Index-1 Saddle Points Research Articles

Solution landscapes of the diblock copolymer-homopolymer model under two-dimensional confinement

We investigate the solution landscapes of the confined diblock copolymer and homopolymer in two-dimensional domain by using the extended Ohta-Kawasaki model. The projection saddle dynamics method is developed to compute the saddle points with mass conservation and construct the solution landscape by coupling with downward and upward search algorithms. A variety of stationary solutions are identified and classified in the solution landscape, including Flower class, Mosaic class, Core-shell class, and Tai-chi class. The relationships between different stable states are shown by either transition pathways connected by index-1 saddle points or dynamical pathways connected by a high-index saddle point. The solution landscapes also demonstrate the symmetry-breaking phenomena, in which more solutions with high symmetry are found when the domain size increases.

A Four-Zone Model and Nonlinear Dynamic Analysis of Solution Multiplicity of Buoyancy Ventilation in Underground Building

The solution multiplicity of natural ventilation in buildings is very important to personnel safety and ventilation design. In this paper, a four-zone model of buoyancy ventilation in typical underground building is proposed. The underground structure is divided to four zones, a differential equation is established in each zone, and therefore, there are four differential equations in the underground structure. By solving and analyzing the equilibrium points and characteristic roots of the differential equations, we analyze the stability of three scenarios and obtain the criterions to determine the stability and existence of solutions for two scenarios. According to these criterions, the multiple steady states of buoyancy ventilation in any four-zone underground buildings for different stack height ratios and the strength ratios of the heat sources can be obtained. These criteria can be used to design buoyancy ventilation or natural exhaust ventilation systems in underground buildings. Compared with the two-zone model in (Liu et al. 2020), the results of the proposed four-zone model are more consistent with CFD results in (Liu et al. 2018). In addition, the results of proposed four-zone model are more specific and more detailed in the unstable equilibrium point interval. We find that the unstable equilibrium point interval is divided into two different subintervals corresponding to the saddle point of index 2 and the saddle focal equilibrium point of index 2, respectively. Finally, the phase portraits and vector field diagrams for the two scenarios are given.

Fate of escaping orbits in barred galaxies

AbstractIn the present work, we have developed a two-dimensional gravitational model of barred galaxies to analyse the fate of escaping stars from the central barred region. For that, the model has been analysed for two different bar profiles viz. strong and weak. Here the phenomena of stellar escape from the central barred region have been studied from the perspective of an open Hamiltonian dynamical system. We observed that the escape routes correspond to the escape basins of the two index-1 saddle points. Our results show that the formation of spiral arms is encouraged for the strong bars. Also, the formation of grand design spirals is more likely for strong bars if they host central super massive black holes (SMBHs). In the absence of central SMBHs, the formation of less-prominent spiral arms is more likely. Again, for weak bars, the formation of inner disc rings is more probable.

A three-dimensional dynamical model for double-barred galaxies, escape dynamics and the role of the NHIMs

A new analytic multi-component gravitational model with three degrees of freedom to describe the orbital properties of stars in a double-barred galaxy is introduced. We assume that the galaxy contains two bars: the primary one, parallel to the disc and the secondary one which is perpendicular to the primary bar. By following the trajectories, belonging to large sets of starting conditions, we manage to distinguish between localized (chaotic, sticky or regular) and escaping motion of stars. The character of orbits is revealed by presenting modern colour-coded diagrams on several choices of planes of two dimensions. Additionally, we investigate the properties of the normally hyperbolic invariant manifolds (NHIMs), associated with the index-1 saddle points of the system. The dynamics near the index-1 saddle points is demonstrated by presenting the bifurcation diagrams of the Lyapunov periodic orbits, and by visualizing the restriction of the Poincaré maps to the NHIMs. Useful conclusions are drawn by comparing our results with previous related ones, from other types of Hamiltonian systems.

Orbital and escape dynamics in barred galaxies – IV. Heteroclinic connections

Continuing the series of papers on a new model for a barred galaxy, we investigate the heteroclinic connections between the two normally hyperbolic invariant manifolds sitting over the two index-1 saddle points of the effective potential. The heteroclinic trajectories and the nearby periodic orbits of similar shape populate the bar region of the galaxy and a neighbourhood of its nucleus. Thereby we see a direct relation between the important structures of the interior region of the galaxy and the projection of the heteroclinic tangle into the position space. As a side result, we obtain a detailed picture of the primary heteroclinic intersection surface in the phase space.

Simplified gentlest ascent dynamics for saddle points in non-gradient systems.

The gentlest ascent dynamics (GAD) [W. E and X. Zhou, Nonlinearity 24, 1831 (2011)] is a time continuous dynamics to efficiently locate saddle points with a given index by coupling the position and direction variables together. These saddle points play important roles in the activated process of randomly perturbed dynamical systems. For index-1 saddle points in non-gradient systems, the GAD requires two direction variables to approximate, respectively, the eigenvectors of the Jacobian matrix and its transposed matrix. In the particular case of gradient systems, the two direction variables are equal to the single minimum mode of the Hessian matrix. In this note, we present a simplified GAD which only needs one direction variable even for non-gradient systems. This new method not only reduces the computational cost for the direction variable by half but also avoids inconvenient transpose operation of the Jacobian matrix. We prove the same convergence property for the simplified GAD as that for the original GAD. The motivation of our simplified GAD is the formal analogy with Hamilton's equation governing the noise-induced exit dynamics. Several non-gradient examples are presented to demonstrate our method, including a two dimensional model and the Allen-Cahn equation in the presence of shear flow.

Chaos in the incommensurate fractional order system and circuit simulations

In this paper, the dynamics of an autonomous three dimensional system with three nonlinearity quadratic terms is investigated. With the help of stability analysis of equilibrium points, the Lyapunov exponent, and the phase portraits we study the dynamical behavior of the fractional order system. We find that system can display double scroll and double-four wing chaotic attractor. The commensurate order has been investigated and we find that the necessary condition for chaos to appear is $$0.84<\alpha \le 1$$ . For the incommensurate order, we show that the instability measure of the equilibrium points for all the saddle points of index 2 must be non-negative for a system with five equilibrium points to be chaotic. Numerical simulations and the analog simulations are carried out in Multisim.

Loss surface of XOR artificial neural networks

Training an artificial neural network involves an optimization process over the landscape defined by the cost (loss) as a function of the network parameters. We explore these landscapes using optimization tools developed for potential energy landscapes in molecular science. The number of local minima and transition states (saddle points of index one), as well as the ratio of transition states to minima, grow rapidly with the number of nodes in the network. There is also a strong dependence on the regularization parameter, with the landscape becoming more convex (fewer minima) as the regularization term increases. We demonstrate that in our formulation, stationary points for networks with N_{h} hidden nodes, including the minimal network required to fit the XOR data, are also stationary points for networks with N_{h}+1 hidden nodes when all the weights involving the additional node are zero. Hence, smaller networks trained on XOR data are embedded in the landscapes of larger networks. Our results clarify certain aspects of the classification and sensitivity (to perturbations in the input data) of minima and saddle points for this system, and may provide insight into dropout and network compression.

Correlating the escape dynamics and the role of the normally hyperbolic invariant manifolds in a binary system of dwarf spheroidal galaxies

We elucidate the escape properties of stars moving in the combined gravitational field of a binary system of two interacting dwarf spheroidal galaxies. A galaxy model of three degrees of freedom is adopted for describing the dynamical properties of the Hamiltonian system. All the numerical values of the involved parameters are chosen having in mind the real binary system of the dwarf spheroidal galaxies NGC 147 and NGC 185. We distinguish between bounded (regular, sticky or chaotic) and escaping motion by classifying initial conditions of orbits in several types of two dimensional planes, considering only unbounded motion for several energy levels. We analyse the orbital structure of all types of two dimensional planes of initial conditions by locating the basins of escape and also by measuring the corresponding escape time of the orbits. Furthermore, the properties of the normally hyperbolic invariant manifolds (NHIMs), located in the vicinity of the index-1 saddle points L1, L2, and L3, are also investigated. These manifolds are of great importance, as they control the flow of stars (between the two galaxies and towards the exterior region) over the different saddle points. In addition, bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of the Poincaré map to the NHIMs are presented for revealing the dynamics in the neighbourhood of the saddle points. Comparison between the current outcomes and previous related results is also made.

Convex splitting method for the calculation of transition states of energy functional

Among numerical methods for partial differential equations arising from steepest descent dynamics of energy functionals (e.g., Allen–Cahn and Cahn–Hilliard equations), the convex splitting method is well-known to maintain unconditional energy stability for a large time step size. In this work, we show how to use the convex splitting idea to find transition states, i.e., index-1 saddle points of the same energy functionals. Based on the iterative minimization formulation (IMF) for saddle points [14], we introduce the convex splitting method to minimize the auxiliary functional at each cycle of the IMF. We present a general principle of constructing convex splitting forms for these auxiliary functionals and show how to avoid solving nonlinear equations. The new numerical scheme based on the convex splitting method allows for large time step sizes. The new methods are tested for the one dimensional Ginzburg–Landau energy functional in the search of the Allen–Cahn or Cahn–Hilliard types of transition states. We provide the numerical results of transition states for the two dimensional Landau–Brazovskii energy functional for diblock copolymers.

Orbital and escape dynamics in barred galaxies – III. The 3D system: correlations between the basins of escape and the NHIMs

The escape dynamics of the stars in a barred galaxy composed of a spherically symmetric central nucleus, a bar, a flat thin disk and a dark matter halo component is investigated by using a realistic three degrees of freedom (3-dof) dynamical model. Modern colour-coded diagrams are used for distinguishing between bounded and escaping motion. In addition, the smaller alignment index (SALI) method is deployed for determining the regular, sticky or chaotic nature of bounded orbits. We reveal the basins of escape corresponding to the escape through the two symmetrical escape channels around the Lagrange points $L_2$ and $L_3$ and also we relate them with the corresponding distribution of the escape times of the orbits. Furthermore, we demonstrate how the stable manifolds, around the index-1 saddle points, accurately define the fractal basin boundaries observed in the colour-coded diagrams. The development scenario of the fundamental vertical Lyapunov periodic orbit is thoroughly explored for obtaining a more complete view of the unfolding of the singular behaviour of the dynamics at the cusp values of the parameters. Finally, we examine how the combination of the most important parameters of the bar (such as the semi-major axis and the angular velocity) influences the observed stellar structures (rings and spirals) which are formed by escaping stars guided by the invariant manifolds near the saddle points.

An interval-matrix branch-and-bound algorithm for bounding eigenvalues

We present and explore the behaviour of a branch-and-bound algorithm for calculating valid bounds on the kth largest eigenvalue of a symmetric interval matrix. Branching on the interval elements of the matrix takes place in conjunction with the application of Rohn's method (an interval extension of Weyl's theorem) in order to obtain valid outer bounds on the eigenvalues. Inner bounds are obtained with the use of two local search methods. The algorithm has the theoretical property that it provides bounds to any arbitrary precision (assuming infinite precision arithmetic) within finite time. In contrast with existing methods, bounds for each individual eigenvalue can be obtained even if its range overlaps with the ranges of other eigenvalues. Performance analysis is carried out through nine examples. In the first example, a comparison of the efficiency of the two local search methods is reported using 4000 randomly generated matrices. The eigenvalue bounding algorithm is then applied to five randomly generated matrices with overlapping eigenvalue ranges. Valid and sharp bounds are indeed identified given a sufficient number of iterations. Furthermore, most of the range reduction takes place in the first few steps of the algorithm so that significant benefits can be derived without full convergence. Finally, in the last three examples, the potential of the algorithm for use in algorithms to identify index-1 saddle points of nonlinear functions is demonstrated.

Enclosure of all index-1 saddle points of general nonlinear functions

Transition states (index-1 saddle points) play a crucial role in determining the rates of chemical transformations but their reliable identification remains challenging in many applications. Deterministic global optimization methods have previously been employed for the location of transition states (TSs) by initially finding all stationary points and then identifying the TSs among the set of solutions. We propose several regional tests, applicable to general nonlinear, twice continuously differentiable functions, to accelerate the convergence of such approaches by identifying areas that do not contain any TS or that may contain a unique TS. The tests are based on the application of the interval extension of theorems from linear algebra to an interval Hessian matrix. They can be used within the framework of global optimization methods with the potential of reducing the computational time for TS location. We present the theory behind the tests, discuss their algorithmic complexity and show via a few examples that significant gains in computational time can be achieved by using these tests.

Gentlest ascent dynamics for calculating first excited state and exploring energy landscape of Kohn-Sham density functionals.

We develop the gentlest ascent dynamics for Kohn-Sham density functional theory to search for the index-1 saddle points on the energy landscape of the Kohn-Sham density functionals. These stationary solutions correspond to excited states in the ground state functionals. As shown by various examples, the first excited states of many chemical systems are given by these index-1 saddle points. Our novel approach provides an alternative, more robust way to obtain these excited states, compared with the widely used ΔSCF approach. The method can be easily generalized to target higher index saddle points. Our results also reveal the physical interest and relevance of studying the Kohn-Sham energy landscape.

An Iterative Minimization Formulation for Saddle Point Search

This paper proposes and analyzes an iterative minimization formulation for searching index-1 saddle points of an energy function. We give a general and rigorous description of eigenvector-following methodology in this iterative scheme by considering an auxiliary optimization problem at each iteration in which the new objective function is locally defined near the current guess. We prove that this scheme has a quadratic local convergence rate in terms of number of iterations, in comparison to the linear rate of the gentlest ascent dynamics [W. E and X. Zhou, Nonlinearity, 24 (2011), pp. 1831--1842] and many other existing methods. We also propose the generalization of the new methodology for saddle points of higher index and for constrained energy functions on the manifold. Preliminary numerical results on the nature of this iterative minimization formulation are presented.

Infinite-Scroll Attractor Generated by the Complex Pendulum Model

We report the finding of the simple nonlinear autonomous system exhibiting infinite-scroll attractor. The system is generated from the pendulum equation with complex-valued function. The proposed system is having infinitely many saddle points of index two which are responsible for the infinite-scroll attractor.

Atomistic simulations of rare events using gentlest ascent dynamics

The dynamics of complex systems often involve thermally activated barrier crossing events that allow these systems to move from one basin of attraction on the high dimensional energy surface to another. Such events are ubiquitous, but challenging to simulate using conventional simulation tools, such as molecular dynamics. Recently, E and Zhou [Nonlinearity 24(6), 1831 (2011)] proposed a set of dynamic equations, the gentlest ascent dynamics (GAD), to describe the escape of a system from a basin of attraction and proved that solutions of GAD converge to index-1 saddle points of the underlying energy. In this paper, we extend GAD to enable finite temperature simulations in which the system hops between different saddle points on the energy surface. An effective strategy to use GAD to sample an ensemble of low barrier saddle points located in the vicinity of a locally stable configuration on the high dimensional energy surface is proposed. The utility of the method is demonstrated by studying the low barrier saddle points associated with point defect activity on a surface. This is done for two representative systems, namely, (a) a surface vacancy and ad-atom pair and (b) a heptamer island on the (111) surface of copper.

Shrinking Dimer Dynamics and Its Applications to Saddle Point Search

Saddle point search on an energy surface has broad applications in fields like materials science, physics, chemistry, and biology. In this paper, we present the shrinking dimer dynamics (SDD), a dynamic system which can be applied to locate a transition state on an energy surface corresponding to an index-1 saddle point where the Hessian has a negative eigenvalue. By searching for the saddle point and the associated unstable direction simultaneously in a single dynamic system defined in an extended space, we show that unstable index-1 saddle points of the energy become linearly stable steady equilibria of the SDD which makes the SDD a robust approach for the computation of saddle points. The time discretization of the SDD is connected to various iterative algorithms, including the popular dimer method used in many practical applications. Our study of these discretization schemes lays a rigorous mathematical foundation for the corresponding iterative saddle point search algorithms. Both linear local asymptotic stability analysis and optimal error reduction (convergence) rates are presented and further confirmed by numerical experiments. Global convergence and nonlinear asymptotic stability are also illustrated for some simpler systems. Applications of the SDD in both finite- and infinite-dimensional energy spaces are discussed.

The gentlest ascent dynamics

Dynamical systems that describe the escape from the basins of attraction of stable invariant sets are presented and analysed. It is shown that the stable fixed points of such dynamical systems are the index-1 saddle points. Generalizations to high index saddle points are discussed. Both gradient and non-gradient systems are considered. Preliminary results on the nature of the dynamical behaviour are presented.

Fractional controller to stabilize fixed points of uncertain chaotic systems: Theoretical and experimental study

This paper proposes a fractional order controller to stabilize saddle points of index 2 in a chaotic system. The proposed controller has a simple structure and can be tuned to stabilize fixed points when unstructured norm bounded perturbations exist on the model of the chaotic system. Numerical simulations confirm the ability of the controller to stabilize the fixed points in uncertain chaotic systems. Also, to illustrate the practical capability of the proposed controller, it is experimentally applied to control chaos in a chaotic circuit.