Period-doubling bifurcation
Encyclopedia
In mathematics
, a period doubling bifurcation in a discrete dynamical system
is a bifurcation
in which the system switches to a new behavior with twice the period of the original system. Period doubling bifurcations can also occur in continuous
dynamical systems, namely when a new limit cycle emerges from an existing limit cycle,
and the period of the new limit cycle is twice that of the old one.
Consider the following logistical map for a modified Phillips curve
:
where
is the actual inflation
,
is the expected inflation, u is the level of unemployment, and 
is the money supply
growth rate. Keeping
and varying 
, the system undergoes period doubling bifurcations, and after a point becomes chaotic
, as illustrated in the bifurcation diagram on the right.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a period doubling bifurcation in a discrete dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
is a bifurcation
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
in which the system switches to a new behavior with twice the period of the original system. Period doubling bifurcations can also occur in continuous
dynamical systems, namely when a new limit cycle emerges from an existing limit cycle,
and the period of the new limit cycle is twice that of the old one.
Examples
- Logistic mapLogistic mapThe logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
- Logistical map for a modified Phillips curvePhillips curveIn economics, the Phillips curve is a historical inverse relationship between the rate of unemployment and the rate of inflation in an economy. Stated simply, the lower the unemployment in an economy, the higher the rate of inflation...
Consider the following logistical map for a modified Phillips curve
Phillips curve
In economics, the Phillips curve is a historical inverse relationship between the rate of unemployment and the rate of inflation in an economy. Stated simply, the lower the unemployment in an economy, the higher the rate of inflation...
:
where
is the actual inflationInflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
,
is the expected inflation, u is the level of unemployment, and is the money supplyMoney supply
In economics, the money supply or money stock, is the total amount of money available in an economy at a specific time. There are several ways to define "money," but standard measures usually include currency in circulation and demand deposits .Money supply data are recorded and published, usually...
growth rate. Keeping
and varying , the system undergoes period doubling bifurcations, and after a point becomes chaoticChaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
, as illustrated in the bifurcation diagram on the right.
- Complex quadratic mapComplex quadratic polynomialA complex quadratic polynomial is a quadratic polynomial whose coefficients are complex numbers.-Forms:When the quadratic polynomial has only one variable , one can distinguish its 4 main forms:...
Period-halving bifurcation
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
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
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
to order.
External links
- The Flip (Period Doubling) Bifurcation in Discrete Time, Dynamic Processes