Gianni Arioli

Politecnico di Milano | Polimi · Department of Mathematics "Francesco Brioschi"

We develop a computer-assisted technique for constructing and analyzing orbits of dissipative evolution equations. As a case study, the methods are applied to the Kuramoto-Sivashinski equation, for which we prove the existence of a hyperbolic periodic orbit.

We consider the spectral stability problem for Floquet-type systems such as the wave equation v_ ττ = γ^2 v_xx − ψv with periodic forcing ψ. Our approach is based on a comparison with finite-dimensional approximations. Specific results are obtained for a system where the forcing is due to a coupling between the wave equation and a time-period solut...

The spectacular collapse of the Tacoma Narrows Bridge has attracted the attention of engineers, physicists, and mathematicians in the last 74 years. There have been many attempts to explain this amazing event, but none is universally accepted. It is however well established that the main culprit was the unexpected appearance of torsional oscillatio...

We construct periodic and quasiperiodic solutions for the nonlinear plate equation on the unit sphere in R 3. Periodic solutions for the nonlinear wave equation are constructed as well. The methods do not involve small parameters. In the case of a cubic nonlinear term, our approach is computer-assisted and thus yields detailed information about eac...

We consider the equation −∆w = w 3 with zero boundary conditions on planar domains that are conformal images of annuli. Starting with an approximate solution, we prove that there exist a true solution nearby. Our approach is computer-assisted. It involves simultaneous and accurate control of the (inverse) Dirichlet Laplacean, nonlinearities, and co...

A physical rattleback is a toy that can exhibit counter-intuitive behavior when spun on a horizontal plate. Most notably, it can spontaneously reverse its direction of rotation. Using a standard mathematical model of the rattleback, we prove the existence of reversing motion, reversing motion combined with rolling, and orbits that exhibit such beha...

A physical rattleback is a toy that can exhibit counter-intuitive behavior when spun on a horizontal plate. Most notably, it can spontaneously reverse its direction of rotation. Using a standard mathematical model of the rattleback, we prove the existence of reversing motion, reversing motion combined with rolling, and orbits that exhibit such beha...

We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions th...

The stationary Navier–Stokes equations under Navier boundary conditions are considered in a square. The uniqueness of solutions is studied in dependence of the Reynolds number and of the strength of the external force. For some particular forcing, it is shown that uniqueness persists on some continuous branch of solutions, when these quantities bec...

We consider the Navier–Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Ou...

We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions th...

We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Ou...

Econophysics is a branch of economics that imports concepts and methods from physics to the financial markets. The present paper focuses on a subfield of econophysics called quantum finance, more precisely on the approaches developed by Ilinski (2001) and Baaquie (2004, 2009, 2018), wherein the quantum formalism is applied to deal with the uncertai...

Traveling waves for the FPU chain are constructed by solving the associated equation for the spatial profile u of the wave. We consider solutions whose derivatives need not be small, may change sign several times, but decrease at least exponentially. This includes multi-bump solutions. Our method of proof is computer-assisted. Unlike other methods,...

The stationary Navier-Stokes equations under Navier boundary conditions are considered in a square. The uniqueness of solutions is studied in dependence of the Reynolds number and of the strength of the external force. For some particular forcing, it is shown that uniqueness persists on some continuous branch of solutions, when these quantities bec...

We consider several breather solutions for FPU-type chains that have been found numerically. Using computer-assisted techniques, we prove that there exist true solutions nearby, and in some cases, we determine whether or not the solution is spectrally stable. Symmetry properties are considered as well. In addition, we construct solutions that are c...

Traveling waves for the FPU chain are constructed by solving the associated equation for the spatial profile u of the wave. We consider solutions whose derivatives u ′ need not be small, may change sign several times, but decrease at least exponentially. Our method of proof is computer-assisted. Unlike other methods, it does not require that the FP...

Traveling waves for the FPU chain are constructed by solving the associated equation for the spatial profile $u$ of the wave. We consider solutions whose derivatives $u'$ need not be small, may change sign several times, but decrease at least exponentially. Our method of proof is computer-assisted. Unlike other methods, it does not require that the...

Starting with approximate solutions of the equation −Δu=wu3 on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a computer,...

We introduce a model for the dynamics of stock prices based on a non quadratic path integral. The model is a generalization of Ilinski's path integral model, more precisely we choose a different action, which can be tuned to different time scales. The result is a model with a very small number of parameters that provides very good fits of some stoc...

We consider several breather solutions for FPU-type chains that have been found numerically. Using computer-assisted techniques, we prove that there exist true solutions nearby, and in some cases, we determine whether or not the solution is spectrally stable. Symmetry properties are considered as well. In addition, we construct solutions that are c...

We consider several breather solutions for FPU-type chains that have been found numerically. Using computer-assisted techniques, we prove that there exist true solutions nearby, and in some cases, we determine whether or not the solution is spectrally stable. Symmetry properties are considered as well. In addition, we construct solutions that are c...

We introduce a model for the short-term dynamics of financial assets based on an application to finance of quantum gauge theory, developing ideas of Ilinski. We present a numerical algorithm for the computation of the probability distribution of prices and compare the results with APPLE stocks prices and the S&P500 index.

Starting with approximate solutions of the equation $-\Delta u=wu^3$ on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a c...

Starting with approximate solutions of the equation −∆u = wu 3 on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a compute...

We consider the nonlinear wave equation $u_{tt}-u_{xx}=\pm u^3$ and the beam equation $u_{tt}+u_{xxxx}=\pm u^3$ on an interval. Numerical observations indicate that time-periodic solutions for these equations are organized into structures that resemble branches and seem to undergo bifurcations. Besides describing our observations, we prove the exis...

In the last 10 years many 3D numerical schemes have been developed for the study the flow of a mixture of liquid and gas in a pipeline (Frank, Numerical simulation of slug flow regime for an air-water two-phase flow in horizontal pipes. In: The 11th international topical meeting on nuclear reactor thermal-hydraulics (NURETH-11), Avignon, 2005; Vall...

All attempts of aeroelastic explanations for the torsional instability of suspension bridges have been somehow criticised and none of them is unanimously accepted by the scientific community. We suggest a new nonlinear model for a suspension bridge and we perform numerical experiments with the parameters corresponding to the collapsed Tacoma Narrow...

We consider the equation on a symmetric bounded domain in with Dirichlet boundary conditions. Here w is a positive function or measure that is invariant under the (Euclidean) symmetries of the domain. We focus on solutions u that are positive and/or have a low Morse index. Our results are concerned with the existence of non-symmetric solutions and...

We suggest a new model for the dynamics of a suspension bridge through a system of nonlinear nonlocal hyperbolic differential equations. The equations are of second and fourth order in space and describe the behavior of the main components of the bridge: the deck, the sustaining cables and the connecting hangers. We perform a careful energy balance...

The FitzHugh–Nagumo model is a reaction–diffusion equation describing the propagation of electrical signals in nerve axons and other biological tissues. One of the model parame- ters is the ratio ε of two time scales, which takes values between 0.001 and 0.1 in typical simulations of nerve axons. Based on the existence of a (singular) limit at ε =...

We present a new numerical method for the computation of the forcing term of minimal norm such that a two-point boundary value problem admits a solution. The method relies on the following steps. The forcing term is written as a (truncated) Chebyshev series, whose coefficients are free parameters. A technique derived from automatic differentiation...

We consider the equation −Δu=wu3−Δu=wu3 on a square domain in R2R2, with Dirichlet boundary conditions, where w is a given positive function that is invariant under all (Euclidean) symmetries of the square. This equation is shown to have a solution u, with Morse index 2, that is neither symmetric nor antisymmetric with respect to any nontrivial sym...

We analyze a model of electric signaling in biological tissues and prove that this model admits a traveling wave solution. Our result is based on a new technique for computing rigorous bounds on the stable and unstable manifolds at an equilibrium point of a dynamical system depending on a parameter.

The coupling between cardiac mechanics and electric signaling is addressed in a nonstandard framework in which the electrical potential dictates the active strain (not stress) of the muscle. The physiological and mathematical motivations leading us to this choice are illustrated. The propagation of the electric signal is assumed to be governed by t...

We prove the existence of the critical fixed point (F, G) for MacKay’s renormalization operator for pairs of maps of the plane. The maps F and G commute, are area-preserving, reversible, real analytic, and they satisfy a twist condition.

Existence results available for the semilinear Brezis-Nirenberg eigenvalue problem suggest that the compactness problems for the corresponding action functionals are more serious in small dimensions. In space dimension n = 3, one can even prove nonexistence of positive solutions in a certain range of the eigenvalue parameter. In the present paper w...

We study a reaction–diffusion system of two parabolic differential equations describing the behavior of a nuclear reactor. We provide existence results for nontrivial periodic solutions, nonexistence results for stationary solutions and we prove that, depending on the value of the parameters, either the system admits a compact global attractor, or...

We consider the Newtonian system -q̈ + B(t) q = W q(q,t) with B, W periodic in t, B positive definite, and show that for each isolated homoclinic solution q0 having a nontrivial critical group (in the sense of Morse theory), multibump solutions (with 2 ≤ k ≤ ∞ bumps) can be constructed by gluing translates of q0. Further we show that the collection...

We prove the existence of a new branch of solutions of Mountain Pass type for the periodic 3-body problem with choreographical constraint. At first we describe the variational structure of the action functional associated to the choreographical three body problem in \mathbbR3\mathbb{R}^{3}. In the second part, using a bisection algorithm, we provi...

We investigate entire radial solutions of the semilinear biharmonic equation Δ2u=λexp(u) in Rn, n⩾5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the n...

We introduce two novel methods for studying periodic solutions of the FPU -model, both numerically and rigorously. One is a variational approach, based on the dual formulation of the problem, and the other involves computer-assisted proofs. These methods are used e.g. to construct a new type of solutions, whose energy is spread among several modes,...

We study a semilinear fourth order elliptic problem with exponential nonlinearity. Motivated by a question raised in (Li), we partially extend known results for the corre- sponding second order problem. Several new diculties arise and many problems still remain to be solved. We list the ones we feel particularly interesting in the nal section. Math...

We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is w...

We describe a method for studying the existence and the linear stability of branches of periodic solutions for a dynamical system with a param- eter. We apply the method to the planar restricted 3-body problem extending the results of (A). More precisely, we prove the existence of some continuous branches of periodic orbits with the energy or the m...

We consider the semilinear stationary Schrdinger equation in a magnetic field: (–i+A)2 u+V(x)u=g(x,|u|)u in N , where V is the scalar (or electric) potential and A is the vector (or magnetic) potential. We study the existence of nontrivial solutions both in the critical and in the subcritical case (respectively g(x,|u|)=|u|2 * –2 and |g(x,|u|)|c(1...

We continue the investigation of the Hénon–Heiles system started in (Arioli G and Zgliczyński P 2001 Symbolic dynamics for the Henon–Heiles Hamiltonian on the critical level J. Diff. Eqns 171 173–202) and we provide new results in four directions. We prove the existence of infinitely many solutions which are homoclinic or heteroclinic to periodic s...

: This paper concerns the restricted 3-body problem. By applying topological methods we give a computer assisted proof of the existence of some classes of periodic orbits, the existence of symbolic dynamics and we give a rigorous lower estimate for the topological entropy.

We present a computer assisted proof of the existence of a rich symbolic dynamic structure for the Hénon–Heiles Hamiltonian system at the critical energy E=1/6.

We build a deformation for a continuous functional defined on a Banach space and invariant with respect to an isometric action of a noncompact group. Under these assumptions the Palais-Smale condition does not hold. When the functional is also invariant with respect to the action of a compact Lie group, we prove that the deformation can be chosen t...

We consider the periodic problem for a class of planar N -body systems in Celestial Mechanics. Our goal is to give a variational characterization of the Hill's (retrograde) orbits as minima of the action functional under some geometrical and topological constraints. The method developed here also turns out to be useful in the study of the full prob...

We use a nonsmooth critical point theory to prove existence results for a variational system of quasilinear elliptic equations in both the sublinear and superlinear cases. We extend a technique of Bartsch to obtain multiplicity results when the system is invariant under the action of a compact Lie group. The problem is rather different from its sca...

We study an eigenvalue problem by a non-smooth critical point theory. Under general assumptions, we prove the existence of at least one solution as a minimum of a constrained energy functional. We apply some results on critical point theory with symmetry to provide a multiplicity result.

We prove the existence of infinitely many homoclinic solutions for a first order Hamiltonian system, symmetric with respect to an action of a compact Lie group, by means of variational methods. We make no convexity assumption on the Hamiltonian.

This paper was written while the first author was visiting the Department of Mathematics at Stockholm University, supported by a CNR scholarship 1 Partially supported by MURST Gruppi di Ricerca 40% The existence of nontrivial solutions of quasilinear elliptic equations at critical growth is proved. The solutions are obtained by variational methods:...

Existence and multiplicity results for a variational quasilinear elliptic equation on unbounded domains are proved; the solutions are obtained as critical points of a nonsmooth functional. We consider the case where the functional is coercive or has a saddle-point geometry.

Equations involving the p-Laplacian with a term in the critical growth range are considered: existence results are obtained under minimal assumptions on the lower order perturbation. The problem is studied by means of variational methods: in particular, a problem with linking geometry is treated thanks to the orthogonalization technique introduced...

We study an autonomous dynamical system of Fermi–Pasta–Ulam type with in-finitely many degrees of freedom by means of variational methods. The functional is strongly indefinite: we apply two different techniques to handle this difficulty and we give a bifurcation result. Abstract. Nou etudions unsys eme dynamique autonome de sorte de Fermi–Pasta–Ul...

We study a class of variational quasilinear elliptic PDE's with resonance at infinity by means of a nonsmooth critical point theory. We prove the existence of a solution in the general case and a multiplicity result when the equation is invariant with respect to a Z2-action.

Nous etudions une certaine classe d'EDP quasi lineaires variationnelles avec resonance a l'infini en utilisant une theorie des points critiques pour fonctionnelles irregulieres. Nous demontrons l'existence d'une solution dans le cas general et un resultat de multiplicite lorsque l'equation est invariante par rapport a une action du groupe Z2.

We consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction. Our main purpose is to prove the existence of periodic solutions.

We prove the existence of periodic motions of an infinite lattice of particles; the proof involves the study of periodic motions for finite lattices by a linking technique and the passage to the limit by means of Lions'' concentration-compactness principle. We also give a numerical picture of the motion of some finite lattices and of the way the so...

We report on some numerical results in classical electrodynamics, described as a Hamiltonian system; we consider the standard Pauli-Fierz model, in which the field obeys Maxwell's equations with current due to a charged particle, while the particle satisfies Newton's equation with the Lorentz force. At variance with the standard treatments where th...