#### Clone of Clone of The Logistic Map

##### Kadar Anwar

The L**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0)
- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of OVERSHOOT GROWTH INTO TURBULENCE

##### Brian Lee

The existing global capitalistic growth paradigm is totally flawed

The chaotic turbulence is the result of the concept of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks

See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)

Environment Economics Finance Mathematics Physics Biology Health Fractals Chaos TURBULENCE Engineering Navier Stokes Science Demographics Population Growth Strategy Weather

- 3 years 2 months ago

#### Clone of Clone of The Logistic Map

##### Rachel Berry

The L**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0)
- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of FORCED GROWTH INTO TURBULENCE

##### Brayan Triminio

**FORCED GROWTH GROWTH GOES INTO TURBULENT CHAOTIC DESTRUCTION**

**BEWARE pushing increased growth blows the system!**

**(governments are trying to push growth on already unstable systems !)**

The existing global capitalistic growth paradigm is totally flawed

The chaotic turbulence is the result of the concept and flawed strategy of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks

See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)

Environment Economics Finance Mathematics Physics Biology Health Fractals Chaos TURBULENCE Engineering Navier Stokes Science Demographics Population Growth BIFURCATIONS MTBF Strategy Weather

- 2 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Erin Donovan

The L**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0)
- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Bahar Sayed

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Sarah Miller

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Victoria Wilson

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Chris Kent

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Sarah Di Croce

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Brian J Elliott

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Troy Gamboa

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Lindsay Proulx

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of The Logistic Map

##### V Krishnan Ramanujan

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 4 years 1 week ago

#### Clone of POPULATION LOGISTIC MAP (WITH FEEDBACK)

##### Pieter van der Ploeg

the maximum population is set to be one million, and the growth rate constant mu = 3. Nj: is the “number of items” in our current generation.

Delta Nj: is the “change in number of items” as we go from the present generation into the next generation. This is just the number of items born minus the number of items who have died.

mu: is the growth or birth rate parameter, similar to that in the exponential growth and decay model. However, as we extend our model it will no longer be the actual growth rate, but rather just a constant that tends to control the actual growth rate without being directly proportional to it.

F(Nj) = mu(1‐Nj/Nmax): is our model for the effective “growth rate”, a rate that decreases as the number of items approaches the maximum allowed by external factors such as food supply, disease or predation. (You can think of mu as the growth or birth rate in the absence of population pressure from other items.) We write this rate as F(Nj), which is a mathematical way of saying F is affected by the number of items, i.e., “F is a function of Nj”. It combines both growth and all the various environmental constraints on growth into a single function. This is a good approach to modeling; start with something that works (exponential growth) and then modify it incrementally, while still incorporating the working model.

Nj+1 = Nj + Delta Nj : This is a mathematical way to say, “The new number of items equals the old number of items plus the change in number of items”.

Nj/Nmax: is what fraction a population has reached of the maximum "carrying capacity" allowed by the external environment. We use this fraction to change the overall growth rate of the population. In the real world, as well as in our model, it is possible for a population to be greater than the maximum population (which is usually an average of many years), at least for a short period of time. This means that we can expect fluctuations in which Nj/Nmax is greater than 1.

This equation is a form of what is known as the logistic map or equation. It is a map because it "maps'' the population in one year into the population of the next year. It is "logistic'' in the military sense of supplying a population with its needs. It a nonlinear equation because it contains a term proportional to Nj^2 and not just Nj. The logistic map equation is also an example of discrete mathematics. It is discrete because the time variable j assumes just integer values, and consequently the variables Nj+1 and Nj do not change continuously into each other, as would a function N(t). In addition to the variables Nj and j, the equation also contains the two parameters mu, the growth rate, and Nmax, the maximum population. You can think of these as "constants'' whose values are determined from external sources and remain fixed as one year of items gets mapped into the next year. However, as part of viewing the computer as a laboratory in which to experiment, and as part of the scientific process, you should vary the parameters in order to explore how the model reacts to changes in them.

Environment MATHS Mathematics Chaos Fractals BIFURCATION Model Economics Finance TURBULENCE Population Growth DECAY STABILITY SUSTAINABLE Engineering Science Demographics Strategy

- 2 years 4 months ago

#### Clone of POPULATION LOGISTIC MAP (WITH FEEDBACK)

##### Andriy Samilyak

the maximum population is set to be one million, and the growth rate constant mu = 3. Nj: is the “number of items” in our current generation.

Delta Nj: is the “change in number of items” as we go from the present generation into the next generation. This is just the number of items born minus the number of items who have died.

mu: is the growth or birth rate parameter, similar to that in the exponential growth and decay model. However, as we extend our model it will no longer be the actual growth rate, but rather just a constant that tends to control the actual growth rate without being directly proportional to it.

F(Nj) = mu(1‐Nj/Nmax): is our model for the effective “growth rate”, a rate that decreases as the number of items approaches the maximum allowed by external factors such as food supply, disease or predation. (You can think of mu as the growth or birth rate in the absence of population pressure from other items.) We write this rate as F(Nj), which is a mathematical way of saying F is affected by the number of items, i.e., “F is a function of Nj”. It combines both growth and all the various environmental constraints on growth into a single function. This is a good approach to modeling; start with something that works (exponential growth) and then modify it incrementally, while still incorporating the working model.

Nj+1 = Nj + Delta Nj : This is a mathematical way to say, “The new number of items equals the old number of items plus the change in number of items”.

Nj/Nmax: is what fraction a population has reached of the maximum "carrying capacity" allowed by the external environment. We use this fraction to change the overall growth rate of the population. In the real world, as well as in our model, it is possible for a population to be greater than the maximum population (which is usually an average of many years), at least for a short period of time. This means that we can expect fluctuations in which Nj/Nmax is greater than 1.

This equation is a form of what is known as the logistic map or equation. It is a map because it "maps'' the population in one year into the population of the next year. It is "logistic'' in the military sense of supplying a population with its needs. It a nonlinear equation because it contains a term proportional to Nj^2 and not just Nj. The logistic map equation is also an example of discrete mathematics. It is discrete because the time variable j assumes just integer values, and consequently the variables Nj+1 and Nj do not change continuously into each other, as would a function N(t). In addition to the variables Nj and j, the equation also contains the two parameters mu, the growth rate, and Nmax, the maximum population. You can think of these as "constants'' whose values are determined from external sources and remain fixed as one year of items gets mapped into the next year. However, as part of viewing the computer as a laboratory in which to experiment, and as part of the scientific process, you should vary the parameters in order to explore how the model reacts to changes in them.

Environment MATHS Mathematics Chaos Fractals BIFURCATION Model Economics Finance TURBULENCE Population Growth DECAY STABILITY SUSTAINABLE Engineering Science Demographics Strategy

- 2 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Julian Hermes

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Aaron Bouchard

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### PJ Volz

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of Clone of The Logistic Map

##### Bryan Potter

**ogistic 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. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

where:

- r is a positive number, and represents a combined rate for reproduction and starvation.

To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

- 3 years 4 months ago

#### Clone of FORCED GROWTH INTO TURBULENCE

##### Thibaud Métral

**FORCED GROWTH GROWTH GOES INTO TURBULENT CHAOTIC DESTRUCTION**

**BEWARE pushing increased growth blows the system!**

**(governments are trying to push growth on already unstable systems !)**

The existing global capitalistic growth paradigm is totally flawed

The chaotic turbulence is the result of the concept and flawed strategy of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks

See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)

Environment Economics Finance Mathematics Physics Biology Health Fractals Chaos TURBULENCE Engineering Navier Stokes Science Demographics Population Growth BIFURCATIONS MTBF Strategy Weather

- 2 years 2 months ago

#### Clone of POPULATION LOGISTIC MAP (WITH FEEDBACK)

##### Roman Knaus

the maximum population is set to be one million, and the growth rate constant mu = 3. Nj: is the “number of items” in our current generation.

Delta Nj: is the “change in number of items” as we go from the present generation into the next generation. This is just the number of items born minus the number of items who have died.

mu: is the growth or birth rate parameter, similar to that in the exponential growth and decay model. However, as we extend our model it will no longer be the actual growth rate, but rather just a constant that tends to control the actual growth rate without being directly proportional to it.

F(Nj) = mu(1‐Nj/Nmax): is our model for the effective “growth rate”, a rate that decreases as the number of items approaches the maximum allowed by external factors such as food supply, disease or predation. (You can think of mu as the growth or birth rate in the absence of population pressure from other items.) We write this rate as F(Nj), which is a mathematical way of saying F is affected by the number of items, i.e., “F is a function of Nj”. It combines both growth and all the various environmental constraints on growth into a single function. This is a good approach to modeling; start with something that works (exponential growth) and then modify it incrementally, while still incorporating the working model.

Nj+1 = Nj + Delta Nj : This is a mathematical way to say, “The new number of items equals the old number of items plus the change in number of items”.

Nj/Nmax: is what fraction a population has reached of the maximum "carrying capacity" allowed by the external environment. We use this fraction to change the overall growth rate of the population. In the real world, as well as in our model, it is possible for a population to be greater than the maximum population (which is usually an average of many years), at least for a short period of time. This means that we can expect fluctuations in which Nj/Nmax is greater than 1.

This equation is a form of what is known as the logistic map or equation. It is a map because it "maps'' the population in one year into the population of the next year. It is "logistic'' in the military sense of supplying a population with its needs. It a nonlinear equation because it contains a term proportional to Nj^2 and not just Nj. The logistic map equation is also an example of discrete mathematics. It is discrete because the time variable j assumes just integer values, and consequently the variables Nj+1 and Nj do not change continuously into each other, as would a function N(t). In addition to the variables Nj and j, the equation also contains the two parameters mu, the growth rate, and Nmax, the maximum population. You can think of these as "constants'' whose values are determined from external sources and remain fixed as one year of items gets mapped into the next year. However, as part of viewing the computer as a laboratory in which to experiment, and as part of the scientific process, you should vary the parameters in order to explore how the model reacts to changes in them.

Environment MATHS Mathematics Chaos Fractals BIFURCATION Model Economics Finance TURBULENCE Population Growth DECAY STABILITY SUSTAINABLE Engineering Science Demographics Strategy

- 1 year 9 months ago

#### Clone of THE BROKEN LINK BETWEEN SUPPLY AND DEMAND CREATES CHAOTIC TURBULENCE (+controls)

##### Marco

The existing global capitalistic growth paradigm is totally flawed

Growth in supply and productivity is a summation of variables as is demand ... when the link between them is broken by catastrophic failure in a component the creation of unpredictable chaotic turbulence puts the controls ito a situation that will never return the system to its initial conditions as it is STIC system (Lorenz)

The chaotic turbulence is the result of the concept of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks

See Guy Lakeman Bubble Theory for more details on keeping systems within finite working containers (villages communities)

Environment Economics Finance Mathematics Physics Biology Health Fractals Chaos TURBULENCE Engineering Navier Stokes Supply Demand Strategy

- 2 years 10 months ago

#### Clone of Lissajous curve

##### Tsogbadrakh Banzragch

- {\displaystyle x=A\sin(at+\delta ),\quad y=B\sin(bt),}

which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous in 1857.

- 1 year 1 month ago