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CHAOS THEORY

CHAOS THEORY. AJITESH VERMA. CHAOS THEORY. Dictionary meaning of chaos- state of confusion lack of any order or control. Chaos theory is a branch of mathematics which studies the behavior of certain dynamical system that maybe highly sensitive to initial system.

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CHAOS THEORY

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  1. CHAOS THEORY AJITESH VERMA

  2. CHAOS THEORY • Dictionary meaning of chaos- state of confusion lack of any order or control. • Chaos theory is a branch of mathematics which studies the behavior of certain dynamical system that maybe highly sensitive to initial system. • On the contrary of its name, it actually tells you that not all that chaos you see is due to chance or random or caused by unknown factors. • Oxymoron term coined “DETERMINISTIC RANDOMNESS” that means finding order in chaos.

  3. PROPERTIES • System must be sensitive to initial condition means that system is arbitrarily closely approximated by other points with significantly different future trajectories. This property give birth to the question that : does the flap of a butterfly’s wings in brazil set off a tornado in Texas. • System must be topologically mixing means that system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region.

  4. CHAOS THEORY INVESTIGATES Chaos theory investigates the behavior of complex non linear systems . These systems have described as: • Dynamic, complex, non linear • Chaotic, unpredictable, sensitive to initial condition • Strange attractor, fractal

  5. CHAOS THEORY CONSTRUCTS • Phase Space • Strange Attractors & Basins of attraction • Self-affinity • Fractals • Bifurcation • unpredictability • Butterfly Effect • Self-Organization

  6. PHASE SPACES • The Phase Space: a conceptualization (often pictorial or geometric) of the possible states a system might take. • Conveys that, at best, we see only a portion of “reality” at one time—that part on which we choose to focus. • Phase space allows scientists to map information from complex systems, make a picture of their moving parts, and allows insight into a dynamic system's possibilities. It is a mathematically constructed conceptual space where each dimension corresponds to one variable of the system

  7. STRANGE ATTRACTORS & BASINS OF ATTRACTION • Strange attractors are focal points for many patterns generated by dynamical, chaotic systems. • An attractor informally described as strange If it has non integer dimension if the dynamics on it are chaotic • Basins of attraction are the areas containing those patterns within their boundaries.

  8. LORENZ’S STRANGE ATTARCTOR

  9. SELF-AFFINITY • Self-affinity denotes the tendency for phenomena to evidence recurring patterns. • The affinity can be over size, time, different angles, or other ways more difficult to see or to grasp, for example, by the process that generates them or probabilistically.

  10. FRACTALS • A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity. • Fractals are related to chaos because they are complex systems that have definite properties. .

  11. BIFURCATION • Bifurcation is a scientific way to say something splits in two—branches. • If patterns bifurcate quickly enough, they can become complex very fast, leading to bifurcation cascade and chaos.

  12. UNPREDICTABILITY • Unpredictability is the inability to state with certainty the next state (or, for that matter, the previous state) of a system given knowledge of its present state.

  13. BUTTERFLY EFFECT • The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. Small variations of the initial condition of a dynamical system may produce large variations in the long term behavior of the system

  14. SELF ORGANIZATION • Self-organization is the inherent tendency for systems in a chaotic state to form a new coherent pattern. • Their innate ability to reorganize is based only on the interactions of their components

  15. WHY IS CHAOS THEORY IMPORTANT • Organisations operate in turbulent and dynamic environments. • This means uncertainty, unease and feelings of powerlessness with people in and around organisations. • This is unfortunate as it often is on the outskirts of chaos that creativity flourishes.

  16. WHY IS CHAOS THEORY IMPORTANT • ”Understanding chaos theory is important because of its significant implications for world systems design, organization design and administrative behaviour, and public policy analysis and implementation.” (Farazmand 2003:341)

  17. APPLICATION OF CHAOS THEORY • In biology, chaos is used in the identification of new evolutionary processes leading to understanding the genetic algorithm, artificial life simulations, better understanding of learning processes in systems including the brain, and studies of such previously areas as consciousness and the mind. • In physics, thermodynamics in particular, chaos is applied in the study of turbulence leading to the understanding of self-organizing systems and system states (equilibrium, near equilibrium, the edge of chaos, and chaos)

  18. FINAL WORDS • Chaos in the world brings uneasiness, but it also allows the opportunity for creativity and growth. (T. Barrett)

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