Arch-Shaped Equilibrium Solutions in Social Force Model

Social force model is one of the well-known approaches that can successfully simulate pedestrians’ movements realistically. However, it is not suitable to simulate high-density crowd movement realistically due to the model having only three basic crowd characteristics which are goal, attraction, and repulsion. Therefore, it does not satisfy the high-density crowd condition which is complex yet unique, due to its capacity, density, and various demographic backgrounds of the agents. Thus, this research proposes a model that improves the social force model by introducing four new characteristics which are gender, walking speed, intention outlook, and grouping to make simulations more realistic. Besides, the high-density crowd introduces irregular behaviours in the crowd flow, which is stopping motion within the crowd. To handle these scenarios, another model has been proposed that controls each agent with two different states: walking and stopping. Furthermore, the stopping behaviour was categorized into a slow stop and sudden stop. Both of these proposed models were integrated to form a high-density crowd simulation framework. The framework has been validated by using the comparison method and fundamental diagram method. Based on the simulation of 45,000 agents, it shows that the proposed framework has a more accurate average walking speed (0.36 m/s) compared to the conventional social force model (0.61 m/s). Both of these results are compared to the real-world data which is 0.3267 m/s. The findings of this research will contribute to the simulation activities of pedestrians in a highly dense population.

It is well known that a model of any electrical circuit can be translated into a system of ordinary differential equations. Usually, the simulation is performed in the time-domain using classical discrete approaches convenient for both linear and non-linear systems. An other method consists in the quantization on the variable magnitude which is based on discrete events. The discrete event system specification formalism is adapted to simulate any electrical circuit described with a system of ordinary differential equations using the system state quantification. In this paper, a solution to make more efficient the magnitude quantification for a system of linear ordinary differential equations is given. The proposed technique is used to simulate a simple three phase coupled power systems and it is compared to other solutions given by the explicit integration using formal language and the time-domain discretization using fixed-step fifth order Runge-Kutta integration method.

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form q ˙ i = ∂ H ∂ p i , p ˙ i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

3 - LINEAR SYSTEMS

Using the Lie symmetry approach, we have examined herein the system of partial differential equations corresponding to the Einstein–Maxwell equations for a static axially symmetric spacetime. The method used reduces the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations is all optimal subgroups. The system of ordinary differential equations is further solved in general to obtain exact solutions. Several new physically important families of exact solutions are derived.

Walking habits can affect the self-organizing movement in pedestrian flow. In China, pedestrians prefer to walk along the right-hand side in the collision-avoidance process, and the same is true for the left-hand preference that is followed in several countries. Through experiments with pedestrian flow, we find that the relative position between pedestrians can affect their moving preferences. We propose a kind of collision-avoidance force based on the social force model, which considers the predictions of potential conflict and the relative position between pedestrians. In the simulation, we use the improved model to explore the effect of moving preference on the collision-avoidance process and self-organizing pedestrian movement. We conclude that the improved model can bring the simulation closer to reality and that moving preference is conducive to the self-adjustment of counterflow.

Simulating pedestrian flow through narrow exits

Using the Lie symmetry approach, the authors have examined exact and numerical solutions of coupled short pulse equation with time-dependent coefficients. The method reduces the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations for all optimal subgroups. The system of ordinary differential equations is further studied in general to obtain exact and numerical solutions. Several new physically important families of exact and numerical solutions are derived.

Human motion models are finding an increasing number of novel applications in many different fields, such as building design, computer graphics and robot motion planning. The Social Force Model is one of the most popular alternatives to describe the motion of pedestrians. By resorting to a physical analogy, individuals are assimilated to point-wise particles subject to social forces which drive their dynamics. Such a model implicitly assumes that humans move isotropically. On the contrary, empirical evidence shows that people do have a preferred direction of motion, walking forward most of the time. Lateral motions are observed only in specific circumstances, such as when navigating in overcrowded environments or avoiding unexpected obstacles. In this paper, the Headed Social Force Model is introduced in order to improve the realism of the trajectories generated by the classical Social Force Model. The key feature of the proposed approach is the inclusion of the pedestrians’ heading into the dynamic model used to describe the motion of each individual. The force and torque representing the model inputs are computed as suitable functions of the force terms resulting from the traditional Social Force Model. Moreover, a new force contribution is introduced in order to model the behavior of people walking together as a single group. The proposed model features high versatility, being able to reproduce both the unicycle-like trajectories typical of people moving in open spaces and the point-wise motion patterns occurring in high density scenarios. Extensive numerical simulations show an increased regularity of the resulting trajectories and confirm a general improvement of the model realism.

71 - Integrable Combinations

In the paper was investigated the a priori estimate and the existence of periodic solutions of a fixed period for a system of second-order ordinary differential equations with the main quasi-homogeneous non-linearity. It is proved that an a priori estimate of periodic solutions takes place if the corresponding unperturbed system does not have non-zero bounded solutions. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, a criterion for the existence of periodic solutions under any perturbation from a given class is formulated and proven. The results obtained differ from earlier results in that the set of zeros of the main non-linear part is not taken into account.

An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations

The material presented below amounts roughly to the so-called fundamental theorems of Lie and their implications concerning Lie algebras, Lie subgroups and subalgebras. By straightforward and fairly elementary steps, we shall extend the concept of Lie group to include groups admitting coordinate systems in which the functions defining ab, the group product of elements a, b, possess continuous first order derivatives in the coordinates of a and satisfy a Lipschitz condition in those of b. That such groups are equivalent to classical (i.e. analytical) Lie groups was announced in 1936 by van Kampen, although apparently van Kampen published nothing by way of proof beyond a certain decisive uniqueness theorem concerning systems of ordinary differential equations. (This is contained in our theorem (12.1); the proof given below is essentially that of van Kampen [3]). It is conceivable that the condition relative to coordinates of b could be weakened or dispensed with entirely. For, the only part the condition in question plays in our development is to make it certain that the inverse of a certain mapping-the mapping into canonical coordinates-is single-valued. In this connection mention should be made of the paper [1J of Garrett Birkhoff in which it is shown that the existence of continuous first order derivatives of ab with respect to the coordinates of both the a's and the b's is sufficient for defining a Lie group. We shall consider only groups with a finite number of real parameters (whereas the groups considered by Birkhoff are not necessarily finite dimensional). This makes it possible to use the standard existence theorems for systems of ordinary real differential equations. We make no use whatever of the theory of partial differential equations. We maintain throughout a purely local point of view in the sense that we consider only what happens in the neighborhood of the identity. Historical notes. The proof of the uniqueness of square roots (8.5) is due to Claude Chevalley (unpublished). The proof given below of Lie's theorem that a linear system of vector fields which is closed under commutation defines a Lie group is, we believe, due to van der Waerden [4]. We have supplied a number of preliminary lemmas which make the proof rigorously applicable to the case in which the vector fields are only assumed to possess continuous first order derivatives. The theorem in ?17 that every local subgroup of a Lie group is a Lie group is due essentially to E. Cartan [2]. 481

We consider mathematical models of infectious diseases built by G. I. Marchuk in his well-known book on immunology. These models are in the form of systems of ordinary delay differential equations. We add a distributed control in one of the equations describing the dynamics of the antibody concentration rate. Distributed control looks here naturally since the change of this concentration rather depends on the corresponding average value of the difference of the current and normal antibody concentrations on the time interval than on their difference at the point t only. Choosing this control in a corresponding form, we propose some ideas of the stabilization in the cases, where other methods do not work. The main idea is to reduce the stability analysis of a given integro-differential system of the order n, to one of the auxiliary systems of the order \(n+m\), where m is a natural number, which is “easy” for this analysis in a corresponding sense. Results for these auxiliary systems allow us to make conclusions for the given integro-differential system of the order n. We concentrate our attempts in the analysis of the distributed control in an integral form. An idea of reducing integro-differential systems to systems of ordinary differential equations is developed. We present results about the exponential stability of stationary points of integro-differential systems using the method based on the presentation of solution with the help of the Cauchy matrix. Various properties of integro-differential systems are studied by this way. Methods of the general theory of functional differential equations developed by N. V. Azbelev and his followers are used. One of them is the Azbelev W-transform. We propose ideas allowing to achieve faster convergence to stationary point using a distributed control. We obtain estimates of solutions using estimates of the Cauchy matrices.