Chaotic systems

For a dynamical system to be classified as chaotic, it must have the following properties:[6]

Assign z to z2 minus the conjugate of z, plus the original value of the pixel for each pixel, then count how many cycles it took when the absolute value of z exceeds two; inversion (borders are inner set), so that you can see that it threatens to fail that third condition, even if it meets condition two.

1. it must be sensitive to initial conditions,

2. it must be topologically mixing, and

3. its periodic orbits must be dense.

Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.

Complex adaptive systems

Complex adaptive systems (CAS) are special cases of complex systems. They are complex in that they are diverse and made up of multiple interconnected elements and adaptive in that they have the capacity to change and learn from experience. Examples of complex adaptive systems include the stock market, social insect and ant colonies, the biosphere and the ecosystem, the brain and the immune system, the cell and the developing embryo, manufacturing businesses and any human social group-based endeavor in a cultural and social system such as political parties or communities. This includes some large-scale online systems, such as collaborative tagging or social bookmarking systems.

Nonlinear system

The behavior of nonlinear systems is not subject to the principle of superposition while that of Linear systems is subject to superposition. Thus, a nonlinear system is one whose behavior can't be expressed as a sum of the behaviors of its parts (or of their multiples).

For a dynamical system to be classified as chaotic, it must have the following properties:[6]

Assign z to z2 minus the conjugate of z, plus the original value of the pixel for each pixel, then count how many cycles it took when the absolute value of z exceeds two; inversion (borders are inner set), so that you can see that it threatens to fail that third condition, even if it meets condition two.

1. it must be sensitive to initial conditions,

2. it must be topologically mixing, and

3. its periodic orbits must be dense.

Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.

Complex adaptive systems

Complex adaptive systems (CAS) are special cases of complex systems. They are complex in that they are diverse and made up of multiple interconnected elements and adaptive in that they have the capacity to change and learn from experience. Examples of complex adaptive systems include the stock market, social insect and ant colonies, the biosphere and the ecosystem, the brain and the immune system, the cell and the developing embryo, manufacturing businesses and any human social group-based endeavor in a cultural and social system such as political parties or communities. This includes some large-scale online systems, such as collaborative tagging or social bookmarking systems.

Nonlinear system

The behavior of nonlinear systems is not subject to the principle of superposition while that of Linear systems is subject to superposition. Thus, a nonlinear system is one whose behavior can't be expressed as a sum of the behaviors of its parts (or of their multiples).

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