Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

A tutorial on modeling and analysis of dynamic social networks. Part I

Most of the distributed protocols for multi-agent consensus assume that the agents are mutually cooperative and “trustful,” and so the couplings among the agents bring the values of their states closer. Opinion dynamics in social groups, however, require beyond these conventional models due to ubiquitous competition and distrust between some pairs of agents, which are usually characterized by repulsive couplings and may lead to clustering of the opinions. A simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by C. Altafini, who examined first-order consensus algorithms over static signed graphs. This protocol establishes modulus consensus, where the opinions become the same in modulus but may differ in signs. In this paper, we extend the modulus consensus model to the case where the network topology is an arbitrary time-varying signed graph and prove reaching modulus consensus under mild sufficient conditions of uniform connectivity of the graph. For cut-balanced graphs, not only sufficient, but also necessary conditions for modulus consensus are given.

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Opinion Dynamics in Social Networks With Hostile Camps: Consensus vs. Polarization

Most of the distributed protocols for multi-agent consensus assume that the agents are mutually cooperative and "trustful," and so the couplings among the agents bring the values of their states closer. Opinion dynamics in social groups, however, require beyond these conventional models due to ubiquitous competition and distrust between some pairs of agents, which are usually characterized by repulsive couplings and may lead to clustering of the opinions. A simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by C. Altafini, who examined first-order consensus algorithms over static signed graphs. This protocol establishes modulus consensus, where the opinions become the same in modulus but may differ in signs. In this paper, we extend the modulus consensus model to the case where the network topology is an arbitrary time-varying signed graph and prove reaching modulus consensus under mild sufficient conditions of uniform connectivity of the graph. For cut-balanced graphs, not only sufficient, but also necessary conditions for modulus consensus are given.

Opinion Dynamics in Social Networks with Hostile Camps: Consensus vs. Polarization

The coordination and control of social systems is the foundational problem of sociology. The discipline was established in Europe in the aftermath of the American and French Revolutions. With the dismantling of the hierarchical controls of European aristocratic systems, the examination of alternative mechanisms of coordination and control became a preoccupation. The Industrial Revolution (c. 1760-1840), which was occurring at the same time, reinforced this preoccupation by decoupling the power of purse and the power of positions of authority. While the hierarchies of church and state retained coercive power, the exercise of such power was increasingly contested. The moral compass of society, its general welfare, and its capacity to adapt to changing circumstances appeared to be aggregated properties of the unconstrained opinions and behaviors of a large collectivity of individuals. The definition of the problem of coordination and control, which emerged in Europe in the new discipline of sociology, still resonates and guides current sociological work. The problem definition [1], broadly stated, is this: if value is placed on nonhierarchical mechanisms of social control, then what mechanisms and structures (consistent with this value) allow a coordination and control of social systems?

The Problem of Social Control and Coordination of Complex Systems in Sociology: A Look at the Community Cleavage Problem

For communities of agents which are not necessarily cooperating, distributed processes of opinion forming are naturally represented by signed graphs, with positive edges representing friendly and cooperative interactions and negative edges the corresponding antagonistic counterpart. Unlike for nonnegative graphs, the outcome of a dynamical system evolving on a signed graph is not obvious and it is in general difficult to characterize, even when the dynamics are linear. In this paper, we identify a significant class of signed graphs for which the linear dynamics are however predictable and show many analogies with positive dynamical systems. These cases correspond to adjacency matrices that are eventually positive, for which the Perron–Frobenius property still holds and implies the existence of an invariant cone contained inside the positive orthant. As examples of applications, we determine cases in which it is possible to anticipate or impose unanimity of opinion in decision/voting processes even in presence of stubborn agents, and show how it is possible to extend the PageRank algorithm to include negative links.

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Predictable Dynamics of Opinion Forming for Networks With Antagonistic Interactions

Generation of high-quality drive paths is a significant issue for automated wheeled vehicles. To achieve this aim for a truck and trailer vehicle, the paper proposes the use of a parameterized curve primitive, the η4-spline. Using this spline, generation and shaping of smooth feasible paths is made possible as well as the transfer between arbitrary dynamic configurations of the articulated vehicle. The η4-spline is a ninth-order polynomial curve that can interpolate given Cartesian points with associated arbitrary unit tangent vector, curvature, and first and second derivatives of curvature. It depends on a set of eight (eta) parameters that can be freely chosen to modify the path shape without changing the interpolations conditions at the path endpoints. Completeness, minimality, and symmetry of the η4-spline are established. An example on a parking maneuver of the articulated vehicle is presented and the pertinent optimal path planning is also discussed.

Path Generation Using ${\mbi \eta}^4$ -Splines for a Truck and Trailer Vehicle

This paper proposes a multi-optimization approach to the autonomous parking of car-like vehicles. It uses a polynomial curve primitive — the η3-spline — to build up intrinsically feasible path maneuvers over which to minimize with a weighted sum method the total length of parking paths and the moduli of the maximum path curvature and curvature derivative. The approach takes into account the mandatory constraint of obstacle avoidance and maximal steering angle and the constraint of maximal curvature derivative which is a selectable limit to ensure the desired smoothness of the parking paths. Simulation results are included for a garage parking example.

Multi-optimization of η 3 -splines for autonomous parking

The paper poses the problem of minimum-time velocity planning subject to a jerk amplitude constraint and to arbitrary velocity/acceleration boundary conditions. This problem which is relevant in the field of autonomous robotic navigation and also for inertial one-dimensional mechatronics systems is dealt with an algebraic approach based on Pontryagin’s Maximum Principle. The exposed complete solution shows how this time-optimal planning can be reduced to the problem of determining the positive real roots of a quartic equation. An algorithm that is suitable for real-time applications is then presented. The paper includes detailed examples also highlighting the special cases of this planning problem.

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Algebraic solution to minimum-time velocity planning

This paper proposes a method for minimum-time velocity planning with velocity, acceleration and jerk constraints and generic initial and final boundary conditions for the velocity and the acceleration. This minimum-time planning problem is relevant in the context of robotic autonomous navigation, where the iterative steering supervisor periodically replans the future mobile robot motion starting from current position, velocity and acceleration conditions. The problem is faced through discretization and its solution is based on a sequence of linear programming feasibility checks, depending on motion constraints and boundary conditions.

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Gabriele Lini

Papers

Predictable Dynamics of Opinion Forming for Networks With Antagonistic Interactions

Path Generation Using ${\mbi \eta}^4$ -Splines for a Truck and Trailer Vehicle

Multi-optimization of η 3 -splines for autonomous parking

Algebraic solution to minimum-time velocity planning

Minimum-time constrained velocity planning