9.5: Exercise Problems

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9.1. Generalize the reasoning of Sec. 1 to an arbitrary 1D map \(q_{n+1}=f\left(q_{n}\right)\), with a function \(f(q)\) differentiable at all points of interest. In particular, derive the condition of stability of an \(N\)-point limit cycle \(q^{(1)} \rightarrow q^{(2)} \rightarrow \ldots \rightarrow q^{(N)} \rightarrow q^{(1)} \ldots\)

9.2. Use the stability condition, derived in the previous problem, to analyze the possibility of the deterministic chaos in the so-called tent map, with \[f(q)=\left\{\begin{array}{ll} r q, & \text { for } 0 \leq q \leq 1 / 2, \\ r(1-q), & \text { for } 1 / 2 \leq q \leq 1, \end{array} \quad \text { with } 0 \leq r \leq 2\right.\] 9.3. A dynamic system is described by the following system of differential equations: \[\begin{aligned} &\dot{q}_{1}=-q_{1}+a_{1} q_{2}^{3}, \\ &\dot{q}_{2}=a_{2} q_{2}-a_{3} q_{2}^{3}+a_{4} q_{2}\left(1-q_{1}^{2}\right) . \end{aligned}\] Can it exhibit chaos at some set of constant parameters \(a_{1}-a_{4}\) ?

9.4. A periodic function of time has been added to the right-hand side of the first equation of the system considered in the previous problem. Is deterministic chaos possible now?