A Series of Saddle-Node Bifurcation and Chaotic Behavior of a Family of a Semi-Triangular Maps

This paper studies the bifurcations in dynamics of a family of semi-triangular maps } : ) sin( ) ( { IR x x x S S       . We will prove that this family has a series of Saddle-node bifurcations and a period doubling bifurcation. Also, we show that for some value of the parameter the functions  S will be chaotic.


Introduction
The term "bifurcation" refers to significant changes in the set of fixed or periodic points or other sets of dynamics interest. In fact, in dynamical systems, the object of bifurcation theorem is to study the changes that maps undergo as parameters changes. There are several types of bifurcations like saddlenode bifurcation ,period doubling bifurcation pitch fork bifurcation, and others.
Our goal in this paper ,is to study how and when the periodic points of the family of maps } : change, i.e. the bifurcation that this family undergoes.
We will prove that our family has a series of saddle node bifurcations which is route to chaos. Also, we will show that this family has a period doubling bifurcation when the parameter meets the value 327295 Finally, we show that this family has a chaotic behavior on IR when the parameter is 2   .

Definitions
Let f be any function .Then, 1. A point x is called fixed point of the function f if x x f = ) ( , [1] .

A point x is called periodic point if
such that x x f n = ) ( .And we say that x of period n .Note that the fixed point is a periodic point of period 1, [1] . is defined, then the common value of ( ) x  is denoted by  , and is the Lyapunov exponent of f , [2] . 9. A function f is chaotic if it satisfies, at least, one of the following : i. f has a positive Lyapunov exponent at each point in its domain that is not eventually periodic. Or ii. f has a sensitive dependence on initial conditions on its domain , [2] .

Properties and the fixed points of the family 
S : which is a maximum for  S and 913 .
which is a minimum for  S and all of them lie in the interval [0, 2] .
The following proposition gives the fixed points of the function in the family } :

3.2: Proposition
Let ,then the fixed points for this family of Let p x be a fixed point for this family, thus ( )

3.3: Remark
The fixed point 0 is the unique fixed point for 1   , (see Figure 1), below: Hence, the function has infinite number of fixed points.
To study the types of these points : Taking the derivative of ( ) ,…….. are not hyperbolic fixed points , (see Figure 2).
is fixed point , hence the number of fixed points is doubled (see Figure3).
x be a fixed point for  S .Then is also implies that Hence, (1)and (2) give the general form of the fixed point of x be attracting fixed points .This implies that By the same theorem for 1   ,two fixed points were born at each interval ,one is attracting and the other is repelling ;this is exactly a saddle-node bifurcation .
The following theorem studies the period doubling bifurcation of the family S :

4.2: Theorem
Let be a family of maps ,then this family has period doubling bifurcation at 327295 . 1   .

Proof
Our earlier experiments showed that the value 327295 .

4.3:Theorem
Functions of the family S are sensitive dependence on initial condition in the , the theorem can be divided into two parts: x y such that  is small positive number and By taking the absolute value of both sides the following is obtained Multiplying both sides by  the following is obtained By taking the absolute value of both sides the following is obtained , and that proves  S is sensitive dependence on initial condition in the interval 2 From (1) and (2) it can be obtained the function  S is sensitive dependence on initial condition in the interval , the theorem can be divided into two parts: Multiplying both sides by  the following is obtained By taking the absolute value of both sides the following is obtained , and that proves  S is sensitive dependence on initial condition in the interval Multiplying both sides by  the following is obtained By taking the absolute value of both sides the following is obtained , and that proves  S is sensitive dependence on initial condition in the interval 2 From (1) and (2) Figure 9).  We will show that the maping  S route to chaos by a series of saddle-node bifurcations .In fact, this is a typical route to chaos . We will show that 2  S has the same " behavior " inside certain box .We conclude that 2  S has a saddle node bifurcation in this box .
Experimentally ,we choose the interval ( ) .Consider the Figures 10,11,12 and 13.  Continuing this process, we have a series of saddle-node bifurcation for  S as  increases . Therefore, the bifurcation diagram for  S must be as in Figure 14 . The above discussion shows ,experimentally ,that this family encountered with chaotic dynamics for certain values of  namely 2 =  .This is called a saddle node bifurcation route to chaos.