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Брошюра написана по материалам лекций, прочитанных автором в Летней школе «Современная математика» в Дубне в июле 2005 г. В первой части описывается возможное поведение типичных динамических систем на плоскости и двумерной сфере, т. е. рассматривается вопрос о том, куда могут накапливаться траектории динамической системы. Вторая часть брошюры рассказывает о том, что многомерный случай принципиально отличается от двумерного — анализируется пример отображения (подкова Смейла) со счётным числом периодических орбит, не исчезающих при малом возмущении. От читателя не потребуется никаких знаний из теории дифференциальных уравнений, предполагается лишь знакомство с понятием производной. Брошюра адресована старшим школьникам и студентам.
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