Most classical chaotic systems, such as the Lorenz system and the Chua circuit, have chaotic attractors in bounded regions. This article constructs and analyzes a different kind of non-smooth impulsive systems, which have growing numbers of attractors in the sense that the number of attractors or the scrolls of an attractor is growing as time increases, and these attractors or scrolls are not located in bounded regions. It is found that infinitely many chaotic attractors can be generated in some of such systems. As an application, both theoretical and numerical analyses of an impulsive Lorenz-like system with infinitely many attractors are demonstrated.
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