Find and classify all fixed points as a function of r. **Analyzing the Long-Term Behavior of a Mathematical Map**This section explores the long-term dynamics of the map given by the equation:\[ x_{n+1} = \frac{rx_n}{1 + x_n^2} \]where \( r > 0 \).**Introduction**The equation describes a recursive sequence where each term \( x_{n+1} \) is determined by the preceding term \( x_n \). The parameter \( r \) is a positive constant that influences the behavior of the sequence over time.**Objective**To understand how the sequence behaves as \( n \) approaches infinity, we investigate various values of \( r \) and initial conditions for \( x_0 \). Specifically, we'll consider:- Whether the sequence converges to a stable value.- The conditions under which the sequence remains bounded.- Any periodic or chaotic behavior exhibited by the sequence.**Discussion**The fraction \(\frac{rx_n}{1 + x_n^2}\) suggests that the values of the sequence are influenced both by scaling with \( r \) and moderated by the presence of \( x_n^2 \) in the denominator. This can lead to interesting dynamics, such as saturation effects where large values of \( x_n \) reduce the change in \( x_{n+1} \).**Conclusion**By analyzing this map for different parameters, one can gain insights into systems that exhibit similar nonlinear behaviors.

Given, the map xn+1=rxn(1+xn2), when r>0To find and classify all fixed points as a function of r: The fixed points are given by solutions of the equation X = rX1+X2 ⇒ X 1-r1+X2=0 ⇒ X=0, 1+X2 =r⇒ X=0, X =± r-1 So, X1*=0, X2*=- r-1, and X3*= r-1.∴ if r > 1, we have Three fixed points otherwise there are two. Let f(x)=rX1+X2 ⇒f'(X) =r1-X21+X22 At X =0; f'(0)=r. At X=± r-1; f'(± r-1)=r2-rr2 =2r-1. So, the fixed point 0 is unstable and the points± r-1 are stable if and only if r > 1. Now, we will analyze the existence of non-trivial periodic orbits in two cases.Case(i): if r ≤1, then |f(X)|= rX1+X2 ≤r|X|≤XSince r≤1 with equality if and only if X = 0.So, f*(X) = |f (fn-1(X))|≤|fn-1(X)|. Continuing by induction we have for all X≠0 that |fn(X)|≤|X| ∴ if r < 1 then there are no periodic orbits. However, Since Xn = fn(X) satisfies |Xn|<|Xn+1| it follows that all sequences are monotonic decresing. Consequently, limn→∞Xn= 0. Therefore, there is no choos in this regine.Case(ii):if r>1, then the nonexistence of periodic orbits is less clear. Graphically, the functions fn(X) is given by: ∴ No periodic orbit exist.