Saddle Value Research Articles

Stable Heteroclinic Channel-Based Movement Primitives: Tuning Trajectories Using Saddle Parameters

Dynamic systems which underlie controlled systems are expected to increase in complexity as robots, devices, and connected networks become more intelligent. While classical stable systems converge to a stable point (a sink), another type of stability is to consider a stable path rather than a single point. Such stable paths can be made of saddle points that draw in trajectories from certain regions, and then push the trajectory toward the next saddle point. These chains of saddles are called stable heteroclinic channels (SHCs) and can be used in robotic control to represent time sequences. While we have previously shown that each saddle is visualizable as a trajectory waypoint in phase space, how to increase the fidelity of the trajectory was unclear. In this paper, we hypothesized that the waypoints can be individually modified to locally vary fidelity. Specifically, we expected that increasing the saddle value (ratio of saddle eigenvalues) causes the trajectory to slow to more closely approach a particular saddle. Combined with other parameters that control speed and magnitude, a system expressed with an SHC can be modified locally, point by point, without disrupting the rest of the path, supporting their use in motion primitives. While some combinations can enable a trajectory to better reach into corners, other combinations can rotate, distort, and round the trajectory surrounding the modified saddle. Of the system parameters, the saddle value provides the most predictable tunability across 3 orders of magnitude.

A delayed subgradient method for nonsmooth convex-concave min–max optimization problems

In this paper, we aim to solve a convex-concave min–max optimization problem, where the convex-concave coupling function is nonsmooth in both variables. We propose a simple subgradient method which is simultaneously updating the iterates through their delayed subgradients at stale iterates of their own variable together with the current iterates of the other variable. We investigate the convergence behavior of the proposed method by providing an upper bound of the function value at the coupling averaged iterates to the saddle value. Specifically, the obtained result refers to the function value at the coupling averaged iterates as the function value of an approximate saddle point at the current iteration. The numerical results demonstrate that, with a suitable strategy of delays’ selections, the delayed subgradient method can achieve better convergence behaviors than its non-delayed counterpart.

Monotonicity and non-monotonicity regions of topological entropy for Lorenz-like families with infinite derivatives

Abstract We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps T c,ɛ (x) = (−1 + c|x| 1−ɛ ) · sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

On the classification of homoclinic attractors of three-dimensional flows

For three-dimensional dynamical systems with continuous time (flows), a classification of strange homoclinic attractors containing an unique saddle equilibrium state is constructed. The structure and properties of such attractors are determined by the triple of eigenvalues of the equilibrium state. The method of a saddle charts is used for the classification of homoclinic attractors. The essence of this method is in the construction of an extended bifurcation diagram for a wide class of three-dimensional flows (whose linearization matrix is written in the Frobenius form). Regions corresponding to different configurations of eigenvalues are marked in this extended bifurcation diagram. In the space of parameters defining the linear part of the considered class of three-dimensional flows bifurcation surfaces bounding 7 regions are constructed. One region corresponds to the stability of the equilibrium states while other 6 regions correspond to various homoclinic attractors of the following types: Shilnikov attractor, 2 types of spiral figure-eight attractors, Lorenz- like attractor, saddle Shilnikov attractor and attractor of Lyubimov-Zaks-Rovella. The paper also discusses questions related to the pseudohyperbolicity of homoclinic attractors of three-dimensional flows. It is proved that only homoclinic attractors of two types can be pseudohyperbolic: Lorenz-like attractors containing a saddle equilibrium with a two-dimensional stable manifold whose saddle value is positive and saddle Shilnikov attractors containing a saddle equilibrium state with a two-dimensional unstable manifold.

Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders

ABSTRACTConsider (m + 1)-dimensional, m ≥ 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold Ws(O), one-dimensional unstable manifold Wu(O), and with the saddle value σ different from 1. We assume that Ws(O) and Wu(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when σ < 1, we prove necessary and sufficient conditions of existence of topological horseshoes near homoclinic tangencies. In the case when σ > 1, we also obtain the criterion of existence of horseshoes under the additional assumption that the homoclinic tangency is simple.

Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model

We give an analytic (free of computer assistance) proof of the existence of a classical Lorenz attractor for an open set of parameter values of the Lorenz model in the form of Yudovich–Morioka–Shimizu. The proof is based on detection of a homoclinic butterfly with a zero saddle value and rigorous verification of one of the Shilnikov criteria for the birth of the Lorenz attractor; we also supply a proof for this criterion. The results are applied in order to give an analytic proof for the existence of a robust, pseudohyperbolic strange attractor (the so-called discrete Lorenz attractor) for an open set of parameter values in a 4-parameter family of 3D Henon-like diffeomorphisms.

Robust Hypothesis Testing with &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\alpha $&lt;/tex-math&gt; &lt;/inline-formula&gt;-Divergence

A robust minimax test for two composite hypotheses, which are determined by the neighborhoods of two nominal distributions with respect to a set of distances - called $\alpha-$divergence distances, is proposed. Sion's minimax theorem is adopted to characterize the saddle value condition. Least favorable distributions, the robust decision rule and the robust likelihood ratio test are derived. If the nominal probability distributions satisfy a symmetry condition, the design procedure is shown to be simplified considerably. The parameters controlling the degree of robustness are bounded from above and the bounds are shown to be resulting from a solution of a set of equations. The simulations performed evaluate and exemplify the theoretical derivations.

Internuclear-distance-dependent ionization of H2+ in strong laser field in a classical perspective

Ionizations of atoms and molecules in strong laser fields are fundamental processes of ultrafast physics. Compared with atom ionization, molecular ionization is very complex due to the existence of multi Coulomb centers. As a simplest molecule, H2+ has been widely used to explore new phenomena of molecules in strong laser fields. One of the notable processes in H2+ ionization is charge resonance enhanced ionization (CREI), in which the ionization rate is enhanced substantially when the internuclear distances are around 6 a.u. and 10 a.u. CREI has been extensively studied by numerically simulating the time-dependent Schrödinger equation. While quantum calculations provide accurate ionization rates, the mechanism governing the CREI is not revealed in such ab-initio calculations. On the contrary, the calculations based on the classical trajectories Monte-Carlo assembly may offer an intuitive picture for CREI though some quantum information is not included. In this paper, we revisit the CREI of H2+ in a strong infrared laser field by Monte-Carlo simulation. By initializing ten-thousand classical points whose initial positions and velocities satisfy the field-free Hamiltonian of H2+, we solve the classical Newtonian equation and obtain the trajectories of all particles, from which one may analyze the particle velocities, energies, etc. We count the ionization events by diagnosing the particle energy after the laser interaction. If the sum of the kinetic energy and potential energy is larger than 0, we set it as an ionization event. The ionization rate is calculated by collecting all ionization events and normalizing it with the total particle number involved in the calculation. By setting the internuclear distances to be different values, we obtain the ionization rate as a function of internuclear distance. Our simulation shows that the ionization probability is greatly enhanced when the internuclear distance is about 5 to 6 a.u. by employing a 1064 nm, 4×1013 W/cm2, five cycles laser pulse. By tracing the particle trajectory, we find that the electron usually gains the energy from the laser field by circulating one nucleus, then passes through the interatomic barrier and moves around the other nucleus before being ionized. By looking into the relationship between the ionization probability and the laser-distorted Coulomb potential at different internuclear distances, we find that the ionization probability is maximum when the energy difference between the ground state and the interatomic Coulomb barrier, or between the ground state and the saddle value of the laser-distorted potential, is minimum. The classical calculation of the ionization of H2+ interacting with intense laser field reproduces the qualitative features of the corresponding quantum-mechanical calculation. It offers an intuitive physical picture of the tunneling ionization of molecules through investigating the classical trajectories and provides a new perspective to inspect the intriguing phenomena in quantum systems.

Necessary and sufficient conditions of the existence of homoclinic trajectories and cascade of bifurcations in Lorenz-like systems: birth of strange attractor and 9 homoclinic bifurcations

Necessary and sufficient conditions of the existence of homoclinic trajectories in Lorenz-like systems are obtained. The Shimizu–Morioka, Lu, and Chen systems are considered. For special Lorenz-like system with zero saddle value, cascade of bifurcations is obtained. Birth of strange attractor, blue sky catastrophe and 9 homoclinic bifurcations are discovered.

Cascade of bifurcations in Lorenz-like systems: Birth of a strange attractor, blue sky catastrophe bifurcation, and nine homoclinic bifurcations

A Lorenz-like system with a single parameter and a zero saddle value is constructed for which a cascade of bifurcations is obtained, namely, the birth of limit cycles and a strange attractor, nine homoclinic bifurcations, and a blue sky catastrophe bifurcation.

On the Existence of a Saddle Value

In this work, we achieve a complete characterization of the existence of a saddle value, for bifunctions which are convex, proper, and lower semi continuous in their first argument, by considering new suitably defined notions of special directions of recession. As special cases, we obtain some recent results of Lagrangian duality theory on zero duality gap for convex programs.

多样极小极大定理在零和动态对局问题(I)

考虑一个两人零和动态对局上的极小极大问题。我们将极小极大定理以一个随机对局系统建立损失和获益的总值函数，并证明极小极大定理在某些条件下存在鞍值函数，导出这个动态对局系统存在一平衡点。本文举一个简易的例子，说明协议的对局运作下，说明这个动态对局的架构与极小极大定理的相互关系。 Considering a minimax problem to a two-person zero-sum dynamic game, we establish the total value function of game losses and gains in a stochastic game system. It could perform a minimax theorem. Moreover, we prove that minimax theorem is established by the stochastic space of their strategy spaces for the two-person zero-sum dynamic game under the law of motion. It is also established that the saddle value function exists under certain natural conditions so that the equilibrium point exists in this dynamic game system. A practical example could be employed to our framework in the context.

On the effect of invisibility of stable periodic orbits at homoclinic bifurcations

We study bifurcations of a homoclinic tangency to a saddle-focus periodic point. We show that the stability domain for single-round periodic orbits which bifurcate near the homoclinic tangency has a characteristic “comb-like” structure and depends strongly on the saddle value, i.e. on the area-contracting properties of the map at the saddle-focus. In particular, when the map contracts two-dimensional areas, we have a cascade of periodic sinks in any one-parameter family transverse to the bifurcation surface that corresponds to the homoclinic tangency. However, when the area-contraction property is broken (while three-dimensional volumes are still contracted), the cascade of single-round sinks appears with “probability zero” only. Thus, if three-dimensional volumes are contracted, chaos associated with a homoclinic tangency to a saddle-focus is always accompanied by stability windows; however the violation of the area-contraction property can make the stability windows invisible in one-parameter families.

On Bifurcations of Three-Dimensional Diffeomorphisms with a Homoclinic Tangency to a “Neutral” Saddle Fixed Point

Bifurcations of single-round periodic orbits of three-dimensional diffeomorphisms with a quadratic homoclinic tangency of manifolds of a saddle fixed point with saddle value equal to 1 are studied. Bibliography: 8 titles.

A note on computing the saddle values in isosurface polygonization

The marching cubes method (Lorensen 1987) is a popular technique for visualizing a set of volume data. This method uses 15 base cases to construct polygons for each cube to approximate the isosurfaces F(x, y, z)"t for various values of t. The advantages of the polygon-based approach make it easy to adopt the technique on hardware platforms and to execute it efficiently. Unfortunately, the method has an ambiguity problem that causes anomalies in the rendered images (Durst 1988). Several improved approaches have been proposed (Montani et al. 1994; Nielson et al. 1991; Pasco 1988; Wilhelms and Gelder 1990). These methods try to resolve ambiguous situations by preventing possible ‘‘holes’’ in the isosurfaces. However, these approaches could introduce the topological inconsistency problem (Natarajan 1994), which confuses our perception of the observed objects, especially when supersampling the original data is necessary. In 1994, Natarajan developed a technique that computes topologically consistent isosurfaces by examining the saddle values of a trilinear interpolant. This technique also improves the method proposed by Pasco in 1988, which investigated saddle values for the bilinear interpolant only. In this note, we analyze the trilinear interpolant. We show that the cost for computing the body saddle value can be reduced.

Information Capacity of Channels with Partially Unknown Noise. II. Infinite-Dimensional Channels

Information capacity is considered for a communication channel in which the noise is the sum of a known Gaussian component and an independent component with unknown statistical distributions. A lower bound on capacity is sought; the unknown noise component is thus assumed to be under the control of an adversary---a jammer. The problem is modeled as a zero-sum two-person game with mutual information as the payoff function. Appropriate constraints are determined on the transmitted signal and the unknown noise component. Although the usual conditions sufficient for application of the general form of the von Neumann minimax theorem are shown not to hold, a solution is obtained for the game: a saddle value, saddle point, and minimax strategy for the jammer are obtained. The essential effect of jamming is to convert the infinite-dimensional channel into a finite-dimensional channel having the same constraints, with the dimensionality depending upon the problem parameters: the covariance of the known Gaussian noise component and the constraints on the transmitted signal and the unknown noise component.

Information Capacity of Channels with Partially Unknown Noise. I. Finite-Dimensional Channels

Information capacity is considered for a finite-dimensional communication channel in which the noise is the sum of a known Gaussian noise and an independent component with unknown statistical properties. The problem is modeled as a zero-sum two-person game between the coder and a jammer, with mutual information as the payoff function. The jammer does not control the ambient Gaussian noise. The constraints employed are appropriate for extension to infinite-dimensional channels and for consideration of coding capacity. The original problem is reformulated to prove existence of both a saddle value and a saddle point. The saddle value and a saddle point are determined, along with the jammer’s minimax strategy.

Sensitivity analysis for generalized linear-quadratic problems

In this paper, we study simple necessary and sufficient conditions for the stability of generalized linear-quadratic programs under perturbations of the data. The concept of generalized linear-quadratic problem was introduced by Rockafellar and Wets and consists of solving saddle points of a linear-quadratic convex concave functionJ onU×V, whereU andV are polyhedral convex sets in ℝ n and ℝ m . This paper also establishes results on the closedness and the uniform boundedness of the saddle-point solution sets. These properties are then used to obtain results on the continuity and the directional derivative of the perturbed saddle value.

ON GEOMETRICAL PROPERTIES OF TWO-DIMENSIONAL DIFFEOMORPHISMS WITH HOMOCLINIC TANGENCIES

Two-dimensional diffeomorphisms with a quadratic tangency of invariant manifolds of a saddle fixed point are considered in the cases where the saddle value σ is either less than 1 or equal to it. A description of the structure of hyperbolic subsets is given. In the case σ=1, it is shown that almost all such diffeomorphisms admit the complete description in distinction with the case σ&lt;1.

NORMAL FORMS AND LORENZ ATTRACTORS

Normal forms for eleven cases of bifurcations of codimension-3 are considered, basically, in systems with a symmetry, which can be reduced to one of the two three-dimensional systems. The first system is the well-known Lorenz model in a special notation, the second is the Shimizu-Morioka model. In contrast with two-dimensional normal forms which admit, in principle, a complete theoretical study, in three-dimensional systems such analysis is practically impossible, except for particular parameter values when a system is close to an integrable system. Therefore, the main method of the investigation is qualitatively-numerical. In that sense, a description of principal bifurcations which lead to the appearance of the Lorenz attractor is given for the models above, and the boundaries of the regions of the existence of this attractor are selected. We pay special attention to bifurcation points corresponding to a formation of a homoclinic figure-8 of a saddle with zero saddle value and that of a homoclinic figure-8 with zero separatrix value. In L. P. Shil'nikov [1981], it was established that these points belong to the boundary of the existence of the Lorenz attractor. In the present paper, the bifurcation diagrams near such points for the symmetric case are given and a new criterion of existence of the nonorientable Lorenz is also suggested.