What is meant by period doubling?
With the doubled period, it takes twice as long (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves. A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system.
What does the logistic map model?
This equation defines the rules, or dynamics, of our system: x represents the population at any given time t, and r represents the growth rate. In other words, the population level at any given time is a function of the growth rate parameter and the previous time step’s population level.
What is the logistic map used for?
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations.
Who discovered doubling?
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.
What is a Feigenbaum number?
λn+1 − λn. This constant, called Feigenbaum’s number, crops up repeatedly in self-similar figures and has an approximate value of… 4. 669201609102990671853203820466201617258185577475768632745651. 343004134330211314737138689744023948013817165984855189815134.
What is a flip bifurcation?
1. A period doubling bifurcation in a discrete dynamical system. It is a bifurcation in which the system switches to a new behavior with twice the period of the original system. That is, there exists two points such that applying the dynamics to each of the points yields the other point.
Is logistic map chaotic?
However, since almost all numbers in [0,1) are irrational, almost all initial conditions of the bit-shift map lead to the non-periodicity of chaos. Since this case of the logistic map is chaotic for almost all initial conditions, all of these finite-length cycles are unstable.
How do you plot a logistics map?
- first we fix λ and choose an initial condition.
- we then iterate the map, say 10000 times for the transient part of the dynamics.
- then we iterating say 1000 more times and plot the array of values vs. λ \lambda. λ
- we then increment λ and repeat the process.
What is Feigenbaum constant?
the Feigenbaum constant is the ratio between the diameters of successive circles on the real axis in the complex plane (see animation on the right). Bifurcation parameter is a root point of period-2n component. The ratio in the last column converges to the first Feigenbaum constant.
Where is the Feigenbaum constant found?
The Feigenbaum universal constant, \delta is discovered in 1978 and it is found to occur in many period doubling bifurcation phenomena in the celebrated logistic map and the Lorenz differential equation system with chaotic (or aperiodic) solutions.
What is a bifurcation value?
Most commonly applied to the mathematical study of 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 behavior.
What is an example of bifurcation?
1a : the point or area at which something divides into two branches or parts : the point at which bifurcating occurs Inflammation may occlude the bifurcation of the trachea.
Which is the logistic map of deterministic chaos?
Historically it has been one of the most important and paradigmatic systems during the early days of research on deterministic chaos. The logistic map is defined by the following equation: x n + 1 = λ x n ( 1 − x n) w i t h n = 0, 1, 2, 3…
What kind of equation is a logistic map?
Logistic Map Also called the logistic difference equation or the quadratic difference equation. Mathematician Paul Stein called the complexity of this iterated equation “frightening”.
Are there negative population sizes in a logistic map?
However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values (for example, if r > 4) lead to negative population sizes. This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics.
What is the bifurcation diagram for a logistic map?
Bifurcation diagram for the logistic map. The attractor for any value of the parameter r is shown on the vertical line at that r.