Submitted by Joelson Dayvison Veloso Hermes (joelson.hermes@ifsuldeminas.edu.br) on 2018-04-23T01:02:53Z No. of bitstreams: 1 Joelson_VFinal.pdf: 1544708 bytes, checksum: 9c429a972747727fd4f21efd9b4bdf5a (MD5) / Rejected by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br), reason: - Falta a capa; - Falta a ficha catalográfica, que deve ser solicitada pelo site da biblioteca e colocada após a página de rosto (na versão online) e no verso da página de rosto (na versão impressa); - Folha de aprovação está sem as assinaturas e o resultado (verificar com a seção de pós-graduação) on 2018-04-24T12:48:21Z (GMT) / Submitted by Joelson Dayvison Veloso Hermes (joelson.hermes@ifsuldeminas.edu.br) on 2018-05-07T17:02:26Z No. of bitstreams: 1 Joelson_Vonline.pdf: 2019330 bytes, checksum: 0b929a27a61310386659770f124bcd52 (MD5) / Approved for entry into archive by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br) on 2018-05-07T17:53:12Z (GMT) No. of bitstreams: 1 hermes_jdv_me_rcla.pdf: 2014132 bytes, checksum: 9b8ffcfc6b30d128e5b4dd50f756a88c (MD5) / Made available in DSpace on 2018-05-07T17:53:12Z (GMT). No. of bitstreams: 1 hermes_jdv_me_rcla.pdf: 2014132 bytes, checksum: 9b8ffcfc6b30d128e5b4dd50f756a88c (MD5) Previous issue date: 2018-02-20 / Neste projeto aplicamos o formalismo de escala com o objetivo de explorar a evolução em direção ao equilíbrio perto de uma bifurcação tangente no mapa logístico. No ponto de bifurcação a órbita segue o caminho descrito por uma função homogênea com expoentes críticos bem definidos. Perto da bifurcação, a convergência para o equilíbrio é exponencial, cujo tempo de relaxação é marcado por uma lei de potência. Para obtermos os expoentes utilizamos dois procedimentos distintos: (1) o primeiro, fenomenológico, envolvendo hipóteses de escala, com o qual determinamos uma lei de escala entre os 3 expoentes críticos; (2) o segundo transforma uma equação de diferenças em uma equação diferencial, sendo resolvida com condições iniciais convenientes. Os resultados analíticos confirmam bem os resultados encontrados numericamente. / In this project we apply the scaling formalism to understand and describe the evolution towards the equilibrium at and near at a tangent bifurcation into logistic map. At the bifurcation the convergence to the steady state is described by a homogeneous function with well de ned critical exponents. Near the bifurcation, the evolution to the equilibrium is described by an exponential function whose relaxation time is described by a power law. We use two di erent approaches to obtain the critical exponents: (1) a phenomenological investigation based on three scaling hypotheses leading to a scaling law relating three critical exponents and; (2) a procedure transforming the di erence equation into a di erential equation which is solved under appropriate conditions. The numerical results give support for the theoretical approach.

The diploma thesis is focused on the use of heuristic and metaheuristic methods to stabilization and controlling the selected systems distinguished by the deterministic chaos behavior. There are discussed parameterization of chosen optimization methods, which are the genetic algorithm, simulated annealing and pattern search. The thesis also introduced the suitable controlling methods and the definition of the objective function. In the theoretical part of the thesis there is a brief introduction to the deterministic chaos theory. The next chapters describes the most common and deployed methods in~the~control theory, especially OGY and Pyragas methods. The practical part of the thesis is divided into two chapters. The first one describes the~stabilization of the artifical chaotic systems with the time delayed Pyragas method - TDAS and its modification ETDAS. The second chapter shows the real chaotic system control. The Duffing oscillator system was chosen to serve this purpose.

This thesis examines the utility of mathematical complexity measures for the analysis of stratospheric dynamics. Through theoretical considerations and tests with artificial data sets, e.g., the iteration of the logistic map, suitable parameters are determined for the application of the statistical entropy measures sample entropy (SE) and Rényi entropy (RE) to methane (a long-lived stratospheric tracer) data from simulations of the SOCOL chemistry-climate model. The SE is shown to be useful for quantifying the variability of recurring patterns in a time series and is able to identify tropical patterns similar to those reported by previous studies of the ``tropical pipe'' region. However, the SE is found to be unsuitable for use in polar regions, due to the non-stationarity of the methane data at extra-tropical latitudes. It is concluded that the SE cannot be used to analyse climate complexity on a global scale. The focus is turned to the RE, which is a complexity measure of probability distribution functions (PDFs). Using the second order RE and a normalisation factor, zonal PDFs of ten consecutive days of methane data are created with a Bayesian optimal binning technique. From these, the RE is calculated for every day (moving 10-day window). The results indicate that the RE is a promising tool for identifying stratospheric mixing barriers. In Southern Hemisphere winter and early spring, RE produces patterns similar to those found in other studies of stratospheric mixing. High values of RE are found to be indicative of the strong fluctuations in tracer distributions associated with relatively unmixed air in general, and with gradients in the vicinity of mixing barriers, in particular. Lower values suggest more thoroughly mixed air masses. The analysis is extended to eleven years of model data. Realistic inter-annual variability of some of the RE structures is observed, particularly in the Southern Hemisphere. By calculating a climatological mean of the RE for this period, additional mixing patterns are identified in the Northern Hemisphere. The validity of the RE analysis and its interpretation is underlined by showing that qualitatively similar patterns can be seen when using observational satellite data of a different tracer. Compared to previous techniques, the RE has the advantage that it requires significantly less computational effort, as it can be used to derive dynamical information from model or measurement tracer data without relying on any additional input such as wind fields. The results presented in this thesis strongly suggest that the RE is a useful new metric for analysing stratospheric mixing and its variability from climate model data. Furthermore, it is shown that the RE measure is very robust with respect to data gaps, which makes it ideal for application to observations. Hence, using the RE for comparing observations of tracer distributions with those from model simulations potentially presents a novel approach for analysing mixing in the stratosphere.

This thesis examines the utility of mathematical complexity measures for the analysis of stratospheric dynamics. Through theoretical considerations and tests with artificial data sets, e.g., the iteration of the logistic map, suitable parameters are determined for the application of the statistical entropy measures sample entropy (SE) and Rényi entropy (RE) to methane (a long-lived stratospheric tracer) data from simulations of the SOCOL chemistry-climate model. The SE is shown to be useful for quantifying the variability of recurring patterns in a time series and is able to identify tropical patterns similar to those reported by previous studies of the ``tropical pipe'' region. However, the SE is found to be unsuitable for use in polar regions, due to the non-stationarity of the methane data at extra-tropical latitudes. It is concluded that the SE cannot be used to analyse climate complexity on a global scale. The focus is turned to the RE, which is a complexity measure of probability distribution functions (PDFs). Using the second order RE and a normalisation factor, zonal PDFs of ten consecutive days of methane data are created with a Bayesian optimal binning technique. From these, the RE is calculated for every day (moving 10-day window). The results indicate that the RE is a promising tool for identifying stratospheric mixing barriers. In Southern Hemisphere winter and early spring, RE produces patterns similar to those found in other studies of stratospheric mixing. High values of RE are found to be indicative of the strong fluctuations in tracer distributions associated with relatively unmixed air in general, and with gradients in the vicinity of mixing barriers, in particular. Lower values suggest more thoroughly mixed air masses. The analysis is extended to eleven years of model data. Realistic inter-annual variability of some of the RE structures is observed, particularly in the Southern Hemisphere. By calculating a climatological mean of the RE for this period, additional mixing patterns are identified in the Northern Hemisphere. The validity of the RE analysis and its interpretation is underlined by showing that qualitatively similar patterns can be seen when using observational satellite data of a different tracer. Compared to previous techniques, the RE has the advantage that it requires significantly less computational effort, as it can be used to derive dynamical information from model or measurement tracer data without relying on any additional input such as wind fields. The results presented in this thesis strongly suggest that the RE is a useful new metric for analysing stratospheric mixing and its variability from climate model data. Furthermore, it is shown that the RE measure is very robust with respect to data gaps, which makes it ideal for application to observations. Hence, using the RE for comparing observations of tracer distributions with those from model simulations potentially presents a novel approach for analysing mixing in the stratosphere.

Made available in DSpace on 2016-12-12T20:15:50Z (GMT). No. of bitstreams: 1 Antonio J Fidelis.pdf: 14922669 bytes, checksum: 6b0a7e53941481a93d4afce451520db9 (MD5) Previous issue date: 2013-03-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we present a study of the logistic map xn+1 = rxn(1 xn) based on the supertracks, a set of continuous functions of the fixed parameter r recursively generated from the map s critical point Xmax = 1/2. This functions determine some iriternal and externa! boundaries of the orbit diagram of the map and provide information about the dynamics of the orbits. The iritersections of these functions can be periodic points or Misiurewicz points. We analyze the dynamics of the orbit in a particular Misiurewicz point, originated from the first coilision between the unstable fixed point and the chaotic attractor. As inedited results, we present algebraically the Lyapunov exponent and the invariant measure for this fixed parameter s value r. Algebraical orbits from the birth and the death of the famous period 3 window are presented as inedited result too. / Neste trabalho apresentamos um estudo do mapa logístico xn + 1 = rxn(1 xn) através do formalismo de supertracks, um conjunto de funções contínuas do parâmetro fixo r geradas recursivamente a partir do ponto crítico do mapa Xmax = 1/2. Essas funções determinam algumas fronteiras internas e externas no diagrama de bifurcação do mapa e fornecem informações sobre a dinâmica das órbitas. As interseções dessas funções podem ser pontos periódicos ou pontos de Misiurewicz. Analisamos a dinâmica da órbita num ponto de Misiurewicz em particular, originado da primeira colisão do ponto fixo instável com o atrator caótico. Como resultados inéditos, apresentamos de forma algébrica o expoente de Lyapunov e a medida invariante para este valor do parâmetro r. As órhitas algébricas do nascimento e da morte da famosa janela de período 3 são também ineditamente apresentados.

[EN] In this thesis, we consider two problems: we first explore the application of visibility graphs for describing the orbits of a discrete dynamical system that is governed by a fractional version of the logistic equation. We also study how to use this type of graphs to study response time series from the perspective of psychology. The preliminaries and introduction of these visibility graphs are presented in Chapter 1, where we revisit some basic facts from network science related to them. In the first part of this thesis, we analyze a phenomenon of mathematical nature. Wu and Baleanu introduced a fractional discrete dynamical system inspired by the fractional difference logistic equation. In order to study the trajectories of this model under this perspective of network science, in Chapter 2, we first review the most used fractional derivatives (Riemann-Liouville, Caputo, and Gründwald-Letnikov). Later, we show how to consider discrete fractional derivatives. Within our work, we present an alternative way of deducing the governing equation with respect to the one shown by Wu and Baleanu. We revisit the Wu-Baleanu equation in Chapter 3, focused on the visibility graphs of trajectories generated under different values of the scaling factor and the fractional exponent. We also study the existing connections between these parameters and the fitting with the degree distribution of the corresponding visibility graphs. When chaos is present, we link them with the exponent obtained when fitting the degree distribution to a power-law of the form x^(¿¿). With this approach, we provide an integrated vision of the dynamics of a family of fractional discrete dynamical systems that cannot be obtained from single Feigenbaum diagrams computed for each scaling factor and fractional exponent. We also connect the power-law exponent of the degree distribution fitting with the Shannon entropy of the visibility graphs degree distribution. In the second part, we analyze the response times of students to a binary decision task from the perspective of network science. We analyze the properties of the natural visibility graphs associated with their reaction time series. We observe that the degree distribution of these graphs usually fits a power-law distribution p(x) = x^(¿¿). We study the range in which parameter ¿ occurs and the changes of this exponent with respect to the age and gender of the students. Besides, we also study the links between the parameter ¿ and the ex-Gaussian distribution parameters that best fits each subject's response times. Finally, we outline some conclusions and perspectives of future research in both parts in Chapter 6. / [ES] En esta tesis, hemos considerado dos problemas: primero exploramos la aplicación de los grafos de visibilidad para describir las órbitas de un sistema dinámico discreto que está gobernado por una versión fraccionaria de la ecuación logística. Además, también estudiamos cómo usar este tipo de grafos para estudiar series temporales de tiempos de respuesta desde una perspectiva psicológica. Los preliminares, así como una introducción a estos grafos de visibilidad, se presentan en el Capítulo 1, donde revisitamos algunos hechos básicos de la ciencia de redes relacionados con dichos grafos. En la primera parte de esta tesis, analizamos un fenómeno de naturaleza matemática. Wu y Baleanu introdujeron un sistema dinámico discreto fraccionario inspirado en la ecuación logística con derivadas fraccionarias. Con el propósito de estudiar las trayectorias de este modelo desde la perspectiva de la ciencia de redes, en el Capítulo 2, primero revisamos las derivadas fraccionarias más utilizadas (Riemann-Liouville, Caputo y Gründwald-Letnikov). Posteriormente, mostramos cómo considerar derivadas fraccionarias discretas. En nuestro trabajo, presentamos una forma alternativa de deducir la ecuación gobernante con respecto a la presentada por Wu y Baleanu. Revisitamos la ecuación de Wu-Baleanu en el Capítulo 3, centrado en los grafos de visibilidad de trayectorias generadas a partir de distintos valores del factor de escala y del exponente fraccionario. También estudiamos la existencia de conexiones entre estos parámetros y el ajuste de la distribución de los grados de los correspondientes grafos de visibilidad. Cuando el caos está presente, los enlazamos con el exponente obtenido al ajustar la distribución de los grados a una ley de potencias de la forma x^(¿¿). A través de este enfoque, proporcionamos una visión integrada de la dinámica de una familia de sistemas dinámicos discretos fraccionarios que no se pueden obtener a partir de diagramas de Feigenbaum individuales calculados para cada factor de escala y exponente fraccionario. Además, relacionamos el exponente de la ley de potencias del ajuste de la distribución de grados con la entropía de Shannon de la distribución de grados de los grafos de visibilidad. En la segunda parte, analizamos el tiempo de respuesta de un grupo de estudiantes que realizaron una tarea de decisión binaria desde la perspectiva de la ciencia de redes. Estudiamos las propiedades de los grafos de visibilidad natural asociados con sus correspondientes series de tiempos de respuesta. Observamos que la distribución de los grados de estos grafos normalmente sigue una distribución ley de potencias p(x) = x^(¿¿). Analizamos el rango en el cual el parámetro ¿ se mueve y los cambios de este exponente con respecto a la edad y el sexo de los estudiantes. Por otro lado, también estudiamos la relación entre el parámetro ¿ y los parámetros de la distribución ex-Gaussiana que mejor se ajusta al tiempo de respuesta de cada sujeto. Finalmente, destacamos algunas conclusiones y perspectivas de investigación futura en ambas líneas de trabajo en el Capítulo 6. / [CAT] En aquesta tesi, hem considerat dos problemes: primer explorem l'aplicació dels grafs de visibilitat per a descriure les òrbites d'un sistema dinàmic discret que està governat per una versió fraccionària de l'equació logística. A més a més, també estudiem com emprar aquest tipus de grafs per a analitzar sèries temporals de temps de resposta des d'una perspectiva psicològica. Els preliminars, així com una introducció a aquests grafs de visibilitat, es presenten al Capítol 1, on revisitem alguns fets bàsics de la ciència de xarxes relacionats amb ells. En la primera part d'aquesta tesi, analitzem un fenomen de naturalesa matemàtica. Wu i Baleanu van introduir un sistema dinàmic discret fraccionari inspirat en l'equació logística amb derivades fraccionàries. Amb el fi d'estudiar les trajectòries d'aquest model des d'una perspectiva de la ciència de xarxes, en el Capítol 2, primer revisem les derivades fraccionàries més utilitzades (Riemann-Liouville, Caputo i Gründwald-Letnikov). Posteriorment, mostrem com considerar derivades fraccionàries discretes. Al nostre treball, presentem una forma alternativa de deduir l'equació governant respecte a la presentada per Wu i Baleanu. Revisitem l'equació de Wu-Baleanu al Capítol 3, focalitzat en els grafs de visibilitat de trajectòries generades a partir de valors diferents del factor d'escala i de l'exponent fraccionari. També estudiem l'existència de connexions entre aquests paràmetres i l'ajust de la distribució dels graus dels corresponents grafs de visibilitat. Quan el caos hi és, els enllacem amb l'exponent que hem obtés en ajustar la distribució dels graus a una llei de potències de la forma x^(¿¿). Des d'aquesta perspectiva, proporcionem una visió integrada de la dinàmica d'una família de sistemes dinàmics discrets fraccionaris que no es poden obtenir a partir de diagrames de Feigenbaum individuals calculats per a cada factor d'escala i exponent fraccionari. A més a més, relacionem l'exponent de la llei de potències de l'ajust de la distribució de graus amb l'entropia de Shannon de la distribució de graus dels grafs de visibilitat. A la segona part, analitzem el temps de resposta d'un grup d'estudiants que realitzaren una tasca de decisió binària des del punt de vista de la ciència de xarxes. Estudiem les propietats dels grafs de visibilitat natural associats amb les seues corresponents sèries temporals de temps de resposta. Observem que la distribució dels graus d'aquests grafs normalment segueix una distribució llei de potències p(x) = x^(¿¿). Analitzem el rang en què el paràmetre ¿ es mou i els canvis d'aquest exponent respecte a l'edat i el sexe dels estudiants. D'altra banda, també estudiem la relació entre el paràmetre ¿ i els paràmetres de la distribució ex-Gaussiana que millor fita el temps de resposta de cada subjecte. Finalment, destaquem algunes conclusions i perspectives d'investigació futura en ambdues línies de treball en el Capítol 6. / Mira Iglesias, A. (2021). Applications of Visibility Graphs for the representation of Time Series [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/176012

The aim of this thesis is to investigate the dynamical behaviors of spatially extended systems with fluctuating time delays. In recent years, the study of spatially extended systems and systems with fluctuating delays has experienced a fast growth. In ubiquitous natural and laboratory situations, understanding the action of time-delayed signals is a crucial for understanding the dynamical behavior of these systems. Frequently, the length of the delay is found to change with time. Spatially extended systems are widely studied in many fields, such as chemistry, ecology, and biology. Self-organization, turbulence, and related nonlinear dynamic phenomena in spatially extended systems have developed into one of the most exciting topics in modern science. The first part of this thesis considers the discrete system. Diffusively coupled map lattices with a fluctuating delay are used in the study. The uncoupled local dynamics of the considered system are represented by the delayed logistic map. In particular, the influences of diffusive coupling and fluctuating delay are studied. To observe and understand the influences, the results for the considered system are compared with coupled map lattices without delay and with a constant delay as well as with the uncoupled logistic map with fluctuating delays. Identifying different patterns, determining the existence of traveling wave solutions, and specifying the fully synchronized stable state are the focus of this part of the study. The Lyapunov exponent, the master stability function, spectrum analysis, and the structure factor are used to characterize the different states and the transitions between them. The second part examines the continuous system. The delay is introduced into the reactionterm of the Fisher-KPP equation. The focus of this part of study is the time-delay-induced Turing instability in one-component reaction-diffusion systems. Turing instability has previously only been found in multiple-component reaction-diffusion systems. However, this work demonstrates with the help of the stability exponent that fluctuating delay can result in Turing instability in one-component reaction-diffusion systems as well. / Ziel der vorliegenden Arbeit ist die Untersuchung der Einflüsse der zeitlich fluktuierenden Verzögerungen in räumlich ausgedehnten diffusiven Systemen. Durch den Vergleich von Systemen mit konstanter Verzögerung bzw. Systemen ohne räumliche Kopplung erhält man ein tieferes Verständnis und eine bessere Beschreibungsweise der Dynamik des räumlich ausgedehnten diffusiven Systems mit fluktuierenden Verzögerungen. Im ersten Teil werden diskrete Systeme in Form von diffusiven Coupled Map Lattices untersucht. Als die lokale iterierte Abbildung des betrachteten Systems wird die logistische Abbildung mit Verzögerung gewählt. In diesem Teil liegt der Fokus auf Musterbildung, Existenz von Multiattraktoren und laufenden Wellen sowie der Möglichkeit der vollen Synchronisation. Masterstabilitätsfunktion, Lyapunov Exponent und Spektrumsanalyse werden benutzt, um das dynamische Verhalten zu verstehen. Im zweiten Teil betrachten wir kontinuierliche Systeme. Hier wird die Fisher-KPP Gleichung mit Verzögerungen im Reaktionsteil untersucht. In diesem Teil liegt der Fokus auf der Existenz der Turing Instabilität. Mit Hilfe von analytischen und numerischen Berechnungen wird gezeigt, dass bei fluktuierenden Verzögerungen eine Turing Instabilität auch in 1-Komponenten-Reaktions-Diffusionsgleichungen gefunden werden kann

Control theory of complex systems by utilization of artificial intelligent algorithms is relatively new science field and it can be used in many areas of technical practise. Best known algorithms to solved similar tasks are genetic algorithm, differential evolution, HC12 Nelder-Mead method, fuzzy logic and grammatical evolution. Complex solution is presented at selected examples from mathematical nonlinear systems to examples of anthems design and stabilization of deterministic chaos. The goal of this thesis is present examples of implementation and utilization of artificial algorithms by multi-objective optimization. To achieve optimal results is used designed software solution by multi-platform application, which used Matlab and Java interfaces. The software solution integrate every algorithms of this thesis to complex solution and it extends possible application of those approaches to real systems and practical world.

Control theory of complex systems by utilization of artificial intelligent algorithms is relatively new science field and it can be used in many areas of technical practise. Best known algorithms to solved similar tasks are genetic algorithm, differential evolution, HC12 Nelder-Mead method, fuzzy logic and grammatical evolution. Complex solution is presented at selected examples from mathematical nonlinear systems to examples of anthems design and stabilization of deterministic chaos. The goal of this thesis is present examples of implementation and utilization of artificial algorithms by multi-objective optimization. To achieve optimal results is used designed software solution by multi-platform application, which used Matlab and Java interfaces. The software solution integrate every algorithms of this thesis to complex solution and it extends possible application of those approaches to real systems and practical world.

The aim of this thesis is to investigate the dynamical behaviors of spatially extended systems with fluctuating time delays. In recent years, the study of spatially extended systems and systems with fluctuating delays has experienced a fast growth. In ubiquitous natural and laboratory situations, understanding the action of time-delayed signals is a crucial for understanding the dynamical behavior of these systems. Frequently, the length of the delay is found to change with time. Spatially extended systems are widely studied in many fields, such as chemistry, ecology, and biology. Self-organization, turbulence, and related nonlinear dynamic phenomena in spatially extended systems have developed into one of the most exciting topics in modern science. The first part of this thesis considers the discrete system. Diffusively coupled map lattices with a fluctuating delay are used in the study. The uncoupled local dynamics of the considered system are represented by the delayed logistic map. In particular, the influences of diffusive coupling and fluctuating delay are studied. To observe and understand the influences, the results for the considered system are compared with coupled map lattices without delay and with a constant delay as well as with the uncoupled logistic map with fluctuating delays. Identifying different patterns, determining the existence of traveling wave solutions, and specifying the fully synchronized stable state are the focus of this part of the study. The Lyapunov exponent, the master stability function, spectrum analysis, and the structure factor are used to characterize the different states and the transitions between them. The second part examines the continuous system. The delay is introduced into the reactionterm of the Fisher-KPP equation. The focus of this part of study is the time-delay-induced Turing instability in one-component reaction-diffusion systems. Turing instability has previously only been found in multiple-component reaction-diffusion systems. However, this work demonstrates with the help of the stability exponent that fluctuating delay can result in Turing instability in one-component reaction-diffusion systems as well. / Ziel der vorliegenden Arbeit ist die Untersuchung der Einflüsse der zeitlich fluktuierenden Verzögerungen in räumlich ausgedehnten diffusiven Systemen. Durch den Vergleich von Systemen mit konstanter Verzögerung bzw. Systemen ohne räumliche Kopplung erhält man ein tieferes Verständnis und eine bessere Beschreibungsweise der Dynamik des räumlich ausgedehnten diffusiven Systems mit fluktuierenden Verzögerungen. Im ersten Teil werden diskrete Systeme in Form von diffusiven Coupled Map Lattices untersucht. Als die lokale iterierte Abbildung des betrachteten Systems wird die logistische Abbildung mit Verzögerung gewählt. In diesem Teil liegt der Fokus auf Musterbildung, Existenz von Multiattraktoren und laufenden Wellen sowie der Möglichkeit der vollen Synchronisation. Masterstabilitätsfunktion, Lyapunov Exponent und Spektrumsanalyse werden benutzt, um das dynamische Verhalten zu verstehen. Im zweiten Teil betrachten wir kontinuierliche Systeme. Hier wird die Fisher-KPP Gleichung mit Verzögerungen im Reaktionsteil untersucht. In diesem Teil liegt der Fokus auf der Existenz der Turing Instabilität. Mit Hilfe von analytischen und numerischen Berechnungen wird gezeigt, dass bei fluktuierenden Verzögerungen eine Turing Instabilität auch in 1-Komponenten-Reaktions-Diffusionsgleichungen gefunden werden kann