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بسم الله الرحمن الرحيم. Chaos in nonlinear dynamics Prepared by: Isra Abdullah Supervisor: Prof. Issam Al- Ashqer. Abstract.

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بسم الله الرحمن الرحيم

Chaos in nonlinear dynamics

Prepared by: Isra Abdullah

Supervisor: Prof. Issam Al-Ashqer

slide2

Abstract

The usual phenomena says : that we’re living by a systems, but part of our life has no system! that phenomena what scientist -of many fields- discovered 400 years before and called it (Chaos of nonlinear Dynamics).

In our Physical family, what does this mean? And how do we use this? Why it still under discussion to these days?

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How Study Chaos?

  • Newton:
  • Supposed to find a solution to the orbits of the solar system, according to Newton's equations if we take two object revolves around each other as the earth revolves around the sun equations of motion allow us to know the nature of their orbits and predict the exact location of each of the objects at any time from the future.
  • If we add another body to become only three objects suddenly we cannot solve the equations and predicted anything
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Henri Poincare:

  • Discovered that the predicted orbits movement is impossible in some cases regardless of how closely tracks the difference was simple in the beginning it is obvious that away and one of them exhibits a different path ,unexpectedly and suddenly here it is discovered the theory of chaos and chaotic behavior.
  • I realize that any slight disorder of no more than butterfly wings flutter leads to the formation of a storm one would not expect this so-called butterfly effect.
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Edward Lorenz (Meteorological expert):

Lorenz interest in chaos came about accidentally through his work on weather prediction. Lorenz was using a simple digital computer, to run his weather simulation. He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.

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To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 was printed as 0.506. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.

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Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modeling cannot in general make long-term weather predictions. Weather is usually predictable only about a week ahead.

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What Is Chaos?

Is the term used to describe the apparently complex behavior of what we consider to be simple, well-behaved systems chaotic behavior, when looked at causally, (looks erratic and almost random –almost like the behavior of a system) strongly influenced by outside, random (noise) or the complicated behavior of the system with many, degrees of freedom, each (doing its own thing).

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This system is essentially deterministic: that is, precise knowledge of the condition of the system at one time allows us, at least in principle to predict exactly the future behavior of that system. The problem of understanding Chaos is to reconcile this apparently conflicting notation: randomness and determinism.

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Chaos

ButterflyEffect

Deterministic + Sensitivity

to initial condition

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Dynamics

Is a branch of physics (specifically classical mechanics) concerned with the study of forces and torques and their effect on motion, as opposed to kinematics, which studies the motion of objects without reference to its causes.

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Linear System

Linear dynamics pertains to objects moving in a line and involves such quantities as force, mass/inertia, displacement (in units of distance), velocity (distance per unit time), acceleration (distance per unit of time squared) and momentum (mass times unit of velocity).

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Nonlinear System

Is a system whose time evolution equations are nonlinear, that is dynamical variable describing the properties of the system (for example, position, velocity, acceleration, pressure, etc.) appear in the equations in a nonlinear form.

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What is the Difference Between Linear and Nonlinear System?

A nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear systemfulfills these conditions.

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Nonlinear Dynamics

The study of nonlinear behavior is called nonlinear dynamics. Nonlinear dynamics is concerned with the study of systems whose time evolution equations are nonlinear

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Examples on Nonlinear Dynamics

Mechanical Vibrations: Vehicular vibration be non-linear unsteady any movement is non-linear

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Types of Nonlinear Behaviors

  • Classical chaos – the behavior of a system cannot be predicted.
  • Multi stability – alternating between two or more exclusive states.
  • Aperiodic oscillations – functions that do not repeat values after some period (otherwise known as chaotic oscillations or chaos).
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Chaos In Nonlinear Dynamics

The key element in this understanding is that notion of nonlinear , we can develop an constitutive idea of nonlinearity by characterizing the behavior of the system in terms of stimulus and response. If we give the system a kick and observe a certain response to that kick then we can ask what happens if we kick the system it was as hard.

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If the response is twice as large then the systems behavior is said to be linear (at least for the range of kick size we have used) if the response is not twice as large (it might be larger or smaller), then we say the system behavior is nonlinear.

Chaotic behavior it shows up in mechanical oscillators, electrical circuit, lasers, nonlinear optical system, chemical reaction, nerve cells, heated flutes, and many other systems.

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What is the Fuss Over Nonlinear?

The basic idea is the following: if a parameter that describes a linear system, such as the spring constant k is change, then the frequency and amplitude of the resulting osculation will change, but the qualitative nature of the behavior (simple harmonic osculate) remains the same. In fact by appropriately rescaling our length and time axis.

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When we can make the behavior for any value of k look just like that for some other value as we shall see, for nonlinear system, a small in a parameter can led to sudden and dramatic changes in both the qualitative and quantitative behavior of the system.

For one value, the behavior might be periodic, for another value only slightly different from the first; the behavior might be completely aperiodic.

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In general we need these three ingredients to determine the behavior of a system:

1-The time –evolution.

2-The values of the parameters describing the system.

3- The initial condition.

A system is said to be deterministic if knowledge of the time-evolution equations, the parameters that describe the system and the initial conditions

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Conclusion

Chaos theory is a field of study in mathematics, with applications in several disciplines including meteorology, physics, engineering, economics and biology. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions

When the present determines the future, but the approximate present does not approximately determine the future

The world which we live in is unexpected and messy

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References

  • 1-Classical Dynamics,Jorje.Jose.Eugenej.Saltan.
  • 2-Strogatz, S. H. - Nonlinear Dynamics And Chaos, Library of Congress cataloging in publication data, 2-5, 1994.
  • 3-www.wikipedia.org.
  • 4- Introduction to non-linear dynamics, Walton hall ,Milton Keynes ,MK6AA