Recently sum-of-squares (SOS) based methods have been used for the stability analysis and control synthesis of polynomial dynamical systems. This analysis framework was also extended to non-polynomial dynamical systems, including power systems, using an algebraic reformulation technique that recasts the system's dynamics into a set of polynomial differential algebraic equations. Nevertheless, for large scale dynamical systems this method becomes inapplicable due to its computational complexity. For this reason we develop a subsystem based stability analysis approach using vector Lyapunov functions and introduce a parallel and scalable algorithm to infer the stability of the interconnected system with the help of the subsystem Lyapunov functions. Furthermore, we design adaptive and distributed control laws that guarantee asymptotic stability under a given external disturbance. Finally, we apply this algorithm for the stability analysis and control synthesis of a network preserving power system.
Stability analysis tools are essential to understanding and controlling any engineering system. Recently sum-of-squares (SOS) based methods have been used to compute Lyapunov based estimates for the region-of-attraction (ROA) of polynomial dynamical systems. But for a real-life large scale dynamical system this method becomes inapplicable because of growing computational burden. In such a case, it is important to develop a subsystem based stability analysis approach which is the focus of the work presented here. A parallel and scalable algorithm is used to infer stability of an interconnected …
We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller flow on the $SL_2(R)$-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M. Viana with respect to the so-called Masur-Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus $\geq 3$. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and…
Traditional Lyapunov based transient stability assessment approaches focus on identifying the stability region (SR) of the equilibrium point under study. When trying to estimate this region using Lyapunov functions, the shape of the final estimate is often limited by the degree of the function chosen, a limitation that results in conservativeness in the estimate of the SR. More conservative the estimate is in a particular region of state space, smaller is the estimate of the critical clearing time for disturbances that drive the system towards that region. In order to reduce this conservativen…
Let f_t be a meromorphic family of endomorphisms of P^N_C of degree at least 2, and let L(f_t) be the sum of Lyapunov exponents associated to f_t. Favre showed that L(f_t)=L(f)\log|t^{-1}|+o(\log|t^{-1}|) as t -> 0, where L(f) is the sum of Lyapunov exponents on the generic fibre, interpreted as an endomorphism of some projective Berkovich space. Under some additional constraints on the family, we provide an explicit error term.
A cyclic cover over the Riemann sphere branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle over the corresponding arithmetic Teichmuller curve. The key technical element is evaluation of degrees of line subbundles of the Hodge bundle, corresponding to eigenspaces of the induced action of deck transformations.
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmueller curves with various geometrically defined divisors on moduli spaces of curves.
We provide an explicit lower bound for the the sum of the non-negative Lyapunov exponents for some cocycles related to the Anderson model. In particular, for the Anderson model on a strip of width $ W $ the lower bound is proportional to $ W^{-ε} $, for any $ ε>0 $. This bound is consistent with the fact that the lowest non-negative Lyapunov exponent is conjectured to have a lower bound proportional to $ W^{-1} $.
We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.
This paper presents numerical methods for computing regions of finite-time invariance (funnels) around solutions of polynomial differential equations. First, we present a method which exactly certifies sufficient conditions for invariance despite relying on approximate trajectories from numerical integration. Our second method relaxes the constraints of the first by sampling in time. In applications, this can recover almost identical funnels but is much faster to compute. In both cases, funnels are verified using Sum-of-Squares programming to search over a family of time-varying polynomial Lya…
We consider normal covers of CP^1 with abelian deck group, branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces. We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov …
We introduce two square-tiled surfaces, one with $8$ squares inside $Ω\mathcal{M}_3(2,2)$, and the other with $9$ squares inside $Ω\mathcal{M}_4(3,3)$, respectively. In these examples, the dimensions of the isotropic subspaces (in absolute homology) generated by the waist curves of the maximal cylinders in any fixed rational direction are $2$ and $3$ respectively. Hence, a geometrical criterion of G. Forni for the non-uniform hyperbolicity of Kontsevich-Zorich (KZ) cocycle can not be applied to these examples. Nevertheless, we prove that there are no vanishing exponents and the spectrum is sim…
Koopman spectral theory has grown in the past decade as a powerful tool for dynamical systems analysis and control. In this paper, we show how recent data-driven techniques for estimating Koopman-Invariant subspaces with neural networks can be leveraged to extract Lyapunov certificates for the underlying system. In our work, we specifically focus on systems with a limit-cycle, beyond just an isolated equilibrium point, and use Koopman eigenfunctions to efficiently parameterize candidate Lyapunov functions to construct forward-invariant sets under some (unknown) attractor dynamics. Additionally…
In this paper, we give a quantitative estimate for the sum of the first $N$ Lyapunov exponents for random perturbations of a natural class $2N$-dimensional volume-preserving systems exhibiting strong hyperbolicity on a large but noninvariant subset of phase space. Concrete models covered by our setting include systems of coupled standard maps, in both `weak' and `strong' coupling regimes.
We study holomorphic families of polynomial-like maps depending on a parameter s. We prove that the partial sums of largest Lyapunov exponents are plurisubharmonic functions of s. We also study their continuity and introduce the bifurcation locus as the support of bifurcation currents.
In this paper we are considering partially hyperbolic diffeomorphims of the torus, with $dim(E^c) > 1.$ We prove, under some conditions, that if the all center Lyapunov exponents of the linearization $A,$ of a \mbox{DA-diffeomorphism} $f,$ are positive and the center foliation of $f$ is absolutely continuous, then the sum of the center Lyapunov exponents of $f$ is bounded by the sum of the center Lyapunov exponents of $A.$ After, we construct a $C^1-$open class of volume preserving \mbox{DA-diffeomorphisms}, far from Anosov diffeomorphisms, with non compact pathological two dimensional center …
We show the sum of the first $k$ Lyapunov exponents of linear cocycles is an upper semicontinuous function in the $L^p$ topologies, for any $1 \le p \le \infty$ and $k$. This fact, together with a result from Arnold and Cong, implies that the Lyapunov exponents of the $L^p$-generic cocycle, $p<\infty$, are all equal.
This paper considers the problem of approximating the "maximal" region of attraction (the set that contains all asymptotically stable sets) of any given set of locally exponentially stable nonlinear Ordinary Differential Equations (ODEs) with a sufficiently smooth vector field. Given a locally exponential stable ODE with a differentiable vector field, we show that there exists a globally Lipschitz continuous converse Lyapunov function whose 1-sublevel set is equal to the maximal region of attraction of the ODE. We then propose a sequence of d-degree Sum-of-Squares (SOS) programming problems th…
We present a new proof, using the log-sum inequality, that the pseudo-Helmholtz free energy function is a Lyapunov function for complex-balanced mass-action systems. This proof is shorter and simpler than previous proofs.
Lyapunov functions provide a tool to analyze the stability of nonlinear systems without extensively solving the dynamics. Recent advances in sum-of-squares methods have enabled the algorithmic computation of Lyapunov functions for polynomial systems. However, for general large-scale nonlinear networks it is yet very difficult, and often impossible, both computationally and analytically, to find Lyapunov functions. In such cases, a system decomposition coupled to a vector Lyapunov functions approach provides a feasible alternative by analyzing the stability of the nonlinear network through a re…
Linear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as~well as on a space of $p$-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.
We introduce a notion of stability for equilibrium measures in holomorphic families of endomorphisms of CP(k) and prove that it is equivalent to the stability of repelling cycles and equivalent to the existence of some measurable holomorphic motion of Julia sets which we call equilibrium lamination. We characterize the corresponding bifurcations by the strict subharmonicity of the sum of Lyapunov exponents or the instability of critical dynamics and analyze how repelling cycles may bifurcate. Our methods deeply exploit the properties of Lyapunov exponents and are based on ergodic theory and on…
The dynamics of many systems from physics, economics, chemistry, and biology can be modelled through polynomial functions. In this paper, we provide a computational means to find positively invariant sets of polynomial dynamical systems by using semidefinite programming to solve sum-of-squares (SOS) programmes. With the emergence of SOS programmes, it is possible to efficiently search for Lyapunov functions that guarantee stability of polynomial systems. Yet, SOS computations often fail to find functions, such that the conditions hold in the entire state space. We show here that restricting th…
In this paper, we examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. A common approach to Lyapunov function synthesis starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive-definiteness of the function, and the negative-definiteness of its derivative over the domain of interest. Therefore, the key primitives for this encoding include: (a) proving that a given polynomial is positive definite over a domain of interest, and (b) encoding the p…
In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form $$g(z,\varepsilon)=g_0(z)+\sum_{i=1}^k \varepsilon^i g_i(z)+\mathcal{O}(\varepsilon^{k+1}),$$ for $|\varepsilon|\neq0$ sufficiently small. Here $g_i:\mathcal{D}\rightarrow\mathbb{R}^n$, for $i=0,1,\ldots,k$, are smooth functions being $\mathcal{D}\subset \mathbb{R}^n$ an open bounded set. Then we use this result to compute the bifurcation functions which controls the periodic solutions of the following $T$-periodic smooth differential system $$ x'=F_0(t,x)+\sum_{i=1}^k \vare…
