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This Concept Map, created with IHMC CmapTools, has information related to: a2. Chaos, Forecasts of long-term behavior are meaningless. The reasons are sensitivity to initial conditions and the impossibility of measuring a variable to infinite accuracy. ???? Short-term predictions can be relatively accurate., The characteristics of chaos are: ???? It happens in feedback systems in which past events affect today’s events, and today’s events affect the future., Increases in the Control Parameter, λc, Beyond 3 drive A Non-Linear, Dynamical, Dissipative (open) System to a State far from Equilibrium, Random or Stochastic ???? Random can be defined as events that are unpredictable in the absence of underlying rules., Truly Chaotic the goal is the a5. Identification of Chaos, a5. Identification of Chaos in order to Identify Chaotic Systems which might be modified, a5. Identification of Chaos ???? Unpredictability emerges in the form of chaos., 1. Chaos occurs only in deterministic, non-linear, dynamical systems. 2.Chaos demonstrates complex, unsystematic motion 3. Chaos is deterministic, follows simple rules. 4. Values at one time are not proportional to values at a previous time. 5. Long-term predictions are valueless 6 Can develop with just one variable (parameter). 7. The pattern is entirely self-generated. 8. There is no influence from sampling or measurement error ???? Chaos is Difficult to identify in real world data unless 1. Systems have no more than three variables; Data sets are very big; 2. There is unrealistically high accuracy in the measurements; 3.The number of relative variables needed to describe such systems is many times smaller than the number of pieces; and 4. the Data have negligible amounts of noise (unexplained variability in the data), The characteristics of chaos are: ???? Chaos results from deterministic, non-linear systems., The characteristics of chaos are: ???? The motion or pattern for the most part looks disorganized and erratic, although sustained. In fact, it can usually pass all the statistical tests for randomness., These systems are composed of many pieces acting collectively. ???? m1. Complexity, Increases in the Control Parameter, λc, Beyond 3 Highly Irregular Systems, Complexity emerges from processes that occur over and over again in an iterative fashion ???? m1. Complexity, Highly Irregular Systems Complexity emerges from processes that occur over and over again in an iterative fashion, Truly Chaotic ???? Chaos is not chance or randomness nor confusion and disorder, but is supremely organized, dictated by rules, and even capable of being brought under control., Highly Irregular Systems with no underlying rules Random or Stochastic, Chaos is defined as events that are also unpredictable, but which have an underlying set of deterministic rules. some overlap Truly Chaotic, A Non-Linear, Dynamical, Dissipative (open) System to a State far from Equilibrium ???? Non-linear means that output is not directly proportional to input; or that change in one variable does not predict a proportional change or reaction in related variables., For given conditions or control parameters, it is entirely self-generated. In other words, changes in other (i.e. external) variables or parameters are not necessary. ???? Any particular value of x can lead to chaos as long as the control parameter is within appropriate range., Chaos results from an increase in a control factor (parameter) to a value high enough that chaotic, disorderly behavior sets in. ???? As a control parameter increases systematically, an initially non-chaotic system follows one of a select few typical scenarios, called routes, to chaos.