What causes period-doubling bifurcation?

In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system’s parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original.

What is meant by period-doubling?

(physics) A characteristic of the transition of a system or process from regular motion to chaos, in which the period of one of its parameters is seen to double.

What is meant by bifurcation and a bifurcation diagram?

So, again, what is a bifurcation? A bifucation is a period-doubling, a change from an N-point attractor to a 2N-point attractor, which occurs when the control parameter is changed. A Bifurcation Diagram is a visual summary of the succession of period-doubling produced as r increases.

What does the bifurcation diagram signify?

The bifurcation diagram shows the forking of the periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant.

Who discovered the period doubling cascade?

The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum.

Is bifurcation periodic?

At the bifurcation point the period of the periodic orbit has grown to infinity and it has become a homoclinic orbit. After the bifurcation there is no longer a periodic orbit. Left panel: For small parameter values, there is a saddle point at the origin and a limit cycle in the first quadrant.

What are the effect of bifurcation?

When bifurcations plate is inserted, the flow is directed into multiple flow paths. Velocity decreases as bifurcation is approached and increases after bifurcation. Combined effect of CD shape and bifurcation on flow structure and its comparison with rectangular microchannel are discussed in this section.

How do you do bifurcation analysis?

All equations that have fold bifurcation can be transformed into one of these normal forms. dt = f(x, c) Assume x∗ is an equilibrium value and c∗ is a bifurcation value. (x∗,c∗) = 0. To anaylse the equilibrium and bifurcation point we need to analyse the normal form.

What is bifurcation in dynamical systems?

In dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden “qualitative” or topological change in its behaviour.

Who discovered the bifurcation diagram?

In the fifties, Myrberg (1958, 1959, 1963) discovered infinite cascades of period doubling bifurcations. The word “bifurcation” means a sudden qualitative change in the nature of a solution, as a parameter is varied. The parameter value at which a bifurcation occurs, is called a bifurcation parameter value.

Is Fibonacci sequence chaotic?

We emphasize that the Fibonacci sequence will be found in all dynamical systems exhibiting the period-doubling route to chaos, as it is directly linked to the Feigenbaum scaling con- stant α. Thus, the convergence to φ, the “most irra- tional number” [8], occurs in concert with the onset of deter- ministic chaos.

What is a period-doubling bifurcation?

A period-doubling bifurcation corresponds to the creation or destruction of a periodic orbit with double the period of the original orbit.

What is a bifurcation diagram?

A Bifurcation Diagramis a visual summary of the succession of period-doubling produced as r increases. The next figure shows the bifurcation diagram of the logistic map, r along the x-axis. For each value of r the system is first allowed to settle down and then the successive values of x are plotted for a few hundred iterations.

What are the two types of bifurcations in dynamical systems?

Period-halving bifurcations (L) leading to order, followed by period doubling bifurcations (R) leading to chaos. 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.

What is the significance of bifurcations?

These bifurcations are especially prominent in the theory of one-dimensional, noninvertible maps, i.e., dynamical systems that are actions of the semi-group \\Z^+ on the unit interval, where infinite cascades of period-doubling bifurcations are typical, and exhibit certain universal properties.