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In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. More importantly, the vector controller is one-dimensional less than the system. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Chen chaotic system and an integer-order T chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order hyperchaotic system, and the synchronization between a fractional-order hyperchaotic system based on Chen's system and an integer-order Lorenz chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.

This paper describes an individual-based model for simulating the swarming behavior of prey in the presence of predators. Predators and prey are represented as agents that interact through radial force laws. The prey form swarms through attractive and repulsive forces. The predators interact with the prey through an anti-newtonian force, which is a nonconservative force that acts in the same direction for both agents. Several options for forces between predators are explored. The resulting equations are solved numerically and the dynamics are described in the context of the swarm's ability to realistically avoid the predators. The goal is to reproduce swarm behavior that has been observed in nature with the simplest possible model.

A particularly simple and mathematically elegant example of chaos in a three-dimensional flow is examined in detail. It has the property of cyclic symmetry with respect to interchange of the three orthogonal axes, a single bifurcation parameter that governs the damping and the attractor dimension over most of the range 2 to 3 (as well as 0 and 1) and whose limiting value b = 0 gives Hamiltonian chaos, three-dimensional deterministic fractional Brownian motion, and an interesting symbolic dynamic.

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