries in the noncommutative chiral bosons theory

Abstract We use the gauge-unfixing (GU) formalism framework in a two-dimensional noncommutative chiral bosons (NCCB) model to disclose new hidden symmetries. That amounts to converting a second-class system to a first-class one without adding any extra degrees of freedom in phase space. The NCCB model has two second-class constraints —one of them turns out as a gauge symmetry generator while the other one, considered as a gauge-fixing condition, is disregarded in the converted gauge-invariant system. We show that it is possible to apply a conversion technique based on the GU formalism direct to the second-class variables present in the NCCB model, constructing deformed gauge-invariant GU variables, a procedure which we name here as modified GU formalism. For the canonical analysis in noncommutative phase space, we compute the deformed Dirac brackets between all original phase space variables. We obtain two different gauge-invariant versions for the NCCB system and, in each case, a GU Hamiltonian is derived satisfying a corresponding first-class algebra. Finally, the phase space partition function is presented for each case allowing for a consistent functional quantization for the obtained gauge-invariant NCCB.

Preservation of Quadratic Invariants by Semiexplicit Symplectic Integrators for Nonseparable Hamiltonian Systems

We prove that the recently developed semiexplicit symplectic integrators for non-separable Hamiltonian systems preserve any linear and quadratic invariants possessed by the Hamiltonian systems. This is in addition to being symmetric and symplectic as shown in our previous work; hence it shares the crucial structure-preserving properties with some of the well-known symplectic Runge--Kutta methods such as the Gauss--Legendre methods. The proof follows two steps: First we show how the extended Hamiltonian system proposed by Pihajoki inherits linear and quadratic invariants in the extended phase space from the original Hamiltonian system. Then we show that this inheritance in turn implies that our integrator preserves the original linear and quadratic invariants in the original phase space. We also analyze preservation/non-preservation of these invariants by Tao's extended Hamiltonian system and the extended phase space integrators of Pihajoki and Tao. The paper concludes with numerical demonstrations of our results using a simple test case and a system of point vortices.

Explicit K-Symplectic and Symplectic-like Methods for Charged Particle System in General Magnetic Field

We propose explicit K-symplectic and explicit symplectic-like methods for the charged particle system in a general strong magnetic field. The K-symplectic methods are also symmetric. The charged particle system can be expressed both in a canonical and a non-canonical Hamiltonian system. If the three components of the magnetic field can be integrated in closed forms, we construct explicit K-symplectic methods for the non-canonical charged particle system; otherwise, explicit symplectic-like methods can be constructed for the canonical charged particle system. The symplectic-like methods are constructed by extending the original phase space and obtaining the augmented separable Hamiltonian, and then by using the splitting method and the midpoint permutation. The numerical experiments have shown that compared with the higher order implicit Runge-Kutta method, the explicit K-symplectic and explicit symplectic-like methods have obvious advantages in long-term energy conservation and higher computational efficiency. It is also shown that the influence of the parameter ε in the general strong magnetic field on the Runge-Kutta method is bigger than the two kinds of symplectic methods.

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit, K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for two non-canonical Hamiltonian systems. Numerical tests show that the proposed methods exhibit good numerical performance in preserving the phase orbit and the energy of the system over long time, whereas higher order Runge–Kutta methods do not preserve these properties. Numerical tests also show that the K-symplectic methods exhibit better efficiency than that of the same order implicit symplectic, explicit and implicit symplectic methods for the original nonseparable non-canonical systems. On the other hand, the fourth order K-symplectic method is more efficient than the fourth order Yoshida’s method, the optimized partitioned Runge–Kutta and Runge–Kutta–Nyström explicit K-symplectic methods for the extended phase space Hamiltonians, but less efficient than the the optimized partitioned Runge–Kutta and Runge–Kutta–Nyström extended phase space symplectic-like methods with the midpoint permutation.

Semiexplicit symplectic integrators for non-separable Hamiltonian systems

We construct a symplectic integrator for non-separable Hamiltonian systems combining an extended phase space approach of Pihajoki and the symmetric projection method. The resulting method is semiexplicit in the sense that the main time evolution step is explicit whereas the symmetric projection step is implicit. The symmetric projection binds potentially diverging copies of solutions, thereby remedying the main drawback of the extended phase space approach. Moreover, our semiexplicit method is symplectic in the original phase space. This is in contrast to existing extended phase space integrators, which are symplectic only in the extended phase space. We demonstrate that our method exhibits an excellent long-time preservation of invariants, and also that it tends to be as fast as and can be faster than Tao’s explicit modified extended phase space integrator particularly for small enough time steps and with higher-order implementations and for higher-dimensional problems.

Geometric Integration of ODEs Using Multiple Quadratic Auxiliary Variables

We present a novel numerical method for solving ODEs while preserving polynomial first integrals. The method is based on introducing multiple quadratic auxiliary variables to reformulate the ODE as an equivalent but higher-dimensional ODE with only quadratic integrals to which the midpoint rule is applied. The quadratic auxiliary variables can subsequently be eliminated yielding a midpoint-like method on the original phase space. The resulting method is shown to be a novel discrete gradient method. Furthermore, the averaged vector field method can be obtained as a special case of the proposed method. The method can be extended to higher-order through composition and is illustrated through a number of numerical examples.

Pullback exponential attractors for a class of non-Newtonian micropolar fluids

In this paper, we consider the pullback asymptotic behaviors of weak solutions for the 2D non-autonomous Non-Newtonian micropolar fluids. Employing the methods of ℓ-trajectories in [31], we first establish the existence of a finite dimensional pullback attractor and a pullback exponential attractor for the process {L(t,τ)}t≥τ associated with (1.2)-(1.4) in the phase space Xℓ. Then we construct a pullback exponential attractor for the process {U(t,τ)}t≥τ associated with (1.2)-(1.4) in the original phase space H×L2(Ω).

On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions

<p style='text-indent:20px;'>The objective of this paper is to study the fractal dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Inspired by the idea of the <inline-formula><tex-math id="M1">\begin{document}$ \ell $\end{document}</tex-math></inline-formula>-trajectory method, we prove the existence of a finite dimensional global attractor in an auxiliary normed space for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions and estimate the fractal dimension of the global attractor in the original phase space for this system by defining a Lipschitz mapping from the auxiliary normed space into the original phase space.</p>

Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds

We recently derived a method, local orthogonal rectification (LOR), that provides a natural and useful geometric frame for analyzing dynamics relative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst., 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simplifies the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories' behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.

On motions of a conservative system on invariant manifolds

In [Irtegov and Burlakova, 2017], the algorithms for the qualitative analysis of conservative systems have been presented. These are based on the Routh-Lyapunov method [Lyapunov, 1954] and some its modifications [Irtegov and Titorenko, 2009] as well as computer algebra methods [Cox, Little, and O’Shea, 1997]. In the paper the application of the algorithms is demonstrated by analysing a conservative system, the study of which is also of interest. We conduct qualitative analysis for the differential equations describing the rotational motion of a rigid body with a fixed point in two constant force fields. Similar problems arise, e.g., in space dynamics [Sarychev and Gutnik, 2015], quantum mechanics [Adler, Marikhin, and Shabat, 2012], [Smirnov, 2008]. In the phase space of the problem, we isolate the invariant manifolds of maximal dimension and study the equations of motion on them. For these equations, solutions (and their families) corresponding in the original phase space of the problem to permanent rotations and pendulum-like oscillations of the body as well as the invariant manifolds of 2nd level, which these solutions belong to, have been found and their Lyapunov’s stability has been investigated. The possibility of stabilization for the motions of conservative systems, whose stability conditions have the form of some constraints on the constants of first integrals, is discussed.

Pullback exponential attractors for the viscous Cahn–Hilliard–Navier–Stokes system with dynamic boundary conditions

This paper is concerned with the existence of pullback exponential attractors for the viscous Cahn–Hilliard–Navier–Stokes system with dynamic boundary conditions. Due to the lack of the Hölder continuity in time of the semigroup and the smoothing property of the difference of two weak solutions, we construct a pullback exponential attractor in the original phase space by the ℓ-trajectories methods.

Remarks on the linking theory of shape dynamics

This short note is an attempt to bring out the geometric structures in the linking theory of shape dynamics. Symplectic induction is applied to give a natural construction of the extended phase space used in the linking theory as a trivial vector bundle over the original phase space for canonical gravity. The geometry of the gauge fixing for shape dynamics is analyzed with the assistance of the Lichnerowicz–York equation lifted to the extended phase space. An alternative description is provided to show how the same geometry simply derives from symplectic induction.

Verification of 5D continuum gyrokinetic code COGENT: Studies of kinetic drift wave instability

COGENT (Continuum Gyrokinetic Edge New Technology) is a kinetic plasma simulation code that is being developed by the edge simulation laboratory (ESL) collaboration. The original version of the code has been developed in a 4D phase space (2D configuration space and 2D velocity space) to address kinetic plasma phenomena in a complex magnetic field geometry including the core, magnetic separatrix, and scrape‐off layer regions. This work is focused on extending the original 4D phase space to a 5D phase space (3D2V) to address full kinetic turbulences in a tokamak edge region. Here, we report on the current status of 5D COGENT, which presently operates in a shear‐less, simple slab geometry. As a verification study, we use the problems of collision‐less drift wave instability (universal instability) including its modification in the presence of collisional effects. The simulation model includes the gyrokinetic equations for ion and electron species coupled to the long‐wavelength limit of the gyro‐Poisson equation. Collisional effects are represented by the Krook collision model. Linear analytical results for the drift mode growth rate and real frequency are recovered, and the non‐linear stage is modelled and analysed as well. In addition, extensive 5D runs have been performed to address the effects of the drift wave instability on blob/filamentary structures characteristic of a tokamak edge. A helical‐shaped potential perturbation is observed to grow exponentially in time while spinning around the filament axis with electron drift frequency.

Gauge unfixing formalism of the O(N) nonlinear sigma model

In a few years back, one of us has proposed a new scheme of embedding constrained systems based mainly on the Gauge Unfixing (GU) formalism and it is known as an extended GU formalism. The proposition was to modify directly the original phase space variables of an arbitrary system in order to turn the system a gauge invariant one. Since the new theory is gauge invariant, we can say that the new system is a first class one in Dirac terminology. In this way, the GU method is a constraint conversion technique. In this work, by using this extended GU formalism we have obtained two different versions of the first class system related on the O(N) nonlinear sigma model

Stability integral manifold of the differential equations system in critical case

We consider the stability problem for non-zero integral manifolds of a non-linear finite-dimensional system of ordinary differential equations, where the right-hand side is a periodic vector function with respect to an independent variable and contains a parameter. It is assumed that the studied system has a trivial integral manifold for all values of the parameter, and the corresponding linear subsystem does not have the property of exponential dichotomy. The aim of the paper is to find sufficient conditions of existence in a neighborhood of the system equilibrium state for stable non-zero integral manifold to be lower dimension than the original phase space. For this purpose, based on the classical method of Lyapunov functions and the transforming matrix method operators are constructed, allowing solve the task by finding their fixed points. Due to the specific nature of the considered systems Lyapunov functions method is modified.

Application of first-order canonical perturbation method with dissipative Hori-like kernel

Lie-Hori canonical perturbation theory provides asymptotic solutions for conservative Hamiltonian systems. This restriction prevents the canonical method from being applied directly to dissipative mechanical systems. There are, however, two main alternatives to overcome this difficulty, enabling the application of canonical perturbation methods. The first one consists in constructing a time-dependent Hamiltonian, through a generating function, related to the energy dissipation pattern of the system. The second embeds the original phase space into a double dimensional one where the dynamics of the system can be formulated in a Hamiltonian way. In this paper, a modified Lie-Hori method that avoid the disadvantages of the former approaches is proposed. Namely, it is not necessary to find out a time-dependent generating function, nor doubling the number of the canonical variables of the original problem. The new algorithm provides first order analytical solutions for a certain set of dissipative non-linear dynamical systems. It is based on a suitable modification of the Hori kernel in the double-dimensional embedding phase space, allowing the inclusion of the dissipative (or generalized) forces. By means of this redefined auxiliary system, the path-integrals of the method can be performed in a domain of the phase space with the same dimensionality as the original problem.

Hierarchical structure of noncanonical Hamiltonian systems

Topological constraints play a key role in the self-organizing processes that create structures in macro systems. In fact, if all possible degrees of freedom are actualized on equal footing without constraint, the state of ‘equipartition’ may bear no specific structure. Fluid turbulence is a typical example—while turbulent mixing seems to increase entropy, a variety of sustained vortical structures can emerge. In Hamiltonian formalism, some topological constraints are represented by Casimir invariants (for example, helicities of a fluid or a plasma), and then, the effective phase space is reduced to the Casimir leaves. However, a general constraint is not necessarily integrable, which precludes the existence of an appropriate Casimir invariant; the circulation is an example of such an invariant. In this work, we formulate a systematic method to embed a Hamiltonian system in an extended phase space; we introduce phantom fields and extend the Poisson algebra. A phantom field defines a new Casimir invariant, a cross helicity, which represents a topological constraint that is not integrable in the original phase space. Changing the perspective, a singularity of the extended system may be viewed as a subsystem on which the phantom fields (though they are actual fields, when viewed from the extended system) vanish, i.e., the original system. This hierarchical relation of degenerate Poisson manifolds enables us to see the ‘interior’ of a singularity as a sub Poisson manifold. The theory can be applied to describe bifurcations and instabilities in a wide class of general Hamiltonian systems (Yoshida and Morrison 2014 Fluid Dyn. Res. 46 031412).