📘 dynamical systems

1. The problem is to draw a bifurcation diagram. 2. A standard example is the logistic map, given by the recurrence formula $$x_{n+1}=rx_n(1-x_n).$$

1. The problem is to generate and draw a bifurcation diagram, which is a visual representation of the possible long-term values (fixed points, periodic orbits, chaos) of a system a

1. The user requests a bifurcation diagram, which is a graphical representation used in dynamical systems to show how the qualitative nature of equilibria changes as a parameter va

1. **Restate the problem:** We want to validate the correctness of the previous derivation about $ abla_{\mathcal{M}} \mathcal{F}$ being a Lyapunov function for the GTA flow

1. **Problem statement:** We want to prove or refute that under the condition $\beta < \alpha$, the functional $\mathcal{F}$ is a Lyapunov function for the GTA flow given by $$\fra

1. **Problem Statement:** We are given the Dequan-Li system of ordinary differential equations (ODEs):

1. **State the problem:** We have a discrete-time dynamical system (DTDS) defined by the updating rule $$N_{t+1} = \frac{2}{4 + N_t^3}$$ for $t=0,1,2,\ldots$. We want to analyze th