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Numerical integrations of the Solar System reveal a remarkable stability of the orbits of the inner planets over billions of years, in spite of their chaotic variations characterized by a Lyapunov time of only 5 million years and the lack of integrals of motion able to constrain their dynamics. To open a window on such long-term behavior, we compute the entire Lyapunov spectrum of a forced secular model of the inner planets. We uncover a hierarchy of characteristic exponents that spans two orders of magnitude, manifesting a slow-fast dynamics with a broad separation of timescales. A systematic analysis of the Fourier harmonics of the Hamiltonian, based on computer algebra, reveals three symmetries that characterize the strongest resonances responsible for the orbital chaos. These symmetries are broken only by weak resonances, leading to the existence of quasi-integrals of motion that are shown to relate to the smallest Lyapunov exponents. A principal component analysis of the orbital solutions independently confirms that the quasi-integrals are among the slowest degrees of freedom of the dynamics. Strong evidence emerges that they effectively constrain the chaotic diffusion of the orbits, playing a crucial role in the statistical stability over the Solar System lifetime., Comment: 24 pages, 11 figures. Published in Physical Review X

We consider a transport network consisting of two classes of users, namely, selfish users whose objective is to minimize their individual travel times, and cooperative users whose objective is to maximize the aggregate throughput of their class, or equivalently, to minimize the total travel time of their class. The selfish users decide the paths from their respective origins to destinations depending on their knowledge of the traffic conditions prevailing up to the beginning of their journeys, and they do not alter their paths once their journeys start. The cooperative users, on the other hand, keep making the decisions of 'the next link to take towards the destination' throughout their journeys based on their current knowledge of the traffic conditions, which evolves with time and the availability of traffic information. We develop a new framework for the problem of informed mixed-user dynamic traffic assignment (IMUDTA) considering the delay and the degree of penetration of information. The proposed framework enables us to study the route choices made by the two classes of users and the resulting network performance. Due to its non-linear formulation with high complexity, we propose a novel incremental solution method to this problem where we present a transformation of this complex IMUDTA problem into a relaxed linear programming (LP) model. We show via numerical results that (i) appropriately postponing (intentionally or unintentionally) the access to information can improve the system performance, (ii) better information penetration increases the robustness of the network performance against the information delay, and (iii) the value of information (in terms of improving network performance) is reduced when the degree of information penetration is high. The applicability of our framework is demonstrated in a large-scale network example, and analyze the computational aspects and the impact of information on network performance with varying information delay and

Apart from being chaotic, the inner planets in the Solar System constitute an open system, as they are forced by the regular long-term motion of the outer ones. No integrals of motion can bound a priori the stochastic wanderings in their high-dimensional phase space. Still, the probability of a dynamical instability is remarkably low over the next 5 billion years, a timescale thousand times longer than the Lyapunov time. The dynamical half-life of Mercury has indeed been estimated recently at 40 billion years. By means of the computer algebra system TRIP, we consider a set of dynamical models resulting from truncation of the forced secular dynamics recently proposed for the inner planets at different degrees in eccentricities and inclinations. Through ensembles of $10^3$ to $10^5$ numerical integrations spanning 5 to 100 Gyr, we find that the Hamiltonian truncated at degree 4 practically does not allow any instability over 5 Gyr. The destabilisation is mainly due to terms of degree 6. This surprising result suggests an analogy to the Fermi-Pasta-Ulam-Tsingou problem, in which tangency to Toda Hamiltonian explains the very long timescale of thermalisation, which Fermi unsuccessfully looked for., Comment: Accepted for publication in MNRAS. 9 pages, 7 figures

The long-term variations of the orbit of the Earth govern the insolation on its surface and hence its climate. The use of the astronomical signal, whose imprint has been recovered in the geological records, has revolutionized the determination of the geological time scales (e.g. Gradstein & Ogg 2020). However, the orbital variations beyond 60 Myr cannot be reliably predicted because of the chaotic dynamics of the planetary orbits in the Solar System (Laskar 1989). Taking into account this dynamical uncertainty is necessary for a complete astronomical calibration of geological records. Our work addresses this problem with a statistical analysis of 120 000 orbital solutions of the secular model of the Solar System ranging from 500 Myr to 5 Gyr. We obtain the marginal probability density functions of the fundamental secular frequencies using kernel density estimation. The uncertainty of the density estimation is also obtained here in the form of confidence intervals determined by the moving block bootstrap method. The results of the secular model are shown to be in good agreement with those of the direct integrations of a comprehensive model of the Solar System. Application of our work is illustrated on two geological data: the Newark-Hartford records and the Libsack core., Comment: Accepted for publication in A&A, 18 pages, 14 figures

The long-term variations in the orbit of the Earth govern the insolation on its surface and hence its climate. The use of the astronomical signal, whose imprint has been recovered in the geological records, has revolutionized the determination of the geological timescales. However, the orbital variations beyond 60 Myr cannot be reliably predicted because of the chaotic dynamics of the planetary orbits in the Solar System. Taking this dynamical uncertainty into account is necessary for a complete astronomical calibration of geological records. Our work addresses this problem with a statistical analysis of 120 000 orbital solutions of the secular model of the Solar System ranging from 500 Myr to 5 Gyr. We obtain the marginal probability density functions of the fundamental secular frequencies using kernel density estimation. The uncertainty of the density estimation is also obtained here in the form of confidence intervals determined by the moving block bootstrap method. The results of the secular model are shown to be in good agreement with those of the direct integrations of a comprehensive model of the Solar System. Application of our work is illustrated on two geological data sets: the Newark-Hartford records and the Libsack core., Comment: Published in A&A, 18 pages, 14 figures. Included the corrigendum for misprints in the Table 2

The dynamic traffic assignment (DTA) problem has been studied intensively in the literature. However, there is no existing linear framework to solve the user equilibrium (UE) DTA problem. In this paper, we develop a novel linear programming framework to solve the UE-DTA problem in a dynamic capacity network that exploits the linkage between the UE and system optimal (SO) solutions underpinned by a first-in-first-out (FIFO) principle. This important property enables us to develop an incremental loading method to obtain the UE solutions efficiently by solving a sequence of linear programs. The proposed solution methodology possesses several nice properties such as a predictable number of iterations before reaching the UE solution, and a linear system of equations to be solved in each of the iterations.In contrast to the related iterative methods, such as Frank-Wolfe algorithm, successive average (MSA) or projection and their extensions where the purpose of iteration is to seek the solution convergence, whereas ours is to solve a linear problem over multiple iterations but only for a single unit of demand in each iteration. Furthermore, we provide a theoretical proof that in the limit, the SO objective can be used to obtain the UE solution as the system time step goes to zero given the satisfaction of the FIFO constraint. We show via numerical examples the significant improvements in the obtained UE solutions both in terms of accuracy and computational complexity. © 2017 Elsevier Ltd.

The dynamic traffic assignment (DTA) problem has been studied intensively in the literature. However, there is no existing linear framework to solve the user equilibrium (UE) DTA problem. In this paper, we develop a novel linear programming framework to solve the UE-DTA problem in a dynamic capacity network that exploits the linkage between the UE and system optimal (SO) solutions underpinned by a first-in-first-out (FIFO) principle. This important property enables us to develop an incremental loading method to obtain the UE solutions efficiently by solving a sequence of linear programs. The proposed solution methodology possesses several nice properties such as a predictable number of iterations before reaching the UE solution, and a linear system of equations to be solved in each of the iterations.In contrast to the related iterative methods, such as Frank-Wolfe algorithm, successive average (MSA) or projection and their extensions where the purpose of iteration is to seek the solution convergence, whereas ours is to solve a linear problem over multiple iterations but only for a single unit of demand in each iteration. Furthermore, we provide a theoretical proof that in the limit, the SO objective can be used to obtain the UE solution as the system time step goes to zero given the satisfaction of the FIFO constraint. We show via numerical examples the significant improvements in the obtained UE solutions both in terms of accuracy and computational complexity. © 2017 Elsevier Ltd.