Computer-assisted Proof Research Articles

Constructing a New Multi-Scroll Chaotic System and Its Circuit Design

Multi-scroll chaotic systems have complex dynamic behaviors, and the multi-scroll chaotic system design and analysis of their dynamic characteristics is an open research issue. This study explores a new multi-scroll chaotic system derived from an asymptotically stable linear system and designed with a uniformly bounded controller. The main contributions of this paper are given as follows: (1) The controlled system can cause chaotic behavior with an appropriate control position and parameters values, and a new multi-scroll chaotic system is proposed using a bounded sine function controller. Meanwhile, the dynamical characteristics of the controlled system are analyzed through the stability of the equilibrium point, a bifurcation diagram, and Lyapunov exponent spectrum. (2) According to the Poincaré section, the existence of a topological horseshoe is proven using the rigorous computer-aided proof in the controlled system. (3) Numerical results of the multi-scroll chaotic system are shown using Matlab R2020b, and the circuit design is also given to verify the multi-scroll chaotic attractors.

Computer-assisted proofs for the many steady states of a chemotaxis model with local sensing

We study the steady states of a system of cross-diffusion equations arising from the modeling of chemotaxis with local sensing, where the motility is a decreasing function of the concentration of the chemical. In order to capture the many different equilibria that sometimes co-exist, we use computer-assisted proofs: Given an approximate solution obtained numerically, we apply a fixed-point argument in a small neighborhood of this approximate solution to prove the existence of an exact solution nearby. This allows us to rigorously study the steady states of this cross-diffusion system much more extensively than what previously possible with purely pen-and-paper techniques. Our computer-assisted argument makes use of Fourier series decomposition, which is common in the literature, but usually restricted to systems with polynomial nonlinearities. This is not the case for the model considered in this paper, and we develop a new way of dealing with some nonpolynomial nonlinearities in the context of computer-assisted proofs with Fourier series.

Proofs for a price: Tomorrow’s ultra-rigorous mathematical culture

Computational tools might tempt us to renounce complete certainty. By forgoing of rigorous proof, we could get (very) probable results for a fraction of the cost. But is it really true that proofs (as we know and love them) can lead us to certainty? Maybe not. Proofs do not wear their correctness on their sleeve, and we are not infallible in checking them. This suggests that we need help to check our results. When our fellow mathematicians will be too tired or too busy to scrutinize our putative proofs, computer proof assistants could help. But feeding a mathematical argument to a computer is hard. Still, we might be willing to undertake the endeavor in view of the extra perks that formalization may bring—chiefly among them, an enhanced mathematical understanding.

Efficient and Validated Numerical Evaluation of Abelian Integrals

Abelian integrals play a key role in the infinitesimal version of Hilbert’s 16th problem. Being able to evaluate such integrals—with guaranteed error bounds—is a fundamental step in computer-aided proofs aimed at this problem. Using interpolation by trigonometric polynomials and quasi-Newton-Kantorovitch validation, we develop a validated numerics method for computing Abelian integrals in a quasi-linear number of arithmetic operations. Our approach is both effective, as exemplified on two practical perturbed integrable systems, and amenable to an implementation in a formal proof assistant, which is key to provide fully reliable computer-aided proofs.

Proof in the time of machines

We are concerned here with the nature of proof and what proof will become in this age of machines. We do so by comparing the values associated with (traditional) community based proof verification to those associated with computer proof verification. We finish by proposing ways that computer proofs might incorporate successful strategies from human experiences.

Another computer-assisted proof of unimodality of solutions for Proudman–Johnson equation

This paper presents a computer-assisted proof of the existence and unimodality of steady-state solutions for the Proudman–Johnson equation which is representative of two-dimensional fluid flow. The proposed approach is based on an infinite-dimensional fixed-point theorem with interval arithmetic, and is another proof by Miyaji and Okamoto (Jpn J Ind Appl Math 36:287–298, 2019). Verification results show the validity of both proofs.

Computer-assisted proofs of existence of fiberwise hyperbolic invariant tori in quasi-periodic systems via Fourier methods

Computer-assisted proofs of existence of fiberwise hyperbolic invariant tori in quasi-periodic systems via Fourier methods

Computer-assisted proofs of existence of KAM tori in planetary dynamical models of [formula omitted]-And [formula omitted

We reconsider the problem of the orbital dynamics of the innermost exoplanet of the υ-Andromedæsystem (i.e., υ-And b) into the framework of a Secular Quasi-Periodic Restricted Hamiltonian model. This means that we preassign the orbits of the planets that are expected to be the biggest ones in that extrasolar system (namely, υ-And c and υ-And d). The Fourier decompositions of their secular motions are injected in the equations describing the orbital dynamics of υ-And b under the gravitational effects exerted by those two exoplanets. By a computer-assisted procedure, we prove the existence of KAM tori corresponding to orbital motions that we consider to be very robust configurations, according to the analysis and the numerical explorations made in our previous article. The computer-assisted assisted proofs are successfully performed for two variants of the Secular Quasi-Periodic Restricted Hamiltonian model, which differs for what concerns the effects of the relativistic corrections on the orbital motion of υ-And b, depending on whether they are considered or not.

Constructive proofs for localised radial solutions of semilinear elliptic systems on

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on , the Swift–Hohenberg equation on , and a three-component FitzHugh–Nagumo system on . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.

Reduction in Degrees of Freedom for Large-Scale Nonlinear Systems in Computer-Assisted Proofs

In many physical systems, it is important to know the exact trajectory of a solution. Relevant applications include celestial mechanics, fluid mechanics, robotics, etc. For cases where analytical methods cannot be applied, one can use computer-assisted proofs or rigorous computations. One can obtain a guaranteed bound for the solution trajectory in the phase space. The application of rigorous computations poses few problems for low-dimensional systems of ordinary differential equations (ODEs) but is a challenging problem for large-scale systems, for example, systems of ODEs obtained from the discretization of the PDEs. A large-scale system size for rigorous computations can be as small as about a hundred ODE equations because computational complexity for rigorous algorithms is much larger than that for simple computations. We are interested in the application of rigorous computations to the problem of proving the existence of a periodic orbit in the Kolmogorov problem for the Navier–Stokes equations. One of the key issues, among others, is the computation complexity, which increases rapidly with the growth of the problem dimension. In previous papers, we showed that 79 degrees of freedom are needed in order to achieve convergence of the rigorous algorithm only for the system of ordinary differential equations. Here, we wish to demonstrate the application of the proper orthogonal decomposition (POD) in order to approximate the attracting set of the system and reduce the dimension of the active degrees of freedom.

Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error analysis

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The paper provides an elementary analysis to derive the Hadamard shape derivative formula for complicated problem settings that allow non-homogeneous Neumann boundary conditions. Moreover, an algorithm to rigorously evaluate the Hadamard formula is proposed by utilizing the guaranteed computation for both eigenvalues and eigenfunctions of differential operators, which is recently developed by the second author. Besides the model homogeneous Dirichlet eigenvalue problem, the eigenvalue problem associated with a non-homogeneous Neumann boundary condition, which is related to the Crouzeix–Raviart interpolation error constant, is considered. The computer-assisted proof tells that among the triangles with the unit diameter, the equilateral triangle minimizes the first eigenvalue for each concerned eigenvalue problem.

Epidemic Waves in a Stochastic SIRVI Epidemic Model Incorporating the Ornstein–Uhlenbeck Process

The worldwide data for COVID-19 for active, infected individuals in multiple waves show that traditional epidemic models with constant parameters are not able to capture this kind of disease behavior. We solved this major open mathematical problem in this report. We first consider the disease transmission rate for the stochastic SIRVI epidemic model, which satisfies the mean-reverting Ornstein–Uhlenbeck (OU) process, and we propose a new stochastic SIRVI model. We then showed the existence and uniqueness of the global solution and obtained sufficient conditions for the persistent mean and exponential extinction of infectious disease, which have not been given before. In the second part, we derive a nonlinear system of differential equations for the time-dependent transmission rate from the deterministic SIRVI model and present an algorithm to compute the time-dependent transmission rate directly from the given active, infected individuals’ data. We then show that the time-dependent transmission obtained from and perturbed by the Ornstein–Uhlenbeck process could be represented after using a smoothing technique using a finite linear combination of a Gaussian radial basis function, which was obtained from our algorithm. This novel computer-assisted proof provides a theoretical basis for other epidemic models and epidemic waves. Finally, some numerical solutions of the stochastic SIRVI model are presented using COVID-19 data from Saudi Arabia and Austria.

Elli Heesch, Heinrich Heesch and Hilbert’s eighteenth problem: collaborative research between philosophy, mathematics and application

This paper examines the hitherto unknown scientific collaboration between the siblings Elli Heesch (1904–1993) and Heinrich Heesch (1906–1995). Heinrich Heesch, a well-known mathematician, was spearheading the early development of the computer-aided proof of the four-colour theorem. Much less is known about his sister Elli Heesch, a philosopher and logician. Together with her brother she investigated tiling problems and worked out a solution of Hilbert’s 18th problem. In 1944, Elli and Heinrich Heesch wrote a joint treatise on the industrial application of the tessellation method, which was of great interest to the German war and armaments industry. The collaboration of the Heesch siblings illustrates individual, disciplinary, cultural, and political aspects of knowledge production. The common interplay of close family relations and socio-political conditions that we find here underlines the fact that women’s contributions to solving mathematical problems often remained invisible.

Persistence of Periodic Orbits under State-dependent Delayed Perturbations: Computer-assisted Proofs

Persistence of Periodic Orbits under State-dependent Delayed Perturbations: Computer-assisted Proofs

Computer-Assisted Proofs Take on Fluid Flow

Researchers use computers to show that equations can "blow up."

Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric Euler Equations Using Neural Networks.

Whether there exist finite-time blow-up solutions for the 2D Boussinesq and the 3D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks, that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate physics-informed neural networks could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.

Philosophical Assumptions Behind the Rejection of Computer-Based Proofs

Abstract In 1977, the first computer-assisted proof of a mathematical theorem was presented by K. Appel and W. Haken. The proof was met with a lot of criticism from both mathematicians and philosophers. In this paper, I present some examples of computer-assisted proofs, including Appel and Haken’s work. Then, I analyze the most famous arguments against the equal acceptance of computer-based and human-based proofs in mathematics and examine the philosophical assumptions behind the presented criticism. In the conclusion, I talk about whether the philosophical assumptions are justified as they are, or one needs to take a specific philosophical position to accept them.

Existence proof of librational invariant tori in an averaged model of HD60532 planetary system

We investigate the long-term dynamics of HD60532, an extrasolar system hosting two giant planets orbiting in a 3:1 mean motion resonance. We consider an average approximation at order one in the masses which results (after the reduction in the constants of motion) in a resonant Hamiltonian with two libration angles. In this framework, the usual algorithms constructing the Kolmogorov normal form approach do not easily apply and we need to perform some untrivial preliminary operations, in order to adapt the method to this kind of problems. First, we perform an average over the fast angle of libration which provides an integrable approximation of the Hamiltonian. Then, we introduce action-angle variables that are adapted to such an integrable approximation. This sequence of preliminary operations brings the Hamiltonian in a suitable form to successfully start the Kolmogorov normalization scheme. The convergence of the KAM algorithm is proved by applying a technique based on a computer-assisted proof. This allows us to reconstruct the quasi-periodic motion of the system, with initial conditions that are compatible with the observations.

Conic Linear Optimization for Computer-Assisted Proofs

From a mathematical perspective, optimization is the science of proving inequalities. In this sense, computational optimization is a method for computer-assisted proofs. Conic (linear) optimization is the problem of minimizing a linear functional over the intersection of a convex cone with an affine subspace of a topological vector space. For many cones this problem is computationally tractable, and as a result there is a growing number of computer-assisted proofs using conic optimization in discrete geometry, (extremal) graph theory, numerical analysis, and other fields, the most famous example perhaps being the proof of the Kepler Conjecture. The aim of this workshop was to bring researchers from these diverse fields together to work towards expanding the current scope of conic optimization as a method of generating proofs, and to identify problems and challenges to work on together.

Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems

We confirm a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. The method of showing the main chaotic property, a positive Lyapunov exponent, is a computer-assisted proof. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains and the modified Furstenberg-Khasminskii formula, the problem boils down to the rigorous computation of eigenfunctions of the Kolmogorov operators describing distributions of the underlying stochastic process.