Polynomial Dynamics Research Articles

Equidistribution of the Zeros of Higher Order Derivatives in Polynomial Dynamics

Equidistribution of the Zeros of Higher Order Derivatives in Polynomial Dynamics

Compositional synthesis of control barrier certificates for networks of stochastic systems against [formula omitted]-regular specifications

This paper is concerned with a compositional scheme for the construction of control barrier certificates for interconnected discrete-time stochastic systems. The main objective is to synthesize switching controllers against ω-regular properties that can be described by accepting languages of deterministic Streett automata (DSA) along with providing probabilistic guarantees for the satisfaction of such specifications. The proposed framework leverages the interconnection topology and a notion of so-called control sub-barrier certificates of subsystems, which are used to compositionally construct control barrier certificates of interconnected systems by imposing some dissipativity-type compositionality conditions. We propose a systematic approach to decompose high-level ω-regular specifications into simpler tasks by utilizing the automata corresponding to the specifications. In addition, we formulate an alternating direction method of multipliers (ADMM) optimization problem in order to obtain suitable control sub-barrier certificates of subsystems while satisfying compositionality conditions. For systems with polynomial dynamics, we provide a sum-of-squares (SOS) optimization problem for the computation of control sub-barrier certificates and local controllers of subsystems. Finally, we demonstrate the effectiveness of our proposed approaches by applying them to a physical case study.

A Timing-Based Framework for Designing Resilient Cyber-Physical Systems under Safety Constraint

Cyber-physical systems (CPS) are required to satisfy safety constraints in various application domains such as robotics, industrial manufacturing systems, and power systems. Faults and cyber attacks have been shown to cause safety violations, which can damage the system and endanger human lives. Resilient architectures have been proposed to ensure safety of CPS under such faults and attacks via methodologies including redundancy and restarting from safe operating conditions. The existing resilient architectures for CPS utilize different mechanisms to guarantee safety, and currently, there is no common framework to compare them. Moreover, the analysis and design undertaken for CPS employing one architecture is not readily extendable to another. In this article, we propose a timing-based framework for CPS employing various resilient architectures and develop a common methodology for safety analysis and computation of control policies and design parameters. Using the insight that the cyber subsystem operates in one out of a finite number of statuses, we first develop a hybrid system model that captures CPS adopting any of these architectures. Based on the hybrid system, we formulate the problem of joint computation of control policies and associated timing parameters for CPS to satisfy a given safety constraint and derive sufficient conditions for the solution. Utilizing the derived conditions, we provide an algorithm to compute control policies and timing parameters relevant to the employed architecture. We also note that our solution can be applied to a wide class of CPS with polynomial dynamics and also allows incorporation of new architectures. We verify our proposed framework by performing a case study on adaptive cruise control of vehicles.

On critical renormalization of complex polynomials

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider invariant continua that are not polynomial-like Julia sets because of extra critical points. However, under certain assumptions, these invariant continua can be identified with Julia sets of lower degree polynomials up to a topological conjugacy. Thus we extend the concept of renormalization.

Improving the Feasibility of Moment-Based Safety Analysis for Stochastic Dynamics

Given a dynamical system modelled via stochastic differential equations (SDEs), we evaluate the safety of the system through its exit time moments. Using appropriate semidefinite positive matrix constraints, an SDP moment-based approach can be used to compute moments of the exit time. However, the approach is impeded when analyzing higher dimensional physical systems as the dynamics are limited to polynomials. Computational scalability is also poor as the dimensionality of the state grows, largely due to the combinatorial growth of the optimization program. We propose methods to make feasible the safety analysis of higher dimensional physical systems. The restriction to polynomial dynamics is lifted by using state augmentation which allows one to generate the optimization for a broader class of nonlinear stochastic systems. We then reformulate the constraints to mitigate the computational limitations associated with an increase in state dimensionality. We employ our methods on two example processes to characterize their safety via exit times and show the ability to handle multi-dimensional systems that were previously unsupported by the existing SDP method of moments.

Mahler measure of polynomial iterates

AbstractGranville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial $f(z)=z^d+\cdots \in {\mathbb C}[z],\ \deg (f)\ge 2,$ we show that the Mahler measure of the iterates $f^n$ grows geometrically fast with the degree $d^n,$ and find the exact base of that exponential growth. This base is expressed via an integral of $\log ^+|z|$ with respect to the invariant measure of the Julia set for the polynomial $f.$ Moreover, we give sharp estimates for such an integral when the Julia set is connected.

Fixed points and orbits in skew polynomial rings

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring [Formula: see text]. We extend results of the first author and Vishkautsan on polynomial dynamics in [Formula: see text]. In particular, we show that if [Formula: see text] and [Formula: see text] satisfy [Formula: see text], then [Formula: see text] for every formal power of [Formula: see text]. More generally, we give a sufficient condition for a point [Formula: see text] to be [Formula: see text]-periodic with respect to a polynomial [Formula: see text]. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.

Polynomial dynamics and local analysis of small and grand orbits

We prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role of the divisible group is taken by the small orbit of a point $\alpha$ where the small orbit by a polynomial $f$ is given by \begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*} Our main theorem is a classification of the algebraic relations that hold between infinitely many pairs of points in $\mathcal {S}_\alpha$ when everything is defined over the algebraic numbers and the degree $d$ of $f$ is at least 2. Our proof relies on a careful study of localisations of the dynamical system and follows an entirely different approach than previous proofs in this area. In particular, we introduce transcendence theory and Mahler functions into this field. Our methods also allow us to classify all algebraic relations that hold for infinitely many pairs of points in the grand orbit \begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*} of $\alpha$ if $|f^{\circ n}(\alpha )|_v \rightarrow \infty$ at a finite place $v$ of good reduction co-prime to $d$.

A novel moving orthonormal coordinate-based approach for region of attraction analysis of limit cycles

<p style='text-indent:20px;'>The paper proposes a Lyapunov theory-based method to compute inner estimates of the <i>region of attraction</i> (ROA) of stable limit cycles. The approach is based on a transformation of the system to transverse coordinates, defined on a <i>moving orthonormal coordinate system</i> (MOC) for which a novel construction is presented. The proposed <i>center point MOC</i> (cp-MOC) is associated with a user-defined center point and provides flexibility to the construction of the transverse coordinates. In particular, compared to the standard approach based on hyperplanes orthogonal to the flow, the new construction allows the analyst to obtain larger regions of the state space where the well-definedness property of the transformation is satisfied. This has important benefits when using transverse coordinates to compute inner estimates of the ROA. To demonstrate these improvements, a sum-of-squares optimization-based formulation is proposed for computing inner estimates of the ROA of limit cycles for polynomial dynamics described in transverse coordinates. Different algorithmic options are explored, taking into account computational and accuracy aspects. Results are shown for three different systems exhibiting increasing complexity. The presented algorithms are extensively compared, and the newly cp-MOC is shown to markedly outperform existing approaches.</p>

Robust Data-Driven Safe Control Using Density Functions

This paper presents a tractable framework for data-driven synthesis of robustly safe control laws. Given noisy experimental data and some priors about the structure of the system, the goal is to synthesize a state feedback law such that the trajectories of the closed loop system are guaranteed to avoid an unsafe set even in the presence of unknown but bounded disturbances (process noise). The main result of the paper shows that for polynomial dynamics, this problem can be reduced to a tractable convex optimization by combining elements from polynomial optimization and the theorem of alternatives. This optimization provides both a rational control law and a density function safety certificate. These results are illustrated with numerical examples.

Robust non-hyperbolic fibred quadratic polynomial dynamics

Robust non-hyperbolic fibred quadratic polynomial dynamics

Hausdorff Dimension of Julia Sets in the Logistic Family

A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the Mandelbrot set \({{\mathcal {M}}}\), the Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we observe that the Hausdorff dimension of quadratic Julia sets depends continuously on c and find explicit bounds at the tip of \({{\mathcal {M}}}\) for most real parameters in the the sense of 1-dimensional Lebesgue measure.

Transcendence of polynomial canonical heights

There are two fundamental problems motivated by Silverman’s conversations over the years concerning the nature of the exact values of canonical heights of \(f(z)\in \bar{\mathbb {Q}}(z)\) with \(d:=\deg (f)\ge 2\). The first problem is the conjecture that \(\hat{h}_f(a)\) is either 0 or transcendental for every \(a\in \mathbb {P}^1(\bar{\mathbb {Q}})\); this holds when f is linearly conjugate to \(z^d\) or \(\pm C_d(z)\) where \(C_d(z)\) is the Chebyshev polynomial of degree d since \(\hat{H}_f(a)\) is algebraic for every a. Other than this, very little is known: for example, it is not known if there exists even one rational number a such that \(\hat{h}_f(a)\) is irrational where \(f(z)=z^2+\displaystyle \frac{1}{2}\). The second problem asks for the characterization of all pairs (f, a) such that \(\hat{H}_f(a)\) is algebraic. In this paper, we solve the second problem and obtain significant progress toward the first problem in the case of polynomial dynamics. These are consequences of our main result concerning algebraic numbers that can be expressed as a multiplicative combination of values of Böttcher coordinates. The proof of our main result uses a certain auxiliary polynomial and the powerful Medvedev–Scanlon classification of preperiodic subvarieties of split polynomial maps.

Splitting Hairs with Transcendental Entire Functions

Abstract In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of their topological dynamics in terms of a simpler model has been given inspired by methods from polynomial dynamics. In this paper, and for the 1st time, we give analogous results in cases when the postsingular set is unbounded. More specifically, we show that if $f$ is of finite order, has bounded criticality on its Julia set $J(f)$, and its singular set consists of finitely many critical values that escape to infinity and satisfy a certain separation condition, then $J(f)$ is a collection of dynamic rays or hairs, which split at critical points, together with their corresponding landing points. In fact, our result holds for a much larger class of functions with bounded singular set. Moreover, this result is a consequence of a significantly more general one: we provide a topological model for the action of $f$ on its Julia set.

Stability and Performance Verification of Dynamical Systems Controlled by Neural Networks: Algorithms and Complexity

This work makes several contributions on stability and performance verification of nonlinear dynamical systems controlled by neural networks. First, we show that the stability and performance of a polynomial dynamical system controlled by a neural network with semialgebraically representable activation functions (e.g., ReLU) can be certified by convex semidefinite programming. The result is based on the fact that the semialgebraic representation of the activation functions and polynomial dynamics allows one to search for a Lyapunov function using polynomial sum-of-squares methods. Second, we remark that even in the case of a linear system controlled by a neural network with ReLU activation functions, the problem of verifying asymptotic stability is undecidable. Finally, under additional assumptions, we establish a converse result on the existence of a polynomial Lyapunov function for this class of systems. Numerical results with code available online on examples of state-space dimension up to 50 and neural networks with several hundred neurons and up to 30 layers demonstrate the method.

Homotopy Hubbard trees for post-singularly finite exponential maps

AbstractWe extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict traditional sense, as invariant compact trees embedded in $\mathbb {C}$ , do not exist even for post-singularly finite exponential maps; the difficulty lies in the existence of asymptotic values. We therefore introduce the concept of a homotopy Hubbard tree that takes care of these difficulties. Specifically for the family of exponential maps, we show that every post-singularly finite map has a homotopy Hubbard tree that is unique up to homotopy, and that post-singularly finite exponential maps can be classified in terms of homotopy Hubbard trees, using a transcendental analogue of Thurston’s topological characterization theorem of rational maps.

Converging outer approximations to global attractors using semidefinite programming

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

Optimal control and learning for cyber‐physical systems

Optimal control and learning for <scp>cyber‐physical</scp> systems

Value distribution of derivatives in polynomial dynamics

AbstractFor every$m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in$\mathbb {C}\setminus \{0\}$under the$m$th order derivatives of the iterates of a polynomials$f\in \mathbb {C}[z]$of degree$d&gt;1$towards the harmonic measure of the filled-in Julia set offwith pole at$\infty $. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number fieldkfor a sequence of effective divisors on$\mathbb {P}^1(\overline {k})$having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of$\mathbb {C}^2$has a given eigenvalue.

Reachability Analysis of Neural Feedback Loops

Neural Networks (NNs) can provide major empirical performance improvements for closed-loop systems, but they also introduce challenges in formally analyzing those systems' safety properties. In particular, this work focuses on estimating the forward reachable set of \textit{neural feedback loops} (closed-loop systems with NN controllers). Recent work provides bounds on these reachable sets, but the computationally tractable approaches yield overly conservative bounds (thus cannot be used to verify useful properties), and the methods that yield tighter bounds are too intensive for online computation. This work bridges the gap by formulating a convex optimization problem for the reachability analysis of closed-loop systems with NN controllers. While the solutions are less tight than previous (semidefinite program-based) methods, they are substantially faster to compute, and some of those computational time savings can be used to refine the bounds through new input set partitioning techniques, which is shown to dramatically reduce the tightness gap. The new framework is developed for systems with uncertainty (e.g., measurement and process noise) and nonlinearities (e.g., polynomial dynamics), and thus is shown to be applicable to real-world systems. To inform the design of an initial state set when only the target state set is known/specified, a novel algorithm for backward reachability analysis is also provided, which computes the set of states that are guaranteed to lead to the target set. The numerical experiments show that our approach (based on linear relaxations and partitioning) gives a $5\times$ reduction in conservatism in $150\times$ less computation time compared to the state-of-the-art. Furthermore, experiments on quadrotor, 270-state, and polynomial systems demonstrate the method's ability to handle uncertainty sources, high dimensionality, and nonlinear dynamics, respectively.