Invariant Surface Research Articles

MULTIMODE Calculations of Vibrational Spectroscopy and 1d Interconformer Tunneling Dynamics in Glycine Using a Full-Dimensional Potential Energy Surface.

A full-dimensional, permutationally invariant polynomial potential energy surface for glycine recently reported (R. Conte et al., J. Chem. Phys. 2020, 153, 244301) is used with the code MULTIMODE to determine the IR absorption spectra for Conformers I and II using a new separable dipole moment function. The calculated spectra agree well with the experimental ones. The full-dimensional nature of the potential allows us also to examine dynamical results, such as tunneling rates. Remarkably, using a one-dimensional path based on the potential energy surface to estimate the tunneling rate from Conformer VI to Conformer I, good agreement is found with the recent experimental measurement. Finally a brief comparison of our potential energy surface with a recently reported sGDML one is made.

Invariant algebraic surfaces of a chronic wasting disease model

Invariant algebraic surfaces of a chronic wasting disease model

Coexistence of chaotic attractor and unstable limit cycles in a 3D dynamical system.

The coexistence of stable limit cycles and chaotic attractors has already been observed in some 3D dynamical systems. In this paper we show, using the T-system, that unstable limit cycles and chaotic attractors can also coexist. Moreover, by completing the characterization of the existence of invariant algebraic surfaces and their associated global dynamics, we give a better understanding on the disappearance of the strange attractor and the limit cycles of the studied system.

Polynomial Vector Fields on the Clifford Torus

First, we characterize all the polynomial vector fields in [Formula: see text] which have the Clifford torus as an invariant surface. Then we study the number of invariant meridians and parallels that such polynomial vector fields can have on the Clifford torus as a function of the degree of these vector fields.

Invariant Algebraic Surfaces and Impasses

Polynomial vector fields $$X:\mathbb {R}^3\rightarrow \mathbb {R}^3$$ that have invariant algebraic surfaces of the form $$\begin{aligned} M = \{ f(x,y)z - g(x,y) = 0\} \end{aligned}$$ are considered. We prove that trajectories of X on M are solutions of a constrained differential system having $$\mathcal {I}=\{f(x,y)=0\}$$ as impasse curve. The main goal of the paper is to study the flow on M near points that are projected on typical impasse singularities. The Falkner–Skan equation (Llibre and Valls in Comput Fluids 86:71–76, 2013), the Lorenz system (Llibre and Zhang in J Math Phys 43:1622–1645, 2002) and the Chen system (Lu and Zhang in Int J Bifurc Chaos 17–8:2739–2748, 2007) are some of the well-known polynomial systems that fit our hypotheses.

Integrability and linearizability of a family of three-dimensional quadratic systems

We consider a three-dimensional vector field with quadratic nonlinearities and in general none of the axis plane is invariant. For our investigation, we are interesting in the case of – resonance at the origin. Hence, we deal with a nine parametric family of quadratic systems and our purpose is to understand the mechanisms of local integrability. By computing some obstructions, knowing as resonant focus quantities, first we present necessary conditions that guarantee the existence of two independent local first integrals at the origin. For this reason Gröbner basis and some other algorithms are employed. Then we examine the cases where the origin is linearizable. Some techniques like existence of invariant surfaces and Jacobi multipliers, Darboux method, properties of linearizable nodes of two dimensional systems and power series arguments are used to prove the sufficiency of the obtained conditions. For a particular three-parametric subfamily, we provide conditions on the parameters to guarantee the non-existence of a polynomial first integral.

Global Dynamical Behavior of FitzHugh–Nagumo Systems with Invariant Algebraic Surfaces

In this paper, we characterize the global dynamical behaviors of FitzHugh–Nagumo system $$\dot{x}=z$$ , $$\dot{y}=b(x-dy)$$ , $$\dot{z}=x(x-1)(x-a)+y+cz$$ which has invariant algebraic surfaces. As byproducts, we obtain some new dynamical phenomena related to the invariant surfaces at the infinity, which does not appear previously in the study of other models. In addition, since the system restricted to the invariant algebraic surfaces is not analytic, we adopt some new techniques to overcome this difficulty.

Position-Tracking Controller for Two-Wheeled Balancing Robot Applications Using Invariant Dynamic Surface

This paper suggests a position-tracking algorithm for the outer-loop through the invariant dynamic surface approach for balancing robot applications. The main feature is to devise a dynamic surface representing the target position-tracking performance with variable cut-off frequency. The proposed controller makes the dynamic surface invariant while updating the closed-loop cut-off frequency accordingly with the self-tuner. The closed-loop properties are rigorously analyzed. The experimental verification result shows that the proposed controller establishes the 48% enhancement of the circular tracking performance in comparison with the feedback linearization method, where the LEGO Mindstorms EV3 is used.

On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems

ABSTRACT For a map T from to C of the form , the dynamical system as a population model is competitive if . A well know theorem for competitive systems, presented by Hirsch [On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn. 2(2) (2008), pp. 169–179] and proved by Ruiz-Herrera [Exclusion and dominance in discrete population models via the carrying simplex, J. Differ. Equ. Appl. 19(1) (2013), pp. 96–113] with various versions by others, states that, under certain conditions, the system has a compact invariant surface that is homeomorphic to , attracting all the points of , and called carrying simplex. The theorem has been well accepted with a large number of citations. In this paper, we point out that one of its conditions requiring all the entries of the Jacobian matrix to be negative is unnecessarily strong and too restrictive. We prove the existence and uniqueness of a modified carrying simplex by reducing that condition to requiring every entry of Df to be nonpositive and each is strictly decreasing in . As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others.

Modeling critical phenomena in a generalized van der Pol model

The van der Pol model has a long history of being used in both the physical and biological sciences. This paper considers modeling critical phenomena in one of three-time scales fast-slow autonomous dynamical systems. We investigate a model that is a generalization of the classical van der Pol model. In the framework of this paper, we construct an invariant manifold with variable stability for the generalized van der Pol model. Expansions of the manifold and corresponding gluing function were constructed using the flow curvature method. We show the existence of a sufficiently smooth invariant surface (a black swan surface) for the model. Computer algebra methods are used for the quantitative analysis of the model.

A Study for MHD Boundary Layer Flow of Variable Viscosity over a Heated Stretching Sheet via Lie-Group Method

The present work deals with the study of Magnetohydrodynamic (MHD) boundary layer flow over a heated stretching sheet with variable fluid viscosity. The fluid viscosity is assumed to vary as a linear function of temperature in the presence of uniform transverse magnetic field. The fluid is assumed to be electric ally conducting. Lie-group method is applied for determining symmetry reductions for the MHD boundary-layer equations. Lie-group method starts out with a general infinitesimal group of tran sformations under which the given partial differential equations are invariant. The determining equations are a set of linear diffe rential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variable s. After the group has been determined, a solution to the given partial differe ntial equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the n umber of independent variables of the system. The effect of the Hartmann number (M ), the viscosity parameter ( A ) and the Prandtl number ( Pr ) on the horizontal and vertical velocities, temperature pr ofiles, wall heat transfer and the wall shear stress (skin friction) , have been studied and the results are plotted.

Second-Order Conditional Lie-Bäcklund Symmetry and Differential Constraint of Radially Symmetric Diffusion System

The classifications and reductions of radially symmetric diffusion system are studied due to the conditional Lie-Bäcklund symmetry method. We obtain the invariant condition, which is the so-called determining system and under which the radially symmetric diffusion system admits second-order conditional Lie-Bäcklund symmetries. The governing systems and the admitted second-order conditional Lie-Bäcklund symmetries are identified by solving the nonlinear determining system. Exact solutions of the resulting systems are constructed due to the compatibility of the original system and the admitted differential constraint corresponding to the invariant surface condition. For most of the cases, they are reduced to solving four-dimensional dynamical systems.

Dynamics of the FitzHugh–Nagumo system having invariant algebraic surfaces

In this paper, we study the dynamics of the FitzHugh–Nagumo system $$\dot{x}=z,\;\dot{y}=b\left( x-dy\right) ,\;\dot{z}=x\left( x-1\right) \left( x-a\right) +y+cz$$ having invariant algebraic surfaces. This system has four different types of invariant algebraic surfaces. The dynamics of the FitzHugh–Nagumo system having two of these classes of invariant algebraic surfaces have been characterized in Valls (J Nonlinear Math Phys 26:569–578, 2019). Using the quasi-homogeneous directional blow-up and the Poincaré compactification, we describe the dynamics of the FitzHugh–Nagumo system having the two remaining classes of invariant algebraic surfaces. Moreover, for these FitzHugh–Nagumo systems we prove that they do not have limit cycles.

Invariant surfaces for toric type foliations in dimension three

A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on (C3, 0) without saddle-nodes has invariant surface. We extend the argument of Cano–Cerveau for the nondicritical caseto the compact dicritical components of the exceptional divisor. These components are projective toric surfaces and the isolated invariant branches of the induced foliation extend to closed irreducible curves. We build the invariant surface as a germ along the singular locus and those closed irreducible invariant curves. The result of OrtizBobadilla–Rosales-Gonzalez–Voronin about the distribution of invariant branches indimension two is a key argument in our proof.

Intra- and intermolecular rovibrational states of HCl-H2O and DCl-H2O dimers from full-dimensional and fully coupled quantum calculations

We report full-dimensional and fully coupled quantum bound-state calculations of the J=1 intra- and intermolecular rovibrational states of two isotopologues of the hydrogen chloride-water dimer, HCl-H2O (HH) and DCl-H2O (DH). The present study complements our recent theoretical investigations of the J=0 nine-dimensional (9D) vibrational level structure of these and two other H/D isotopologues of this noncovalently bound molecular complex, and employs the same accurate 9D permutation invariant polynomial-neural network potential energy surface. The calculations yield all intramolecular vibrational fundamentals of the HH and DH dimers and the low-energy intermolecular rovibrational states in these intramolecular vibrational manifolds. The results are compared with those of the 9D J=0 calculations of the same dimers. The energy differences between the K=1 and K=0 eigenstates exhibit pronounced variations with the intermolecular rovibrational states, for which a qualitative explanation is provided.

Nonexistence of invariant tori transverse to foliations: An application of converse KAM theory.

Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend MacKay's converse Kolmogorov-Arnol'd-Moser condition to obtain a sufficient condition for the nonexistence of invariant surfaces that are transverse to a chosen 1D foliation. We show how useful foliations can be constructed from approximate integrals of the system. This theory is implemented numerically for two models: a particle in a two-wave potential and a Beltrami flow studied by Zaslavsky (Q-flows). These are both 3D volume-preserving flows, and they exemplify the dynamics seen in time-dependent Hamiltonian systems and incompressible fluids, respectively. Through both numerical and theoretical considerations, it is revealed how to choose foliations that capture the nonexistence of invariant tori with varying homologies.

Study for particular solutions of cylindrical shock waves in magnetogasdynamics

In this research article, a problem of shock waves propagation in magnetogasdynamics is considered in ideal gas. The problem can be represented by hyperbolic non-linear system of partial differential equations (PDEs). It is solved by similarity method under invariant surface conditions. I reduced non-linear system of magnetogasdynamics into first order system of ordinary differential equations (ODEs) and it is non-linear. The ratio of heat capacity at constant pressure to heat capacity at constant volume is also known as the adiabatic constant and their values for monatomic, diatomic gasses are more than one in ideal gas. So, I have obtained exact solution of magnetogasdynamics system in ideal gas for particular value of adiabatic indexγ=2. The effect of the ambient density exponent θ on flow quantities density, velocity, pressure, and magnetic field are determined before the shock.

Full-dimensional, ab initio potential energy surface for glycine with characterization of stationary points and zero-point energy calculations by means of diffusion Monte Carlo and semiclassical dynamics.

A full-dimensional, permutationally invariant potential energy surface (PES) for the glycine amino acid is reported. A precise fit to energies and gradients calculated at the density functional theory (DFT)/B3LYP level of electronic-structure theory with Dunning's aug-cc-pVDZ basis set is performed involving 20 000 low-energy points and associated Cartesian gradients plus about 50 000 additional higher-energy points. The fact that newly calculated DFT/B3LYP energies for the main stationary points are close to the coupled-cluster single-double-triple [CCSD(T)] values, recently reported in the literature, provides reassurance about the accuracy of the constructed PES. Eight conformers and numerous saddle points are identified and characterized by describing geometries, relative stability, and harmonic frequencies. Stochastic and dynamical approaches are employed to study the vibrational ground state. Specifically, diffusion Monte Carlo simulations and approximate quantum dynamics, performed by means of the adiabatic switching semiclassical initial value representation technique, provide zero-point energies in excellent agreement with each other. The PES we report is sufficiently complete to permit spectroscopic and dynamical studies on glycine, which may be of interest to the biochemical and astrochemistry communities.

On invariant surfaces in gene network models

On invariant surfaces in gene network models

Warped Translation Surfaces of Finite Type in Simply Isotropic 3-Spaces

In this paper, we classify warped translation surfaces being invariant surfaces of i-type, that is, the generating curve has formed by the intersection of the surface with the isotropic xz-plane in the three-dimensional simply isotropic space $\mathbb I^3$ under the condition $\Delta^{J}x_i=\lambda_i x_i,$ with J=I,II. Here, $\Delta^{J}$ is the Laplace operator with respect to first and second fundamental form and $\lambda_i$, $i=1,2,3$ are some real numbers. Also, as an application, we give some examples for these surfaces and also some explicit graphics of them. All graphics have been plotted with Maple14.