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The aim of this paper is to present a new, analytical, method for computing the exact number of relative equilibria in the planar, circular, restricted 4-body problem of celestial mechanics. The new approach allows for a very efficient computer-aided proof, and opens a potential pathway to proving harder instances of the n-body problem. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow. [ABSTRACT FROM AUTHOR]

We define a family of C 1 functions, which we call "nowhere coexpanding functions," that is closed under composition and includes all C 3 functions with non-positive Schwarzian derivatives. We establish results on the number and nature of the fixed points of these functions, including a generalization of a classic result of Singer. [ABSTRACT FROM AUTHOR]

We consider the family of destabilized Kuramoto–Sivashinsky equations in one spatial dimension u t + ν u x x x x + β u x x + γ u u x = α u for α, ν ≥ 0 and β , γ ∈ R . For certain parameter values, shock-like stationary solutions have been numerically observed. In this work we verify the existence of several such solutions using the framework of self-consistent bounds and validated numerics. [ABSTRACT FROM AUTHOR]

By performing a systematic study of the Hénon map, we find low-period sinks for parameter values extremely close to the classical ones. This raises the question whether or not the well-known Hénon attractor--the attractor of the Hénon map existing for the classical parameter values--is a strange attractor, or simply a stable periodic orbit. Using results from our study, we conclude that even if the latter were true, it would be practically impossible to establish this by computing trajectories of the map. [ABSTRACT FROM AUTHOR]

• Development of a set of interval arithmetic based tools for rigorous study of planar polynomial vector fields. • A computer assisted proof that the Songling system has exactly four limit cycles, • Precise bounds for the positions of these limit cycles. Determining how many limit cycles a planar polynomial system of differential equations can have is a remarkably hard problem. One of the main difficulties is that the limit cycles can reside within areas of vastly different scales. This makes numerical explorations very hard to perform, requiring high precision computations, where the necessary precision is not known in advance. Using rigorous computations, we can dynamically determine the required precision, and localize all limit cycles of a given system. We prove that the Songling system of planar, quadratic polynomial differential equations has exactly four limit cycles. Furthermore, we give precise bounds for the positions of these limit cycles using rigorous computational methods based on interval arithmetic. The techniques presented here are applicable to the much wider class of real-analytic planar differential equations. [ABSTRACT FROM AUTHOR]

The question of coexisting attractors for the Hénon map is studied numerically by performing an exhaustive search in the parameter space. As a result, several parameter values for which more than two attractors coexist are found. Using tools from interval analysis, we show rigorously that the attractors exist. In the case of periodic orbits, we verify that they are stable, and thus proper sinks. Regions of existence in parameter space of the found sinks are located using a continuation method; the basins of attraction are found numerically. [ABSTRACT FROM AUTHOR]

Abstract: We develop techniques for the verification of the Chebyshev property of Abelian integrals. These techniques are a combination of theoretical results, analysis of asymptotic behavior of Wronskians, and rigorous computations based on interval arithmetic. We apply this approach to tackle a conjecture formulated by Dumortier and Roussarie in [F. Dumortier, R. Roussarie, Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009) 723–781], which we are able to prove for . [Copyright &y& Elsevier]

An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four. [ABSTRACT FROM AUTHOR]

We show that, for a certain class of systems, the problem of establishing the existence of periodic orbits can be successfully studied by a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate positions of periodic points, and the existence of periodic orbits in a neighborhood of these approximations is proved using an interval operator. As an example, the Lorenz system is studied; a theoretical argument is used to prove that each periodic orbit has a distinct symbol sequence. All periodic orbits with the period p ≤ 16 of the Poincaré map associated with the Lorenz system are found. Estimates of the topological entropy of the Poincaré map and the flow, based on the number and flow-times of short periodic orbits, are calculated. Finally, we establish the existence of several long periodic orbits with specific symbol sequences. [ABSTRACT FROM AUTHOR]

The limit cycle bifurcations of a Z2 equivariant planar Hamiltonian vector field of degree 7 under Z2 equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert's 16th problem for degree 7, i.e. on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation. [ABSTRACT FROM AUTHOR]

The limit cycle bifurcation of a Z2 equivariant quintic planar Hamiltonian vector field under Z2 equivariant quintic perturbation is studied. We prove that the given system can have at least 27 limit cycles. This is an improved lower bound on the possible number of limit cycles that can bifurcate from a quintic planar Hamiltonian system under quintic perturbation. [ABSTRACT FROM AUTHOR]

Abstract: We establish a lower bound on the measure of the set of stable parameters for the quadratic map . For these parameters, we prove that either has a single stable periodic orbit or a period-doubling bifurcation. From this result, we also obtain a non-trivial upper bound on the set of stochastic parameters for . [Copyright &y& Elsevier]

We consider a hyper-elliptic Hamiltonian of degree five, chosen from a generic set of parameters, and study what configurations of limit cycles can bifurcate from the corresponding differential system under quartic perturbations. Perturbations of Lienard type are considered separately. Several different configurations with seven (four) limit cycles, bifurcating from the given system for general (Lienard type) quartic perturbations, are constructed. We also discuss how to construct perturbations yielding a given configuration, and how to validate the correctness of such a candidate perturbation. [ABSTRACT FROM AUTHOR]

Abstract: We present a method to find all zeros of an analytic function in a rectangular domain. The approach is based on finding guaranteed enclosures rather than approximations of the zeros. Well-isolated simple zeros are determined fast and with high accuracy. Clusters of zeros can in many cases be distinguished from multiple zeros by applying the argument principle to sufficiently high-order derivatives of the function. We illustrate the proposed method through five examples of varying levels of complexity. [Copyright &y& Elsevier]

Abstract: We present a method that–given a data set, a finitely parametrized system of ordinary differential equations (ODEs), and a search space of parameters–discards portions of the search space that are inconsistent with the model ODE and data. The method is completely rigorous as it is based on validated integration of the vector field. As a consequence, no consistent parameters can be lost during the pruning phase. For data sets with moderate levels of noise, this yields a good reconstruction of the underlying parameters. Several examples are included to illustrate the merits of the method. [Copyright &y& Elsevier]

Abstract: In this paper we prove that the spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in is within 10−2 from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture, cf., e.g., Problem 3.2.12 in [C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1994]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón–Zygmund theory, as well as interval analysis—resulting in a computer-aided proof. [Copyright &y& Elsevier]

We present a numerical method particularly suited for computing Poincare´ maps for systems of ordinary differential equations. The method is a generalization of a stopping procedure described by He´non [Physica D 5 (1982) 412], and it applies to a wide family of systems. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]