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Mathematical introduction to dynamical systems
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accordance with equ appertaining applied approximation Arnol'd tongues basins of attraction bifurcation diagram calculation centre manifold chaotic behaviour characteristics circle map colour plate control parameter convection coordinates corresponding curve deduce denoted dependence determine differential equations Duffing equation dynamical systems eigenvalues emerge equations of motion equilibrium evolution example expressed Feigenbaum fixed point flow Fourier fractal frequency function Henon homoclinic Hopf bifurcation initial conditions integral interval invariant Julia sets laminar phases limit cycle linear linearised logistic map long-term behaviour Lorenz attractor Lorenz system Lyapunov exponents neighbourhood normal form observe obtain one-dimensional orbit oscillations period doublings periodic motion perturbation phase portraits phase space pitchfork bifurcation Poincare map Poincare section possesses quasi-periodic motion recognise respectively saddle-node bifurcation scaling self-similarity sequence solution stability behaviour strange attractor structure system equ temperature temporal theorem theory torus trajectories transcritical bifurcation transformation turbulence two-dimensional unstable manifold variables vector winding number xr(t zero