arXiv subjects
Publications and source records attributed to Matthew D. Kvalheim.
We exhibit a configuration of five point charges in Euclidean space whose electrostatic potential admits at least 24 critical points all of which are non-degenerate. Maxwell's conjecture that the field of \(n\) point charges has at most \((n-1)^2\) critical points which are all non-degenerate is therefore false.
We study whether second-order systems can be made to behave like prescribed first-order dynamical systems through feedback control. More precisely, we study whether prescribed vector fields on compact smooth manifolds, viewed geometrically as sections of the tangent bundle, can be asymptotically stabilized in a strong sense by second-order control systems on the base manifold. Our class of second-order systems includes most Lagrangian systems, and we obtain both positive and negative results. The positive result asserts that, for fully actuated systems, the section corresponding to any smooth vector field can be made globally exponentially stable, normally hyperbolic, and more. In particular, not only does each closed-loop solution asymptotically have the prescribed velocities, but it also converges to a trajectory of the first-order dynamics generated by the prescribed vector field at an exponential rate. Thus, the closed-loop second-order system asymptotically reproduces the prescribed first-order dynamics. In contrast, the negative result asserts that, for underactuated systems on manifolds with nonzero Euler characteristic, sections corresponding to "almost all" smooth vector fields cannot even be locally asymptotically stabilized. This includes, in particular, all vector fields with only isolated zeros. An example shows that the Euler characteristic assumption is necessary for the negative result.
We consider the problem of determining the class of continuous-time dynamical systems that can be globally linearized in the sense of admitting an embedding into a linear system on a higher-dimensional Euclidean space. We solve this problem for dynamical systems on connected state spaces that are either compact or contain at least one nonempty compact attractor, obtaining necessary and sufficient conditions for the existence of linearizing $C^k$ embeddings for $k\in \mathbb{N}_{\geq 0}\cup \{\infty\}$. Corollaries include (i) several checkable necessary conditions for global linearizability and (ii) extensions of the Hartman-Grobman and Floquet normal form theorems beyond the classical settings. Our results open new perspectives on linearizability by establishing relationships to symmetry, topology, and invariant manifold theory.
Given a "data manifold" $M\subset \mathbb{R}^n$ and "latent space" $\mathbb{R}^\ell$, an autoencoder is a pair of continuous maps consisting of an "encoder" $E\colon \mathbb{R}^n\to \mathbb{R}^\ell$ and "decoder" $D\colon \mathbb{R}^\ell\to \mathbb{R}^n$ such that the "round trip" map $D\circ E$ is as close as possible to the identity map $\mbox{id}_M$ on $M$. We present various topological limitations and capabilites inherent to the search for an autoencoder, and describe capabilities for autoencoding dynamical systems having $M$ as an invariant manifold.
We give a proof of an extension of the Hartman-Grobman theorem to nonhyperbolic but asymptotically stable equilibria of vector fields. Moreover, the linearizing topological conjugacy is (i) defined on the entire basin of attraction if the vector field is complete, and (ii) a $C^{k\geq 1}$-diffeomorphism on the complement of the equilibrium if the vector field is $C^k$ and the underlying space is not $5$-dimensional. We also show that the $C^k$ statement in the $5$-dimensional case is equivalent to the $4$-dimensional smooth Poincaré conjecture.
We study the topology of the space of all smooth asymptotically stable vector fields on $\mathbb{R}^n$, as well as the space of all proper smooth Lyapunov functions for such vector fields. We prove that both spaces are path-connected and simply connected when $n\neq 4,5$ and weakly contractible when $n\leq 3$. Moreover, both spaces have the weak homotopy type of the nonlinear Grassmannian of submanifolds of $\mathbb{R}^n$ diffeomorphic to the $n$-disc. The proofs rely on Lyapunov theory and differential topology, such as the work of Smale and Perelman on the generalized Poincaré conjecture and results of Smale, Cerf, and Hatcher on the topology of diffeomorphism groups of discs. Applications include a partial answer to a question of Conley, a parametric Hartman-Grobman theorem for nonyperbolic but asymptotically stable equilibria, and a parametric Morse lemma for degenerate minima. We also study the related topics of hyperbolic equilibria, Morse minima, and relative homotopy groups of the space of asymptotically stable vector fields inside the space of those vanishing at a single point.
The author's extensions of Brockett's and Coron's necessary conditions for stabilizability are shown to be independent in the fiber bundle picture of control, but the latter is shown to be stronger in the vector bundle picture if the state space is orientable and the Cech-Euler characteristic of the set to be stabilized is nonzero.
Deep neural network autoencoders are routinely used computationally for model reduction. They allow recognizing the intrinsic dimension of data that lie in a $k$-dimensional subset $K$ of an input Euclidean space $\mathbb{R}^n$. The underlying idea is to obtain both an encoding layer that maps $\mathbb{R}^n$ into $\mathbb{R}^k$ (called the bottleneck layer or the space of latent variables) and a decoding layer that maps $\mathbb{R}^k$ back into $\mathbb{R}^n$, in such a way that the input data from the set $K$ is recovered when composing the two maps. This is achieved by adjusting parameters (weights) in the network to minimize the discrepancy between the input and the reconstructed output. Since neural networks (with continuous activation functions) compute continuous maps, the existence of a network that achieves perfect reconstruction would imply that $K$ is homeomorphic to a $k$-dimensional subset of $\mathbb{R}^k$, so clearly there are topological obstructions to finding such a network. On the other hand, in practice the technique is found to "work" well, which leads one to ask if there is a way to explain this effectiveness. We show that, up to small errors, indeed the method is guaranteed to work. This is done by appealing to certain facts from differential topology. A computational example is also included to illustrate the ideas.
A frequently repeated claim in the "applied Koopman operator theory'' literature is that a dynamical system with multiple isolated equilibria cannot be linearized in the sense of admitting a smooth embedding as an invariant submanifold of a linear dynamical system. This claim is sometimes made only for the class of super-linearizations, which additionally require that the embedding "contain the state''. We show that both versions of this claim are false by constructing (super-)linearizable smooth dynamical systems on $\mathbb{R}^k$ having any countable (finite) number of isolated equilibria for each $k>1$.
In this work, we give sufficient conditions for the almost global asymptotic stability of a cascade in which the subsystems are only almost globally asymptotically stable. The result is extended to upper triangular systems of arbitrary size. In particular, if the unforced subsystems are almost globally asymptotically stable and their only chain recurrent points are hyperbolic equilibria, then the boundedness of forward trajectories is sufficient for the almost global asymptotic stability of the full upper triangular system. We show that unboundedness of such cascades is prohibited by growth rate conditions on the interconnection term and a Lyapunov function for the unforced outer subsystem, and the required structure for the chain recurrent set is enjoyed by classes of systems common in geometric control e.g. dissipative mechanical systems. Our results stand in contrast to prior works that require either time scale separation, prohibitively strong disturbance robustness properties, or global asymptotic stability in the subsystems.
Mechanical systems naturally evolve on principal bundles describing their inherent symmetries. The ensuing factorization of the configuration manifold into a symmetry group and an internal shape space has provided deep insights into the locomotion of many robotic and biological systems. On the other hand, the property of differential flatness has enabled efficient, effective planning and control algorithms for various robotic systems. Yet, a practical means of finding a flat output for an arbitrary robotic system remains an open question. In this work, we demonstrate surprising new connections between these two domains, for the first time employing symmetry directly to construct a flat output. We provide sufficient conditions for the existence of a trivialization of the bundle in which the group variables themselves are a flat output. We call this a geometric flat output, since it is equivariant (i.e. maintains the symmetry) and is often global or almost-global, properties not typically enjoyed by other flat outputs. In such a trivialization, the motion planning problem is easily solved, since a given trajectory for the group variables will fully determine the trajectory for the shape variables that exactly achieves this motion. We provide a partial catalog of robotic systems with geometric flat outputs and worked examples for the planar rocket, planar aerial manipulator, and quadrotor.
Many real-world systems are well-modeled by Brownian particles subject to gradient dynamics plus noise arising, e.g., from the thermal fluctuations of a heat bath. Of central importance to many applications in physics and biology (e.g., molecular motors) is the net steady-state particle current or "flux" enabled by the noise and an additional driving force. However, this flux cannot usually be calculated analytically. Motivated by this, we investigate the steady-state flux generated by a nondegenerate diffusion process on a general compact manifold; such fluxes are essentially equivalent to the stochastic intersection numbers of Manabe (1982). In the case that noise is small and the drift is "gradient-like" in an appropriate sense, we derive a graph-theoretic formula for the small-noise asymptotics of the flux using Freidlin-Wentzell theory. When additionally the drift is a local gradient sufficiently close to a generic global gradient, there is a natural flux equivalent to the entropy production rate -- in this case our graph-theoretic formula becomes Morse-theoretic, and the result admits a description in terms of persistent homology. As an application, we provide a mathematically rigorous explanation of the paradoxical "negative resistance" phenomenon in Brownian transport discovered by Cecchi and Magnasco (1996).
Necessary conditions for asymptotic stability and stabilizability of subsets for dynamical and control systems are obtained. The main necessary condition is homotopical and is in turn used to obtain a homological one. A certain extension is ruled out. Questions are posed.
The Hopf theorem states that homotopy classes of continuous maps from a closed connected oriented smooth $n$-manifold $M$ to the $n$-sphere are classified by their degree. Such a map is equivalent to a section of the trivial $n$-sphere bundle over $M$. A generalization of the Hopf theorem is obtained for sections of nontrivial oriented $n$-sphere bundles over $M$.
Brockett's necessary condition yields a test to determine whether a system can be made to stabilize about some operating point via continuous, purely state-dependent feedback. For many real-world systems, however, one wants to stabilize sets which are more general than a single point. One also wants to control such systems to operate safely by making obstacles and other "dangerous" sets repelling. We generalize Brockett's necessary condition to the case of stabilizing general compact subsets having a nonzero Euler characteristic in general ambient state spaces (smooth manifolds). Using this generalization, we also formulate a necessary condition for the existence of "safe" control laws. We illustrate the theory in concrete examples and for some general classes of systems including a broad class of nonholonomically constrained Lagrangian systems. We also show that, for the special case of stabilizing a point, the specialization of our general stabilizability test is stronger than Brockett's.
Oscillators are ubiquitous in nature, and usually associated with the existence of an asymptotic phase that governs the long-term dynamics of the oscillator. % We show that asymptotic phase can be estimated using a carefully chosen series expansion which directly computes the phase response curve and provide an algorithm for estimating the co-efficients of this series. Unlike all previously available data driven phase estimation methods, our algorithm can: (i) use observations that are much shorter than a cycle; (ii) recover phase within any forward invariant region for which sufficient data are available; (iii) recover the phase response curves (PRC-s) that govern weak oscillator coupling; (iv) show isochron curvature, and recover nonlinear features of isochron geometry. Our method may find application wherever models of oscillator dynamics need to be constructed from measured or simulated time-series.
The "Phase Response Curve" (PRC) is a common tool used to analyze phase resetting in the natural sciences. We make the observation that the PRC with respect to a coordinate $y\in\mathbb{R}$ actually depends on the full choice of coordinates $(x,y)$, $x\in\mathbb{R}^d$. We give a coordinate-free definition of the PRC making this observation obvious. We show how by controlling $y$, using delay coordinates of $y$, and postulating the dynamics of $x$ as a function of $x$ and $y$, we can sometimes reconstruct the PRC with respect to the $(x,y)$ coordinates. This suggests a means for obtaining the PRC of, e.g., a neuron via a voltage clamp.
We give a necessary and sufficient condition ensuring that any function which is separately Lipschitz on two fixed compact sets is Lipschitz on their union.