We notice that all the results obtained concerning the fixed points coordinates and stability, i. For instance, if , besides the washout, there is only one nontrivial steady state corresponding to the low value of. This steady state is stable because. The other value of verifies , thus i.

Here, the washout is a saddle point. Finally, if , nontrivial solutions are not possible. Hence, we get only one solution i. All of our findings for this case study agree perfectly with their counterparts obtained by Bequette 2. Here, the instructor can assign a term project leading to the investigation the effect of either adopting other growth rate expressions or using two or more than nutrients.

Some students can also study the effect of the newly formed products e. Indeed, as we will be seen below, this model exhibits a rich variety of behaviors, encompassing simple oscillations, period doubling, and chaos; when the bifurcation parameter, , is varied between 0. The steps for this hypothetical reaction system are as follows: Reaction Rate expression.

Here, P is a chemical precursor with constant concentration, and D , the final product. A , B , and C are intermediate chemical species. If the reactions occur in an isothermal closed system, then the material balances for the three intermediate species write: 16 17 After introducing the new dimensionless concentrations of A , B , and C , and the dimensionless time , we get the dimensionless governing equations 19 20 The four parameters and depend on the rates of the individual reactions and the concentration of the precursor as follows:.

We now proceed to the investigation of the dynamic behavior of this autocatalator model for various values of the bifurcation parameter by fixed values of the rest of parameters. The time series i. The bifurcation diagram a remerging Feigenbaum tree given in Figure 25 illustrates these results.

Finally, it is possible to draw the power spectrums for various values of the bifurcation parameter,. The power spectra for and are depicted in Figure 26 for comparison, showing increasing frequency for larger values. On the other side, chaotic behavior is observed in the power versus frequency diagram for Figure Let us draw some conclusion after careful observation of these figures.

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First, the signal is periodic when period one , thus the power spectrum is constituted of a main frequency, i. Secondly, when period 2 , we observe an important phenomena called period doubling. Indeed, the main frequency is at but there is a peak at. Since the new frequency is divided by 2 and , we called this phenomena period doubling.

Finally, Figure 27 is a typical power spectrum of a chaotic signal. Indeed, it is a continuous power spectrum in opposition to that of a white noise i. The principal component spectrum of a chaotic signal is a straight line with a negative slope while that of the white noise is a horizontal line. Indeed, this criteria yields valuable information about the sign of the real part of the eigenvalues.

Hence, students can easily investigate the stability of the corresponding steady states without actually computing these eigenvalues. Indeed, a positive maximum Lyapunov exponent is an indication of chaotic behavior. Furthermore, if the estimate of the maximum Lyapunov exponent is approximately equal to zero then all other Lyapunov exponents are less than zero and we recover periodic behavior. In order to illustrate several typical calculations in the field of nonlinear dynamics, we have selected four case studies.

Indeed, a blend of theoretical derivations e. In addition, due to its historical importance, the Lorenz system 13 appears prominently among the material covered in class. Moreover, a few moments are usually allocated to play in the classroom some of the videos, available online e. Grades assigned for these projects are based on several criteria such as the level scientific difficulty and the novelty of the tackled problem, the accuracy of the obtained results, and the neatness and clarity of the written report. Final scores obtained by the graduate students, who enrolled in this course, were very satisfactory indicating that the four outcomes listed in the introduction section have been achieved.

Several calculations shown in the present paper can be adapted to any other nonlinear dynamics problem without substantial efforts. Indeed, only the governing equations will need to be tailored to the new problem that reader want to solve. His research interests include the applications of computers in Chemical Engineering. His research interests mainly focus on process thermodynamics, absorption refrigeration and solar cooling, modeling and simulation of unit operations and processes.

He has supervised 35 master degree and 20 PhD theses. Professor Bellagi has written over publications. Binous and Professor B. Volume 27 , Issue 1. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.

If the address matches an existing account you will receive an email with instructions to retrieve your username. Housam Binous Corresponding Author E-mail address: binoushousam yahoo. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Abstract In the present paper, we present the solution of four problems drawn from the chemical and biochemical engineering field of study.

Figure 1 Open in figure viewer PowerPoint.

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Figure 2 Open in figure viewer PowerPoint. Figure 3 Open in figure viewer PowerPoint. Figure 4 Open in figure viewer PowerPoint. Figure 5 Open in figure viewer PowerPoint. Orange dots corresponds to values of b where Hopf bifurcation occurs. Figure 6 Open in figure viewer PowerPoint. Figure 7 Open in figure viewer PowerPoint. Reaction Rate expression.

Figure 8 Open in figure viewer PowerPoint. Figure 9 Open in figure viewer PowerPoint. Figure 10 Open in figure viewer PowerPoint. Figure 11 Open in figure viewer PowerPoint. Figure 12 Open in figure viewer PowerPoint. Figure 13 Open in figure viewer PowerPoint. In such case, the biochemical reactor is governed by two coupled equations 2 , 7 : Figure 14 Open in figure viewer PowerPoint.

Only a single stable steady state is possible: washout case. Figure 15 Open in figure viewer PowerPoint. Two steady states are possible: the washout unstable—green dot and the nontrivial steady state stable—blue dot. For example, we can identify a specific value of p , say p 0 , for which the values of F are approximately equal for different system sizes. From Fig.

### Beschreibung

The number of network configurations for each fixed value of p is The spatiotemporal behaviors of the oscillator system under random perturbations to the network structure can be assessed by direct numerical integration of the original system Eq. The state of the system can then be described by the function h x , t. Taking into account random perturbations to the links, we obtain the time evolution function of h x i , t associated with the order parameter Z i of the oscillator at x i see Methods as Figures 5 a1—e1 show, for five values of p , the spatiotemporal behavior of R x i , t , the module of the order parameter Z i obtained by solving Eq.

The corresponding results obtained from direct simulation of Eq. As p is increased from zero, the period of breathing becomes longer and the value of R x i , t is reduced. From the scaling behavior of F p e. However, Figs.

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Contour plots representing spatiotemporal evolution of R x , t for five values of the link removal probability p from left to right: 0, 0. The five patterns in the top row are obtained by the PDE in the continuum limit the PDE approach , and the corresponding patterns in the bottom row are from direct numerical calculations of the original dynamical system. The trajectories of h t in its own complex plane for some representative oscillators are shown in Fig.

For the two top panels, the oscillator is selected from a region of high coherence, where the trajectory of h t moves about the unit circle in the complex plane. For the two bottom panels, the oscillator is from a subset among which the coherence is much weaker. We see that the corresponding trajectories are somewhat random.

These results suggest that, for a nonlocally-coupled array of identical oscillators, the self-organized mode characterized by the coexistence of spatially high-coherent and weak-coherent domains, as well as temporally breathing behavior stand out as a general type of spatiotemporal pattern, with or without random structural perturbations. The chimera state is effectively a particular case of the spatiotemporal breathing pattern where the oscillators in the high-coherent domain happen to be synchronized or phase-locked.

Panels a1 and b1 correspond to a representative oscillator selected from the region of high coherence, while panels a2 and b2 are for an oscillator from the region of weak coherence. Motivated by the growing recent interest in chimera state in non-locally coupled network of identical oscillators, we address the fundamental issue of robustness of chimera state against random structural perturbations to the network. Using direct numerical simulation and a self-consistency equation, we find that chimera state can persist in a probabilistic sense: the probability of the occurrence of chimera state can be finite even when a large fraction of the links in the networks are removed.

The probability to observe chimera state exhibits critical behaviors with the variation in the link-removal probability. Utilizing direct numerical computation and an analytic approach based on the PDE model derived in the continuum limit, we study the spatiotemporal pattern of the system of non-locally coupled identical oscillators. Especially, by varying the link-removal probability, we uncover a rather striking phenomenon: regardless of whether chimera state can emerge, the system exhibits a general breathing pattern in its spatiotemporal evolution.

Associated with such a pattern, the oscillators in the system can be qualitatively classified into two groups: one group of high coherence and another of weak coherence. The particular breathing pattern stipulates that this division holds even for large link-removal probability where chimera state is ruled out. The implication is that the breathing pattern in the spatiotemporal evolution of the system is general and robust, and chimera state is a particular phenomenon where the oscillators in the highly coherent group happen to be phase synchronized.

Our work thus provides deeper insights into the dynamical origin of chimera state, a phenomenon of continuous interest and subject to intense recent investigation 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , By introducing a complex order parameter depending on space and time as given in Eq. To calculate the contribution from the phase-locked oscillators to the order parameter, we note that any fixed point of Eq.

Hereafter, for convenience, we use subscripts to denote the spatial position of the oscillators, e. The normalization constant is chosen such that. To calculate the contribution to the order parameter from the drifting oscillators, we replace by its statistical average. Using this approximation in Eq. Note that the summation is exactly the same as that found earlier in Eq. In general, there are two equations [corresponding to the real and imaginary parts of Eq. Since Eq. The resulting three equations can then be solved numerically in a self-consistent manner to yield the three quantities of interest.

The order parameter is Equation 22 can be rewritten as Substituting Eq. Inserting Eq. Equation 30 then becomes Equations 29 and 31 can be combined to yield the following equation in terms of h x , t and the order parameter: From Eq. Kuramoto, Y. Strogatz, S. From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators. Physica D , 1 Restrepo, J. Onset of synchronization in large networks of coupled oscillators. E 71 , Guan, S. Transition to global synchronization in clustered networks.

E 77 , Shima, S. Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. E 69 , Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom.

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## DynamIC Seminars – Dynamical Systems at Imperial College London (DynamIC)

Complex Syst. Abrams, D. Chimera states for coupled oscillators. Solvable model for chimera states of coupled oscillators. Sethia, G. Clustered chimera states in delay-coupled oscillator systems. Sheeba, J. Globally clustered chimera states in delay-coupled populations.

E 79 , R Laing, C. Chimera states in heterogeneous networks. Chaos 19 , The dynamics of chimera states in heterogeneous Kuramoto networks. Physica D , Martens, E. Solvable model of spiral wave chimeras. Omel'chenko, O. Chimera states as chaotic spatiotemporal patterns.

E 81 , R Omelchenko, I. Loss of coherence in dynamical networks: spatial chaos and chimera states. Wolfrum, M. Chimera states are chaotic transients. E 84 , R Chimeras in random non-complete networks of phase oscillators.

## Academic Bibliography

Buch kartoniert. Dynamical systems of course originated from ordinary differential equations. Bitte reservieren lassen. Unsere zentrale Filialhotline: - 30 75 75 Bewerten Empfehlen Merkzettel. Expositions in Dynamical Systems. Sprache: Englisch. Springer Berlin Heidelberg September - kartoniert - Seiten. Dynamical systems have involved remarkably in recent years. A wealth of new phenomena, new ideas and new techniques are proving to be of considerable interest to scientists in rather different fields.

It is not surprising that thousands of publications on the theory itself and on its various applications are appearing. Topics are advanced, while detailed exposition of ideas, restriction to typical result- rather than the most general ones - and, last but not least, lucid proofs help to gain the utmost degree of clarity. Elements of the Theory of Minimal States. Second Perturbation: Nondegeneracy of Homoclinic Orbits. Application to Mather Sets. Special Classes of Diffeomorphisms.

Basic Notions and Results. The Reduction Procedure. Asymptotic Periodicity of Constrictive Marcov Operators. Weakly Almost Periodic Operators. Asymptotic Periodicity of Power Bounded Operators.