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This paper gives an up-to-date account of chaos and fractals, in a popular pictorial style for the general scientific reader. A brief historical account covers the development of the subject from Newton’s laws of motion to the astronomy of Poincaré and the weather forecasting of Lorenz. Emphasis is given to the important underlying concepts, embracing the fractal properties of coastlines and the logistics of population dynamics. A wide variety of applications include: NASA’s discovery and use of zero-fuel chaotic “superhighways” between the planets; erratic chaotic solutions generated by Euler’s method in mathematics; atomic force microscopy; spontaneous pattern formation in chemical and biological systems; impact mechanics in offshore engineering and the chatter of cutting tools; controlling chaotic heartbeats. Reference is made to a number of interactive simulations and movies accessible on the web.

The production of oxygen through phytoplankton photosynthesis is a crucial phenomenon in the dynamics of marine ecosystems. A generic oxygen-phytoplankton interaction model is considered to comprehend its underlying mechanism. This paper investigates the discrete-time dynamics of oxygen and phytoplankton in aquatic ecosystems, incorporating factors that cause phytoplankton mortality due to external influences. We explore the conditions for the local stability of steady states concerning the oxygen content in dissolved water and phytoplankton density. The analysis reveals that the model undergoes a co-dimension one bifurcation, encompassing flip and Neimark–Sacker bifurcations, utilizing the center manifold theorem and bifurcation theory. To manage the chaos resulting from the Neimark–Sacker bifurcation, we apply the OGY feedback control method and a hybrid control methodology. Finally, we present numerical simulations to validate the theoretical discussion.