Saddle Dynamics Research Articles

Semi‐implicit method of high‐index saddle dynamics and application to construct solution landscape

AbstractWe analyze the semi‐implicit scheme of high‐index saddle dynamics, which provides a powerful numerical method for finding the any‐index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi‐implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi‐implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi‐implicit scheme via, for example, technical splittings and multi‐variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi‐implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi‐implicit scheme in constructing solution landscapes of complex systems.

Probabilistic Error Estimate for Numerical Discretization of High-Index Saddle Dynamics with Inaccurate Models

Probabilistic Error Estimate for Numerical Discretization of High-Index Saddle Dynamics with Inaccurate Models

Existence and Number of Figure-Eight Loops in Planar Sector-Wise Linear Systems with Saddle–Saddle Dynamics

In this paper, we investigate the existence of one type of homoclinic double loops (i.e. figure-eight loops) in a family of planar sector-wise linear systems with saddle–saddle dynamics. We obtain necessary and sufficient conditions for the existence of a figure-eight loop. Moreover, we prove that such systems can have simultaneously three types of invariant sets: a figure-eight loop, a homoclinic loop and three different types of periodic orbits. We also provide an example to show that a crossing limit cycle can bifurcate from this figure-eight loop.

Revealing excited states of rotational Bose-Einstein condensates

Rotational Bose-Einstein condensates can exhibit quantized vortices as topological excitations. In this study, the ground and excited states of the rotational Bose-Einstein condensates are systematically studied by calculating the stationary points of the Gross-Pitaevskii energy functional. Various excited states and their connections at different rotational frequencies are revealed in solution landscapes constructed with the constrained high-index saddle dynamics method. Four excitation mechanisms are identified: vortex addition, rearrangement, merging, and splitting. We demonstrate changes in the ground state with increasing rotational frequencies and decipher the evolution of the stability of ground states.

A model-free shrinking-dimer saddle dynamics for finding saddle point and solution landscape

A model-free shrinking-dimer saddle dynamics for finding saddle point and solution landscape

Mathematical and Numerical Analysis to Shrinking-Dimer Saddle Dynamics with Local Lipschitz Conditions

Mathematical and Numerical Analysis to Shrinking-Dimer Saddle Dynamics with Local Lipschitz Conditions

Discretization and index-robust error analysis for constrained high-index saddle dynamics on the high-dimensional sphere

We develop and analyze numerical discretization to the constrained high-index saddle dynamics, the dynamics searching for the high-index saddle points confined on the high-dimensional unit sphere. Compared with the saddle dynamics without constraints, the constrained high-index saddle dynamics has more complex dynamical forms, and additional operations such as the retraction and vector transport are required due to the constraint, which significantly complicate the numerical scheme and the corresponding numerical analysis. Furthermore, as the existing numerical analysis results usually depend on the index of the saddle points implicitly, the proved numerical accuracy may be reduced if the index is high in many applications, which indicates the lack of robustness with respect to the index. To address these issues, we derive the error estimates for numerical discretization of the constrained high-index saddle dynamics on the high-dimensional sphere, and then improve it by providing an index-robust error analysis in an averaged norm by adjusting the relaxation parameters. The developed results provide mathematical supports for the accuracy of numerical computations.

Energy landscape analysis for two-phase multi-component NVT flash systems by using ETD type high-index saddle dynamics

With the increasingly accurate modeling and simulation demands and techniques, obtaining the knowledge of the multi-component phase equilibrium states holds the bottleneck challenge in the study of multi-phase fluid flow. In order to resolve the shortages in computational efficiency and stability in the conventional iterative flash calculation, a new phase equilibrium prediction method is proposed by solving the saddle point of the multi-phase system. In this paper, the HiSD (High-index saddle point dynamics) algorithm is used for the first time to calculate the saddle points on the energy landscape of the two-phase two-component NVT flash model based on the Peng-Robinson equation of state, and the up-down search algorithm of HiSD is applied to generate the solution landscape of the flash system. The Rosenbrock-Euler ETD (exponential time differencing) format is involved to reduce the interference of system rigidity to the calculation. It can be referred from the numerical analysis that there are at most the 1st-order saddle points in the energy landscape of the two-component two-phase NVT flash system, and all these saddle points are located on one straight line of the hyperplane, where the energy is equal everywhere. All these 1st-order saddle points can converge to the same or equivalent local minima, which indicates that the two-component two-phase flash system is a system with only one single solution with physical meanings. In addition, the saddle points also obey a linear relationship and the energy remains the same at different temperatures. Therefore, using the method proposed in this paper, the conventional two-step efforts of phase stability test and phase separation calculation can be simplified. The 1st-order saddle points of the system can be directly calculated, reducing the need for an initial guess. The local minima can be directly searched through the downward direction of the saddle point, which greatly reduces the calculation amount of phase equilibrium calculations. Furthermore, the minimum states at different temperatures can be calculated in batch by using one certain initial value, which significantly improves the adaptability and reliability for complex engineering problems with drastic temperature changes.

E-Z Isomerization in Guanidine: Second-order Saddle Dynamics, Non-statisticality, and Time-frequency Analysis.

Our recent work on the E-Z isomerization reaction of guanidine using ab initio chemical dynamics simulations [Rashmi et al., Regul. Chaotic Dyn. 2021, 26, 119] emphasized the role of second-order saddle (SOS) in the isomerization reaction; however, we could not unequivocally establish the non-statistical nature of the dynamics followed in the reaction. In the present study, we performed thousands of on-the-fly trajectories using forces computed at the MNDO level to investigate the influence of second-order saddle in the E-Z isomerization reaction of guanidine and the role of intramolecular vibrational energy redistribution (IVR) on the reaction dynamics. The simulations reveal that while majority of the trajectories follow the traditional transition state pathways, 15 % of the trajectories follow the SOS path. The dynamics was found to be highly non-statistical with the survival probabilities of the reactants showing large deviations from those obtained within the RRKM assumptions. In addition, a detailed analysis of the dynamics using time-dependent frequencies and the frequency ratio spaces reveal the existence of multiple resonance junctions that indicate the existence of regular dynamics and long-lived quasi-periodic trajectories in the phase space associated with non-RRKM behavior.

Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one- and two-dimensional rectangular domains. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.

Constrained High-Index Saddle Dynamics for the Solution Landscape with Equality Constraints

We propose a constrained high-index saddle dynamics (CHiSD) method to search for index-$k$ saddle points of an energy functional subject to equality constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax framework, and its linear stability at an index-$k$ saddle point is proved. To ensure the manifold property, the CHiSD is numerically implemented using retractions and vector transport. Then we present a numerical approach by combining CHiSD with downward and upward search algorithms to construct the solution landscape in the presence of equality constraints. We apply the Thomson problem and the Bose-Einstein condensation as numerical examples to demonstrate the efficiency of the proposed method.

Transition Pathways in Cylinder-Gyroid Interface

When two distinct ordered phases contact, the interface may exhibit rich and fascinating structures. Focusing on the Cylinder-Gyroid interface system, transition pathways connecting various interface morphologies are studied armed with the Landau--Brazovskii model. Specifically, minimum energy paths are obtained by computing transition states with the saddle dynamics. We present four primary transition pathways connecting different local minima, representing four different mechanisms of the formation of the Cylinder-Gyroid interface. The connection of Cylinder and Gyroid can be either direct or indirect via Fddd with three different orientations. Under different displacements, each of the four pathways may have the lowest energy.

Transition pathways connecting crystals and quasicrystals

Due to structural incommensurability, the emergence of a quasicrystal from a crystalline phase represents a challenge to computational physics. Here, the nucleation of quasicrystals is investigated by using an efficient computational method applied to a Landau free-energy functional. Specifically, transition pathways connecting different local minima of the Lifshitz-Petrich model are obtained by using the high-index saddle dynamics. Saddle points on these paths are identified as the critical nuclei of the 6-fold crystals and 12-fold quasicrystals. The results reveal that phase transitions between the crystalline and quasicrystalline phases could follow two possible pathways, corresponding to a one-stage phase transition and a two-stage phase transition involving a metastable lamellar quasicrystalline state, respectively.

Solution landscape of the Onsager model identifies non-axisymmetric critical points

We investigate critical points of the Onsager free-energy model on a sphere with different potential kernels, including the dipolar potential, the Maier–Saupe potential, the coupled dipolar/Maier–Saupe potential, and the Onsager potential. A uniform sampling method is implemented for the discretization of the Onsager model, and solution landscapes of the Onsager model are constructed using saddle dynamics coupled with downward/upward search algorithms. We first construct the solution landscapes with the dipolar and Maier–Saupe potentials, for which all critical points are axisymmetric. For the coupled dipolar/Maier–Saupe potential, the solution landscape shows a novel non-axisymmetric critical point, named tennis, which exists for a wide range of parameters. We further demonstrate various non-axisymmetric critical points in the Onsager model with the Onsager potential, including square, hexagon, octahedral, cubic, quarter, icosahedral, and dodecahedral states. The bifurcation diagram is presented to show the primary and secondary bifurcations of the isotropic state and reveal the emergence of the critical points. The solution landscape provides an efficient approach to show the global structure of the model system as well as the bifurcations of critical points, which can not only support the previous theoretical conjectures but also propose new conjectures based on the numerical findings.

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.

Second-order Saddle Dynamics in Isomerization Reaction

The role of second-order saddle in the isomerization dynamics was investigated by considering the $$E-Z$$ isomerization of guanidine. The potential energy profile for the reaction was mapped using the ab initio wavefunction method. The isomerization path involved a torsional motion about the imine (C-N) bond in a clockwise or an anticlockwise fashion resulting in two degenerate transition states corresponding to a barrier of 23.67 kcal/mol. An alternative energetically favorable path ( $$\sim$$ 1 kcal/mol higher than the transition states) by an in-plane motion of the imine (N-H) bond via a second-order saddle point on the potential energy surface was also obtained. The dynamics of the isomerization was investigated by ab initio classical trajectory simulations. The trajectories reveal that isomerization happens via the transition states as well as the second-order saddle. The dynamics was found to be nonstatistical with trajectories exhibiting recrossing and the higher energy second-order saddle pathway preferred over the traditional transition state pathway. Wavelet based time-frequency analysis of internal coordinates indicate regular dynamics and existence of long-lived quasi-periodic trajectories in the phase space.

Computing Solution Landscape of Nonlinear Space-Fractional Problems Via Fast Approximation Algorithm

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.

Searching the solution landscape by generalized high-index saddle dynamics

We introduce a generalized numerical algorithm to construct the solution landscape, which is a pathway map consisting of all stationary points and their connections. Based on the high-index optimization-based shrinking dimer (HiOSD) method for gradient systems, a generalized high-index saddle dynamics (GHiSD) is proposed to compute any-index saddles of dynamical systems. Linear stability of the index-$k$ saddle point can be proved for the GHiSD system. A combination of the downward search algorithm and the upward search algorithm is applied to systematically construct the solution landscape, which not only provides a powerful and efficient way to compute multiple solutions without tuning initial guesses, but also reveals the relationships between different solutions. Numerical examples, including a three-dimensional example and the phase field model, demonstrate the novel concept of the solution landscape by showing the connected pathway maps.

Periodic orbits in hysteretic systems with real eigenvalues

Planar piecewise linear systems with hysteresis coming from a dimensional reduction in symmetric 3D systems with slow–fast dynamics are considered. We concentrate our attention on the cases of saddle and node dynamics, determining the existence and stability of periodic orbits as well as possible bifurcations. Our analysis rigorously shows, apart from standard bifurcations, as saddle-node bifurcation of periodic orbits and homoclinic and heteroclinic connections, the existence of specific grazing bifurcation. A pitchfork bifurcation of periodic orbits has been also detected, being responsible for the coexistence of up to four different periodic orbits. We illustrate the usefulness of the achieved theoretical results by justifying the appearance of periodic orbits in a concrete 3D system.

Real-Time Identification of Vortex Shedding in the Wake of a Circular Cylinder

Lagrangian saddles, identified as nonparallel intersections of finite-time Lyapunov exponent (FTLE) ridges, have previously been shown to indicate the timing of von Kármán vortex shedding in the wake of a circular cylinder using two-dimensional simulation data. To expand the application of this method, two-dimensional, two-component velocity data were collected in the wake of a circular cylinder using particle image velocimetry at a Reynolds number of 19,000. This experimental data, as well as three-dimensional simulation data at a Reynolds number of 400, were used to calculate FTLE fields. The Lagrangian saddle found upstream of a forming vortex was shown to accelerate away from the cylinder surface as the vortex begins to shed for both cases, in agreement with the previous work. These saddles are impossible to track in real time, however, because future flowfield data are needed for the computation of the FTLE fields. To detect the Lagrangian saddle acceleration without using FTLE, the saddle dynamics are connected to physical quantities measurable in real time. The acceleration of the Lagrangian saddle occurs simultaneously with a maximum in lift in the numerical case and with a minimum in the static pressure at a location slightly upstream of the mean separation location in both cases, albeit with a lag time in the experimental case. This allows the time at which the von Kármán vortex sheds, determined objectively by the acceleration of the Lagrangian saddle away from the circular cylinder, to be identified by distinct signatures in the spatiotemporal evolution of the pressure distribution on the cylinder surface.