Chaotic behavior

Dynamics is chaotic the number of transient steps necessary before the steady-state chaotic behavior appears decreases. 

A(0) 0.70 Fully d( velopcd chaotic behavior appears after only about two time steps. 

Strange Attractors The motion on strange attractors exhibits many of the properties normally associated with completely random or chaotic behavior, despite being well-defined at all times and fully deterministic. More formally, a strange attractor S is an attractor (meaning that it satisfies properties (i)-(iii) above) that also displays sensitivity to initial conditions. In the case of a one-dimensional map, Xn+i = for example, this means that there exists a <5 > 0 such that for... 

Let P a a ) be the probability of transition from state a to state a. In general, the set of transition probabilities will define a system that is not describ-able by an equilibrium statistical mechanics. Instead, it might give rise to limit cycles or even chaotic behavior. Fortunately, there exists a simple condition called detailed balance such that, if satisfied, guarantees that the evolution will lead to the desired thermal equilibrium. Detailed balance requires that the average number of transitions from a to a equal the number of transitions from a to a ... 

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

Note that while a system s static complexity certainly influences its dynamical complexity, the two measures are clearly not equivalent. A system may be structurally rather simple (i.e. have a low static complexity), but have a complex dynamical behavior. (Think of the chaotic behavior of Feigenbaum s logistic equation, for example). 

A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1994)... 

Schouten, J. C., and VandenBleek, C. M., Chaotic Behavior in the Hydrodynamic Model of a Fluidized Bed Reactor, Proc. llthlnt. Conf. on Fluidized Bed Comb., 1 459(1991)... 

Many realistic systems and their models have been considered to study dynamical chaos phenomenon. Such systems as, kicked rotor and various billiard geometries allow one to treat chaotic behavior of deterministic systems successfully. 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

More recently, the problem of self-oscillation and chaotic behavior of a CSTR with a control system has been considered in others papers and books [2], [3], [8], [9], [13], [14], [20], [21], [27]. In the previously cited papers, the control strategy varies from simple PID to robust asymptotic stabilization. In these papers, the transition from self-oscillating to chaotic behavior is investigated, showing that there are different routes to chaos from period doubling to the existence of a Shilnikov homoclinic orbit [25], [26]. It is interesting to remark that in an uncontrolled CSTR with a simple irreversible reaction A B it does not appear any homoclinic orbit with a saddle point. Consequently, Melnikov method cannot be applied to corroborate the existence of chaotic dynamic [34]. 

In the present chapter, steady state, self-oscillating and chaotic behavior of an exothermic CSTR without control and with PI control is considered. The mathematical models have been explained in part one, so it is possible to use a simplified model and a more complex model taking into account the presence of inert. When the reactor works without any control system, and with a simple first order irreversible reaction, it will be shown that there are intervals of the inlet flow temperature and concentration from which a small region or lobe can appears. This lobe is not a basin of attraction or a strange attractor. It represents a zone in the parameters-plane inlet stream flow temperature-concentration where the reactor has self-oscillating behavior, without any periodic external disturbance. 

Self-oscillation and Chaotic Behavior of a CSTR Without Feedback Control... 

Equations (4) and (8) can be used to simulate the reactor at point P3 of Figure 5 in [1]. Remember that point P2 is unstable, so if the initial conditions are those corresponding to this point, it is easy to show [16], [28], the reactor evolves to points P or P3. Then, two forcing actions on the reactor are considered 1) when the coolant flow rate and the inlet stream temperature are varied as sine waves, and 2) reactor being in self-oscillating mode, an external disturbance in the coolant flow rate can drive it to chaotic behavior. 

Chaotic Behavior with Double External Periodic Disturbance... 

It is well known that a nonlinear system with an external periodic disturbance can reach chaotic dynamics. In a CSTR, it has been shown that the variation of the coolant temperature, from a basic self-oscillation state makes the reactor to change from periodic behavior to chaotic one [17]. On the other hand, in [22], it has been shown that it is possible to reach chaotic behavior from an external sine wave disturbance of the coolant flow rate. Note that a periodic disturbance can appear, for instance, when the parameters of the PID controller which manipulates the coolant flow rate are being tuned by using the Ziegler-Nichols rules. The chaotic behavior is difficult to obtain from normal... 

While the conditions 1,2 can be verified approximately by simulation, proving the condition 3 is very difficult. Note that in many studies of chaotic behavior of a CSTR, only the conditions 1,2 are verified, which does not imply chaotic d3mamics, from a rigorous point of view. Nevertheless, the fulfillment of conditions 1,2, can be enough to assure the long time chaotic behavior i.e. that the chaotic motion is not transitory. From the global bifurcations and catastrophe theory other chaotic behavior can be considered throughout the disappearance of a saddle-node fixed point [10], [19], [26]. 

The simulation result (Figure 4) shows that when two initial conditions are very close, after a dimensionless time of 40 units the concentration of reactant A and the reactor temperature are completely different. This means that the system has a chaotic behavior and their d3mamical states diverge from each other very quickly, i.e. the system has high sensitivity to initial conditions. This separation increases with time and the exponential divergence of adjacent phase points has a very important consequence for the chaotic attractor, i.e. 

Exercise 1. From the values of Table 1 and Eq.(lO), write a computer program using a fourth order Runge-Kutta or fifth order Runge-Kutta-Fehlberg method and reproduce Figures 2, 3, 4, 5. In order to check that the chaotic behavior has been reached, it is necessary to run the program with two initial conditions very close, for example ... 

Fig. 6. Power spectrnm of reactant concentration A for chaotic behavior. The peak is located at the distnrbance angular frequency.

Another interesting aspect of the self-oscillating behavior is the following one. If the values of xo,yo) are inside the lobe, an external periodic disturbance of the coolant flow rate can drive the reactor to chaotic behavior. 

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