Titlebook: Dynamics: Numerical Explorations; Accompanying Compute Helena E. Nusse,James A. Yorke,Eric J. Kostelich Book 19941st edition Springer-Verla

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Working with Assumptions: An Overview,l a great deal about the dynamics. The . of a fixed point p of a map F may be defined as the set of points q(0) that have a backward orbit coming from p, that is, a sequence of points q(i) with i = ?1, ?2, ?, so that F(q(i?1)) = q(i) for which q(i) → p as i → ∞. The . of a periodic orbit may be defi

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Challenges of Examining Assumptionsd later in this section. Newton→s method uses the initialization point y1, marked by the small cross, as its initial point. These methods converge to a periodic point with the desired period if the initial condition is reasonably close to it, provided +1 is not an eigenvalue of the Jacobian matrix o

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https://doi.org/10.1007/978-3-030-34456-6 The orbit-following capability is for tracing their behavior as a parameter is varied. The orbit following routine runs for two dimensional maps, including for example the time-2π maps of periodically forced differential equations like the forced damped pendulum (P) and the Duffing equation (D). Th

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Alexandra S. Moore,Samantha Pintoou want to make changes, for example . a . map or differential equation, you will need two things. First, you need the source code, and secondly, a Microsoft C compiler. After compiling, you will also need the files whose names end in “.txt” and the file “y.pic”. Everyone should be able to adapt it

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Changing the Program,ou want to make changes, for example . a . map or differential equation, you will need two things. First, you need the source code, and secondly, a Microsoft C compiler. After compiling, you will also need the files whose names end in “.txt” and the file “y.pic”. Everyone should be able to adapt it to her/his needs.

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Getting the Program Running,ed with the volumes written on how to study small perturbations of a stable equilibrium point. The field of chaotic dynamics did not come alive sufficiently to affect scientists until chaotic systems could be studied on small interactive computers with good graphics. Edward Lorenz’s studies in 1963

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Dimension and Lyapunov Exponents,s which describe the average behavior of the derivative of a map along a trajectory. Let F be a differentiable map from the n-dimensional phase space to itself. For each point x in the phase space, the . (or .) of x is the sequence x, F(x), F(F(x)), F(F(F(x))), … . For each point x in the phase spac

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Straddle Trajectories,tors, but all important elements of dynamics. This chapter presents newly developed techniques for observing trajectories that lie in invariant sets which are not attractors. When trying to understand the global dynamics of a pendulum, a pendulum with friction, one must be familiar with the unstable