Bifurcation

The present study deals with the analysis of a Lotka-Volterra model describing competition between tumor and immune cells. The model consists of differential equations with piecewise constant arguments and based on metamodel constructed by Stepanova. Using the method of reduction to discrete equations, it is obtained a system of difference equations from the system of differential equations. In order to get local and global stability conditions of the positive equilibrium point of the system, we use Schur-Cohn criterion and Lyapunov function that is constructed. Moreover, it is shown that periodic solutions occur as a consequence of Neimark-Sacker bifurcation.

In urban communities, infrastructures that support living are indispensable. There is increased interest in alternative ways of providing such support systems, including semi-autonomous infrastructures resulting from the self-organization of local actors. In this study, we analyze the emergence and management of such infrastructures in light of the theory of complex adaptive systems, within which they are called ‘inverse infrastructures’. Empirical evidence is drawn from the case of water cooperatives in the town of Ikaalinen, Finland. Our analysis shows that, with favorable preconditions in place, inverse infrastructures may contribute significantly to local infrastructure services and so also to the functioning of society.

This paper analyses the nonlinear phenomena in current mode controlled buck-boost converter. The system undergoes various operating regions whenever there is a change in non linear elements presented in the system as well as in the loads. In this work, inductor current is considered as the reference for analysis.The results show that the converter enters into period-1, periodic doubling and chaotic regions as per the parameter (I L) variations. The proposed control strategy, weak periodic perturbation (WPP) method tries to stabilize the system from the chaotic behavior.The buck boost converter along with the control system is simulated using MATLAB/SIMULINK software tool and the results are presented.The hardware implementation of the system is done and the results are verified with the simulation results. It shows that the WPP can transform the system behaviour from the chaotic region to the periodic one.

A numerical study based on the nonlinear Schrödinger equation, as applied to nonlinear resonant standing waves excited directly by a wavemaker in a rectangular tank, is presented. The stationary solutions of the problem serve as a starting point of the investigation. Bifurcations from a single steady state to multiple stationary solutions are obtained for several values of damping coefficients along the tank and at the wavemaker. The stability of the latter solutions is tested. Limit-cycle or fixed-point solutions are obtained. The results of the numerical study are discussed in connection with experimental data. The necessity of incorporation of dissipation at the wavemaker in the theoretical model in order to obtain qualitative agreement with experiment is demonstrated.

Borodkin, Leonid. CHALLENGES OF INSTABILITY: THE CONCEPTS OF SYNERGETICS  IN STUDYING THE HISTORICAL DEVELOPMENT OF RUSSIA The article discusses the big challenges of Russia’s historical development associated with the unstable nature of the social and political processes of the 20th century, the unpredictability of radical changes that capture different countries, and the alternatives of their development. An adequate approach to studying such processes, analyzing mechanisms of their functioning became possible on the basis of the multidisciplinary concept of synergetics related to the doctrine of self-organization, the methods of nonlinear dynamics and chaos theory. An important issue this paper addresses is relationship between regularity and randomness in the development of an unstable social process, in particular, the role of the individual in this process. This issue acquires special significance in researching transition epochs and radical transformations of society, which require an adequate response to the big challenges posed by the risk of a system falling into the trajectory of turbulent development. Connected with this question are new approaches to exploring alternatives of social development that arise at the points of bifurcation, considered in the article on the example of the development of the revolutionary situation in 1917 Russia. It is shown that the challenges of instability which impede sustainable development that have emerged at different periods of Russia’s history and that are possible in its foreseeable future can be analyzed and modeled using synergetic concepts. The methodological tools of historical research are expanding and include now methods of studying unstable historical processes, dynamics of revolutionary changes and protest movements.

В статье рассмотрены большие вызовы исторического развития России, связанные с неустойчивым характером социально-политических процессов ХХ в., непредсказуемость радикальных перемен, захватывающих различные страны, альтернативность их развития. Адекватный подход к изучению таких процессов, к анализу механизмов их функционирования стал возможен на основе мультидисциплинарной концепции синергетики, связанной с учением о самоорганизации, с методами нелинейной динамики и теории хаоса. Обсуждается в работе и соотношение закономерности и случайности в развитии неустойчивого процесса, в частности роль личности в этом процессе. Этот вопрос приобретает особое значение при изучении переходных эпох, радикальных трансформаций общества, требующих адекватного ответа на большие вызовы, порождаемые риском схода системы на траекторию турбулентного развития. С этим вопросом связаны и новые подходы к изучению альтернатив общественного развития, возникающих в точках бифуркации, рассмотренные в статье на примере развития революционной ситуации в России 1917 г. Показано, что вызовы нестабильности, препятствующие устойчивому развитию, возникавшие в разные периоды истории России и возможные в ее обозримом будущем, могут подвергаться анализу и моделироваться с помощью концепций синергетики. Методологический инструментарий исторической науки расширяется, включая теперь методы изучения неустойчивых исторических процессов, динамики революционных изменений и протестных движений.

CONTENTS

FROM THE AUTHOR . . . 6

1. WORLDVIEWS BEFORE SYNERGETICS . . . 9

1.1. Worldview definition . . . 9
1.2. Worldviews in archaic and religious consciousness . . . 13
1.3. From religions to science: development of methodology . . . 20
1.4. Worldview in classical science . . . 33
1.5. Worldview as revised by structuralism and the cybernetics . . . 43

2. MAIN CONCEPTS OF SYNERGETICS . . . 51

2.1. Synergetics vs. theory of self-organization . . . 51
2.2. Phase space and phase parameters . . . 53
2.3. Attractors and their types . . . 55
2.4. Nonlinear condition category . . . 60
2.5. Fractals and their meaning . . . 64
2.6. The concept and essence of bifurcations . . . 73
2.7. Structure and chaos: most important categories of synergetics . . . 80

3. THE PROBLEM OF HUMAN TIME IN HISTORY . . . 86

3.1. The crisis of historical knowledge on the turn of the 19th—20th centuries . . . 86
3.2. Absolute and relative time . . . 91
3.3. The “time for history” and the “time for historians” . . . 96
3.4. The historical time and its human perception . . . 100
3.5. The object’s own time . . . 109

3.5.1. Formation of rhythm . . . 109
3.5.2. Ptolemy’s system as a mathematical model of nonlinear processes . . . 114

3.6. Individual time in H. Bergson’s concept . . . 119
3.7. F. Braudel’s types of historical time . . . 124
3.8. The problem of substantiality of historical fact . . . 129
3.9. The structure of fact . . . 134

4. SYNERGETICS AND EVOLUTIONISM . . . 146

4.1. ‘Synergetics of nature’ before Prigozhin and Hacken . . . 146
4.2. Principles of nomogenesis theory . . . 149
4.3. Selectionists’ objections . . . 156
4.4. Nomogenesis from the point of view of synergetics . . . 159

5.APPLICATION OF SYNERGETICS IN HISTORY . . . 164

5.1. Is ‘theoretical history’ possible? . . . 164
5.2. Problems at the joint of humanities and natural sciences . . . 168
5.3. Variants of post-nonclassical newest historical concepts: rehabilitation of historical idealism . . . 173
5.4. L.N. Gumilev’s ‘Synergetics of ethnos’ . . . 175
5.5. “Theoretical history” of S. P. Kapitsa, S. P. Kurdyumov and G. G. Malinetsky . . . 179
5.6. “Social synergetics’ and “ethnological akmeology” by V. P. Bransky and S. D. Pozharsky . . . 181
5.7. History and ecology in E. S. Kulpin’s concept . . . 195
5.8. “Techno-humanitarian balance” and anthropogenic crises in A. P. Nazaretyan’s concept . . . 205
5.9. Pierre Teilhard de Chardin . . . 220
5.10. History as alternating meanings in M. E. Tkachuk’s concept . . . 224
5.11. Conclusions: possible scope of synergetics in historical sciences . . . 231

6. SYNERGETICS, THE TRUTH AND THE MODELS . . . 234

6.1. The truth, or a model-based thinking? Problem definition . . . 236
6.2. «What is the truth?» . . . 239
6.3. The problem of values: ‘the truth’, or ‘our truth’? . . . 246
6.4. Can the truth be cancelled by interdisciplinarity? . . . 256
6.5. Classics or postmodernity? . . . 258
6.6. Do the truth-based and model-based thinking contradict each other? . . . 272

7. THE SYNERGETIC WORLDVIEW . . . 277

7.1. Synergetics as a methodology of myth? . . . 277
7.2. The landscape of the virtual world, or ‘the unity of place’ . . . 282

7.2.1. The dimensions and the structure of the virtual space . . . 282
7.2.2. A simplified version of cultural space: A. J. Toynbee’s case . . . 287
7.2.3. The problem of geographic determinism . . . 290

7.3. The time for the virtual world, or ‘the unity of time’ . . . 298
7.4. The character of natural processes, or ‘the unity of action’ . . . 300
7.4.1. Chaos as a factor of choice . . . 300
7.4.2. Trends in development or teleology? . . . 302
7.4.3. Does development have an ‘Omega spot’? . . . 308

8. EPILOGUE: WILL THE PROPHET TELL HIS OWN FORTUNE? . . . 312

8.1. Debates on applicability of synergetics . . . 312
8.2. A ‘new science’, or a new methodology for ‘old’ sciences? . . .314
8.3. Variability of the past . . . 316
8.4. The ambiguity of the concept of ‘order’. . . 317
8.5. Prophecies, predictions and freedom of actions . . . 322
8.6. Towards freedom via synergetics, or the outline of the desired future . . . 324
8.7. Conclusions (Twelve years after) . . . 337

REFERENCES (IN RUSSIAN) . . . 341
TRANSLITERATED REFERENCES . . . 352
SUBJECT INDEX . . . 364
NAME INDEX . . . 370
SUMMARY . . . 377

О Г Л А В Л Е Н И Е

ОТ АВТОРА . . . 6

1. КАРТИНЫ МИРА ДО СИНЕРГЕТИКИ . . . 9

      1.1. Понятие картины мира . . . 9
      1.2. Картины мира в архаическом и религиозном сознании . . . 13
      1.3. От религии к науке: формирование метода . . . 20
      1.4. Картина мира классической науки . . . 33
      1.5. Структуралистская ревизия картины мира и кибернетика . . . 43

2. ОСНОВНЫЕ ПОНЯТИЯ СИНЕРГЕТИКИ . . . 51

      2.1. Синергетика или теория самоорганизации . . . 51
      2.2. Фазовое пространство и фазовые параметры . . . 53
      2.3. Аттракторы и их виды . . . 55
      2.4. Категория нелинейности . . . 60
      2.5. Понятие фракталов и их значение . . . 64
      2.6. Понятие и сущность бифуркаций . . . 73
      2.7. Структура и хаос — важнейшие категории синергетики . . . 80

3. ПРОБЛЕМА ЧЕЛОВЕЧЕСКОГО ВРЕМЕНИ В ИСТОРИИ . . . 86

      3.1. Кризис исторического знания на рубеже XIX—XX вв. . . . 86
      3.2. Время абсолютное и относительное . . . 91
      3.3. «Время истории» и «время историков» . . . 96
      3.4. Историческое время и его человеческое ощущение . . . 100
      3.5. Собственное время объекта . . . 109

          3.5.1. Образование ритма . . . 109
          3.5.2. Система Птолемея как математическая модель нелинейного процесса . . . 114

      3.6. Индивидуальное время в концепции А. Бергсона . . . 119
      3.7. Виды исторического времени у Ф. Броделя . . . 124
      3.8. Проблема обоснованности исторического факта . . . 129
      3.9. Проблема структуры факта . . . 134

4. СИНЕРГЕТИКА И ЭВОЛЮЦИОНИЗМ . . . 146

      4.1. «Синергетика природы» до Пригожина и Хакена . . . 146
      4.2. Принципиальные положения теории номогенеза . . . 149
      4.3. Возражения селекционистов . . . 156
      4.4. Номогенез с точки зрения синергетики . . . 159

5. ОПЫТ ПРИМЕНЕНИЯ СИНЕРГЕТИКИ В ИСТОРИИ . . . 164

      5.1. Возможна ли «теоретическая история»? . . . 164
      5.2. Проблемы на стыке гуманитарных и точных наук . . . 168
      5.3. Варианты постнеклассических новейших исторических концепций: реабилитация исторического идеализма . . . 173
      5.4. «Синергетика этноса» у Л. Н. Гумилёва . . . 175
      5.5. «Теоретическая история» С. П. Капицы, С. П. Курдюмова, Г. Г. Малинецкого . . . 179
      5.6. «Социальная синергетика» и «этнологическая акмеология» В. П. Бранского и С. Д. Пожарского . . . 181
      5.7. История и экология в концепции Э. С. Кульпина . . . 195
      5.8. «Техно-гуманитарный баланс» и антропогенные кризисы в концепции А. П. Назаретяна . . . 205
      5.9. Пьер Тейяр де Шарден . . . 220
      5.10. История как смена смыслов в концепции М. Е. Ткачука . . . 224
      5.11. Выводы: возможности синергетики в исторических науках . . . 231

6. СИНЕРГЕТИКА, ИСТИНА И МОДЕЛИ . . . 234

      6.1. Истинность или модельное мышление? Постановка проблемы . . . 236
      6.2. «Что есть истина?» . . . 239
      6.3. Проблема ценностей: «истина» или «наша правда»? . . . 246
      6.4. Отменяется ли истина междисциплинарностью? . . . 256
      6.5. Классика или постмодерн? . . . 258
      6.6. Противоречат ли друг другу истинностное и модельное мышление? . . . 272

7. СИНЕРГЕТИЧЕСКАЯ КАРТИНА МИРА . . . 277

      7.1. Синергетика как методология мифа? . . . 277
      7.2. Пространство виртуального мира, или «Единство места» . . . 282

          7.2.1. Размеры и структура виртуального пространства . . . 282
          7.2.2. Упрощённое понимание культурного пространства: случай А. Дж. Тойнби . . . 287
          7.2.3. Проблема географического детерминизма . . . 290

      7.3. Время виртуального мира, или «Единство времени» . . . 298
      7.4. Характер естественных процессов, или «Единство действия» . . . 300

          7.4.1. Хаос как фактор отбора . . . 300
          7.4.2. Тенденции развития или телеология?. . . 302
          7.4.3. Есть ли у развития «точка Омега»? . . . 308

8. ЗАКЛЮЧЕНИЕ: ПРЕДСКАЖЕТ ЛИ ПРОРОК СОБСТВЕННУЮ СУДЬБУ? . . . 312

      8.1. Споры вокруг применимости синергетики . . . 312
      8.2. «Новая наука» или новая методология «старых» наук? . . . 314
      8.3. Вариативность прошлого . . . 316
      8.4. Неоднозначность понятия «порядок» . . . 317
      8.5. Пророчества, предсказания и свобода действий . . . 322
      8.6. К свободе — через синергию, или Контуры желаемого будущего . . . 324
      8.7. Выводы (Двенадцать лет спустя) . . . 337

ЛИТЕРАТУРА . . . 341
REFERENCES . . . 352
ПРЕДМЕТНЫЙ УКАЗАТЕЛЬ . . . 364
УКАЗАТЕЛЬ ИМЁН . . . 370
SUMMARY . . . 377

In this expository and resources chapter we review selected aspects of the mathematics of dynamical systems, stability, and chaos, within a historical framework that draws together two threads of its early development: celestial mechanics and control theory, and focussing on qualitative theory. From this perspective we show how concepts of stability enable us to classify dynamical equations and their solutions and connect the key issues of nonlinearity, bifurcation, control, and uncertainty that are common to time-dependent problems in natural and engineered systems. We discuss stability and bifurcations in three simple model problems, and conclude with a survey of recent extensions of stability theory to complex networks.

The daily-case numbers of Covid-19 were studied and an oscillatory pattern in the numbers was identified. And this oscillatory nature has been present throughout the pandemic data. Although the US has had four waves of varying amplitude, the division of those waves into smaller units also indicate a repetitive, oscillatory pattern. The study of the smaller waves are important and they can be mathematically modelled using fitting techniques in order to accurately forecast daily cases over a short term. Further, the amplitudes of the prior waves {in a holistic sense (an aggregation of small waves)} can be examined to forecast a range for the amplitude of a future wave. As new data becomes available, it can be incorporated into the paradigm and be used to mutate the model to increase prediction accuracy {short term and long term}. During this time, political, social, and economic decisions can be made to curb any behavior indicative of potential spikes or anomalies {either through quarantine and restriction of human movement or by planning a protocol of an effective vaccination schedule}.

Initially published (July 4th) and retrieved from—
https://smanivann.wordpress.com/2021/07/04/mathematical-model-to-fit-and-forecast-daily-case-numbers-and-to-predict-wave-magnitude-scenarios-covid-19/

Depuis plus de vingt ans une branche de la sociologie pragmatique française, appelée pragmatique de la complexité ou pragmatique des transformations, s’attache à décrire et analyser les processus d’alertes et les controverses ou les mobilisations auxquelles ils donnent lieu. Partant d’un diagnostic sociopolitique de l’époque contemporaine marquée par des chocs et des crises aux multiples conséquences, ce texte examine les conditions de l’enquête d’inspiration pragmatiste face à des processus complexes, non-linéaires, multi-scalaires et riches en rebondissements et en rétroactions. Un conséquentialisme ouvert, non-borné, s’impose pour relier l’exploration des phénomènes et la découverte graduelle, au fil des enquêtes, de caractéristiques ignorées, invisibles ou incertaines. L’enjeu est à la fois de rendre intelligibles les trajectoires suivies par les causes les plus diverses et de penser les incommensurabilités et les irréductibilités liées aux jeux d’échelles. La convergence d’une sociologie pragmatique argumentative, d’une ethnographie des activités dans les milieux et d’une théorie ouverte des systèmes dynamiques, permet d’adapter les concepts et les outils issus du pragmatisme aux situations contemporaines et de réarmer les capacités critiques nécessaires à la pratique des sciences sociales.


For more than twenty years, a current of French pragmatic sociology, called pragmatics of complexity or pragmatics of transformations, has focused on describing and analysing alert processes and the controversies or mobilisations to which they give rise. Starting from a socio-political diagnosis of contemporary times marked by shocks and crises with multiple consequences, this text examines the conditions of pragmatist-inspired enquiry in front of complex, non-linear, multi-scalar processes full of twists and turns and feedbacks. An open, non-bounded consequentialism is needed to link the exploration of phenomena and the gradual discovery, over the course of investigations, of ignored, invisible or uncertain characteristics. The challenge is both to make intelligible the trajectories followed by many causes and to consider the incommensurabilities and irreducibilities linked to the play of scales. The convergence of a pragmatic argumentative sociology, an ethnography of activities in different milieus and an open theory of dynamic systems, leads to adapt the concepts and tools of pragmatism to contemporary situations, helping us to rearm the critical capacities necessary for the practice of social sciences.

This paper investigates the dynamical behaviour of a duffing oscillator using bifurcation diagrams. There has been growing interest and challenges in engineering dynamics to characterize dynamical systems that are chaotic using bifurcation diagrams. The relevant second order differential equations using Runge-Kutta method were solved for ranges of appropriate parameters. The solutions obtained were used to produce the bifurcation diagrams using Microsoft excel 2007. Since an average estimate of 5 = 4.668 from the bifurcation diagrams produced is an approximate value of the Feigenbaum .constant as widely reported in the literatures, it can be deduced that the bifurcation diagrams conforms to the expected results. While the bifurcation diagrams revealed the dynamics of the duffing oscillator, it also shows that the dynamics depend strongly on the initial conditions.

Mémoire de Master 2 soutenu à l'Université Paris V René Descartes, sous la direction de Madame Birgitta ORFALI. Le mémoire porte sur l'étude d'une forme particulière de l'action collective, celle de l'occupation des locaux universitaires. L'occupation est ici problématisée dans une perspective socio-historique, faisant partie d'un répertoire d'action qui date du 19 siècle. Traçant l'histoire de l'occupation, passant par le mouvement ouvrier et les occupations des années 1930 et 1960s, je vise à démontrer les transformations que cette forme d'action a subi au fil du temps, avec l'apparition des nouveaux médias d'information et de communication. Après avoir cerné un contexte historique et théorique, je m'intéresse plus précisement à la réappropriation de cette forme d'action par les mouvements étudiants français contemporains. Mon terrain porte sur les mouvements du 2006, 2009, 2010-2011. Se basant sur 31 interview et des observations non-participantes et participantes lors des occupations de plusieurs universités parisiennes et l'Université de Rennes 2, je dessine une analyse de l'influence de cette forme sur les carrières des étudiants. Je postule que l'occupation devient un "point de bifurcation" qui transforme les modes d'engagement des étudiants autant dans leurs études que dans les mobilisations. L'espace public de l'Université se voit également transformé, repensé, réapproprié. Je porte une attention aux abus et dérives de cette forme d'action, ainsi que à sa banalisation.

This article surveys and synthesizes dynamic systems models of development from biology, neuroscience, and psychology in order to propose an integrated account of growth, learning, and behavior. Key to this account is the concept of self-differentiation or symmetry-breaking. I argue that development can be understood as a cascade of symmetry-breaking events brought about by the ongoing interactions of multiple, nested, nonlinear dynamic systems whose self-organizing behaviors gradually alter their own anatomical conditions.

Physicists are very familiar with forced and parametric resonance, but usually not with self-oscillation, a property of certain dynamical systems that gives rise to a great variety of vibrations, both useful and destructive. In a self-oscillator, the driving force is controlled by the oscillation itself so that it acts in phase with the velocity, causing a negative damping that feeds energy into the vibration: no external rate needs to be adjusted to the resonant frequency. The famous collapse of the Tacoma Narrows bridge in 1940, often attributed by introductory physics texts to forced resonance, was actually a self-oscillation, as was the swaying of the London Millennium Footbridge in 2000. Clocks are self-oscillators, as are bowed and wind musical instruments. The heart is a "relaxation oscillator," i.e., a non-sinusoidal self-oscillator whose period is determined by sudden, nonlinear switching at thresholds. We review the general criterion that determines whether a linear system can self-oscillate. We then describe the limiting cycles of the simplest nonlinear self-oscillators, as well as the ability of two or more coupled self-oscillators to become spontaneously synchronized ("entrained"). We characterize the operation of motors as self-oscillation and prove a theorem about their limit efficiency, of which Carnot's theorem for heat engines appears as a special case. We briefly discuss how self-oscillation applies to servomechanisms, Cepheid variable stars, lasers, and the macroeconomic business cycle, among other applications. Our emphasis throughout is on the energetics of self-oscillation, often neglected by the literature on nonlinear dynamical systems.

From his earliest work forward, Merleau-Ponty attempted to develop a new ontology of nature that would avoid the antinomies of realism and idealism by showing that nature has its own endogenous sense which is prior to reflection. The key to this new ontology was the concept of form, which he appropriated from Gestalt psychology. However, Merleau-Ponty struggled to give a positive characterization of the phenomenon of form which would clarify its ontological status. Evan Thompson has recently taken up Merleau-Ponty’s ontology as the basis for a new, “enactive” approach to cognitive science, synthesizing it with concepts from dynamic systems theory and Francisco Varela’s theory of autopoiesis. However, Thompson does not quite succeed in resolving the ambiguities in Merleau-Ponty’s account of form. This article builds on an indication from Thompson in order to propose a new account of form as asymmetry, and of the genesis of form in nature as symmetry-breaking. These concepts help us to escape the antinomies of Modern thought by showing how nature is the autoproduction of a sense which can only be known by an embodied perceiver.

The ecological emergency can be described as a combination of four categories: climate change, degradation of biodiversity, increasing pollution and scarcity of natural resources. As a consequence of the industrial revolution, these phenomena appear to be linked to the many technological innovations industrialized by engineers during the 19th century. This profession is still based, to some extent, on an anthropocentric universe where nature must be controlled in the name of progress and where "Science" appears as a neutral entity. While surveys indicate a largely depoliticized public, there has nevertheless been a growing militant and ecological commitment among engineers in recent years. It is therefore possible to question the trajectories that lead these individuals from their training to their participation in collective actions. We can also seek to understand the social determinants of these ecological commitments. We sought to answer these questions through a qualitative method based on some twenty semi-directive interviews with committed engineers, students and graduates, and teachers in engineering schools. We also conducted a study of documents – press articles, minutes of meetings, manifestos, engineering school brochures. It appears that engineers face challenges during their training and careers, both on the job market and in terms of raising awareness of environmental issues. It is their patterns of perception and interpretation of the world which are then transformed and which give rise to a new repertoire of action in the service of the ecological cause, the « cause engineering ».

The Bogdanov-Takens (BT) bifurcation, which has attracted increasing attention in the past few decades, is encountered in the study of autonomous dynamical systems arising in different branches of science and engineering. The first part of this talk gives a brief introduction to dynamical systems and bifurcation theory, including a discussion of linearization, the classification of linear systems, and structural stability. In the second part, the normal form of the BT bifurcation and its equilibria will be discussed. This will be followed by an examination of the qualitative behaviour of solutions and the way they change as the parameters of the system are varied.
Mohammed

In this paper we give sucient conditions for constructing Dulac functions of a sum of planar vector elds. We give some applications and examples in order to illustrate the results.

Key words: Bendixson-Dulac criterion, Dulac functions, limit cycles

The paper focuses on the emerging demands for new states in India. In the process an effort has been made to find out the reasons that led to the rise of such demands. Among all the factors explored, the most significant has been the lack of balanced development of the regions. The paper also focuses on the response of the union and the state governments towards the new states demands. The later part of the paper undertakes the study of two sets of demands-for and against reorganisation of states.

Dopo alcuni richiami sulla stabilità degli archi, il problema viene specializzato a tre diversi sistemi, caratterizzati da differenti rapporti di ribassamento f/L. ciascuno di tali sistemi possiede lo stesso vincolo orizzontale cedevole, che vuole simulare il comportamento di una palificata. Per tener conto delle incertezze associate a tali strutture, la rigidezza del vincolo a terra è considerata aleatoria ed è descritta da una distribuzione gaussiana. Mediante simulazioni Monte Carlo vengono ricavati i percorsi di equilibrio associati ad ogni singola rigidezza. Come ente della simulazione si assume il primo punto limite del percorso di equilibrio. Si costruisce in tal modo la statistica dei moltiplicatori di carico. I risultati ottenuti consentono di valutare gli effetti sia del ribassamento dell’arco sia delle incertezze di vincolo.

This article presents a summary of applications of chaos and fractals in robotics. Firstly, basic concepts of determin‐ istic chaos and fractals are discussed. Then, fundamental tools of chaos theory used for identifying and quantifying chaotic dynamics will be shared. Principal applications of chaos and fractal structures in robotics research, such as chaotic mobile robots, chaotic behaviour exhibited by mobile robots interacting with the environment, chaotic optimization algorithms, chaotic dynamics in bipedal locomotion and fractal mechanisms in modular robots will be presented. A brief survey is reported and an analysis of the reviewed publications is also presented.