Period-doubling bifurcation

In mathematics, a period doubling bifurcation in a discrete dynamical system is a bifurcation in which a slight change in a parameter value in the system's equations leads to the system switching to a new behavior with twice the period of the original system. With the doubled period, it takes twice as many iterations as before for the numerical values visited by the system to repeat themselves.Consider the following simple dynamics: x n + 1 = r x n ( 1 − x n ) {displaystyle x_{n+1}=rx_{n}(1-x_{n})}   where x n {displaystyle x_{n}}  , the value of x {displaystyle x}   at time n {displaystyle n}  , lies in the [ 0 , 1 ] {displaystyle }   interval and changes over time according to the parameter r ∈ ( 0 , 4 ] {displaystyle rin (0,4]}  .This classic example is a simplified version of the logistic map.A period halving bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with half the period of the original system. A series of period-halving bifurcations leads the system from chaos to order.