Exotic bifurcations in three connected populations with Allee effect

We consider three connected populations with strong Allee effect, and give a complete classification of the steady state structure of the system with respect to the Allee threshold and the dispersal rate, describing the bifurcations at each critical point where the number of steady states change. One may expect that by increasing the dispersal rate between the patches, the system would become more well-mixed hence simpler. However, we show that it is not always the case, and the number of steady states may (temporarily) increase by increasing the dispersal rate. Besides sequences of pitchfork and saddle-node bifurcations, we find triple-transcritical bifurcations and also a sun-ray shaped bifurcation where twelve steady states meet at a single point then disappear. The major tool of our investigations is a novel algorithm that decomposes the parameter space with respect to the number of steady states and find the bifurcation values using cylindrical algebraic decomposition with respect to the discriminant variety of the polynomial system.

Kinematics and Workspace Analysis of a 3PPPS Parallel Robot With U-Shaped Base

This paper presents the kinematic analysis of the 3-PPPS parallel robot with an equilateral mobile platform and a U-shape base. The proposed design and appropriate selection of parameters allow to formulate simpler direct and inverse kinematics for the manipulator under study. The parallel singularities associated with the manipulator depend only on the orientation of the end-effector, and thus depend only on the orientation of the end effector. The quaternion parameters are used to represent the aspects, i.e. the singularity free regions of the workspace. A cylindrical algebraic decomposition is used to characterize the workspace and joint space with a low number of cells. The discriminant variety is obtained to describe the boundaries of each cell. With these simplifications, the 3-PPPS parallel robot with proposed design can be claimed as the simplest 6 DOF robot, which further makes it useful for the industrial applications.

Motion Planning of a Group of Agents Using the Homotopy Approach

In this paper motion planning of a group of agents is done to move the group from an initial configuration to a final configuration through obstacles in 2-D. Also we introduce a new homotopy approach which uses potential fields to find paths in polynomial space. We use the homotopy approach for changing the group shape of the mobile agents and at the same time treat the group as a single agent by finding a bounding disc for it to plan the motion of the group through obstacles. A time varying polynomial is constructed, the roots of which represent the current positions of the mobile agents in a frame attached to the bounding disc. The real and imaginary parts of the roots of this polynomial represent the x and y coordinates of the mobile agents in this frame. This polynomial is constructed such that it avoids the discriminant variety or the set of polynomials having multiple roots. This is equivalent to saying that the mobile agents do not collide with each other at all times. The bounding disc is then used to plan the motion of the agents through obstacles such that the group avoids the obstacles at all times.

A New Method of Motion Coordination of a Group of Mobile Agents

This paper presents a new method for coordinated motion planning of multiple mobile agents. The position in 2-D of each mobile agent is mapped to a complex number and a time varying polynomial contains information regarding the current positions of all mobile agents, the degree of the polynomial being the number of mobile agents and the roots of the polynomial representing the position in 2-D of the mobile agents at a given time. This polynomial is constructed by finding a path parameterized in time from the initial to the goal polynomial which represent the initial and goal positions of the mobile agents so that the discriminant variety or the set of polynomials with multiple roots is avoided in polynomial space. This is equivalent to saying that there is no collision between any two agents in going from initial position to goal position.