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# LECTURE NOTES

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1 Introduction We shall consider continuous maps f of an interval I into itself and the discrete dynamical system xn 1 f xn n 0 1 1 We apply 1 1 to an initial condition x0 to generate its orbit the sequence x0 x1 x2 A convenient way to visualize the orbit is to draw the cobweb diagram bouncing between the graph of f and the line y x see the figures below The orbit is generated by repeated composition of functions We denote k fold composition by f k that is f 2 x f f x f 3 x f f f x etc A fixed point or equilibrium is a solution to x f x 1 2 A periodic orbit with minimal or prime period p is an orbit such that xk xk mod p and x0 x1 xp 1 are distinct Clearly f p xk xk 0 k p 1 Graphically fixed points are given by the intersection between the graph of f and the line y x An intersection between the graph of f p and the line y x give points with period not greater than p We typically study questions of asymptotic behavior such as convergence to a fixed point or periodic orbit Let us note at the outset that the dynamics are trivial if f is monotone and I is bounded In this case the orbit is a bounded monotone sequence and every initial condition converges to a fixed point We thus consider unimodal maps ie maps with a single hump The fundamental example is the family of logistic maps f x rx 1 x r 0 4 1 3 f takes a unique maximum value r 4 at the critical point x 1 2 thus the restriction to r 4 The orbit diagram of the logistic map has become an icon for chaos see Figure 1 1 This is a numerically generated picture that describes the asymptotic behavior of a typical initial condition as r is varied The complexity of the orbit diagram for this simple example is striking 1 1 Stability and bifurcations Linear maps f x ax admit the exact solution xn an x0 The origin is an attracting fixed point if a 1 and repelling if a 1 Observe that if a 0 the dynamics are oscillatory that is xk and xk 1 have opposite sign In particular there are two distinct forms of neutral stability if a 1

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