Bu çalışma, memristörlerin ve kaotik sistemlerin kapsamlı bir incelemesini sunarak, memristör tabanlı hiperkaotik Lorenz sistemini teorik olarak inceleyerek önemli katkılar sağlamayı amaçlamaktadır. Memristörler, elektriksel dirençleri akım ve gerilime bağlantılı olarak değişen ve bu bilgiyi kalıcı olarak saklayabilen devre elemanları olarak, bilgi depolama ve işleme teknolojilerinde devrim yaratma potansiyeline sahiptir. Çalışmada, memristörlerin çeşitli modelleri, üretim yöntemleri ve makro modelleri incelenirken, bununla beraber bu teknolojilerin matematiksel temellerini ve uygulama alanlarını da incelemektedir. Kaotik sistemler üzerindeki araştırmalar, hem bilinçli hem de tesadüfi olarak uzun yıllar boyunca devam etmektedir. 1890 yılında Henri Poincaré, sade dinamik kuralların kompleks ve rastgele görünümler sergileyebileceğini keşfetmiştir. Bu gözlemi kaos olarak adlandırmamış olsa da, modern kaos teorisinin temellerini atmıştır. Kaotik sistemler, küçük başlangıç koşulu değişikliklerinin büyük, öngörülemez sonuçlara yol açtığı dinamik sistemlerdir. Kaotik sistemlerin incelenmesinde, kararlılık analizi önemli bir rol oynar. Ancak, bir sistemin kaotik olup olmadığını belirlemek için yalnızca kararlılık analizi yeterli değildir. Kaotik sistemler, düzensiz ve öngörülemez davranışlar sergileyebilir, bu yüzden bu sistemlerin dinamiklerini tam olarak anlamak için Lyapunov üst sınırları gibi diğer analiz yöntemlerine de başvurulması gereklidir. Lyapunov üst sınırları, bir sistemin başlangıç koşullarına duyarlılığını ve zamanla nasıl geliştiğini belirlemeye yardımcı olur. Hiperkaos kavramı, ilk kez 1979 yılında Otto E. Rössler tarafından önerilen bir sistem ile literatüre kazandırılmıştır. Hiperkaotik sistemler, daha karmaşık yapıları ve öngörülmesi zor davranışları nedeniyle bilim insanlarının ilgisini çekmektedir. Rössler 1979 yılındaki çalışmasında, 3 boyutlu sistemlerin yanı sıra, 4 boyutlu bir kaotik sistem geliştirmiştir. Bu sistem, diğer kaotik sistemlere kıyasla daha karmaşık bir dinamik davranış sergilediği gözlemlenmiştir. Karmaşık dinamiklerin incelenmesinde sıkça başvurulan Lorenz sistemi, hiperkaotik davranış sergileyen pek çok sistemin temelini oluşturur. Farklı boyutlarda ve özelliklerde sayısız Lorenz benzeri sistemler literatürde yerini almıştır. Ele alınan memristör tabanlı hiperkaotik Lorenz sisteminin farklı kontrol yöntemleri ile simülasyonları gerçekleştirilmiştir. Bu çalışma, memristör tabanlı hiperkaotik Lorenz sisteminin; pasif, aktif ve kayan kipli kontrol metodları kullanarak senkronizasyonunu sağlamayı ve kullanılan metodların birbirleri arasındaki avantajları ve dezavantajları ile performansları hakkında bilgi sunmayı hedeflemektedir. Simülasyon sonuçları analiz edildiğinde memristör tabanlı hiperkaotik Lorenz sisteminin senkronizasyonunda kullanılan metodlar içinde kayan kipli kontrol metodunun, pasif kontrol metodu ile aktif kontrol metodundan oturma süreleri açısından daha iyi sonuçlar elde edildiği görülmüştür.
This study aims to make significant contributions by providing a comprehensive review of memristors and chaotic systems, while theoretically analyzing the memristor-based hyperchaotic Lorenz system. Memristors, which are circuit elements whose electrical resistance changes in relation to current and voltage and can permanently store this information, have the potential to revolutionize information storage and processing technologies. The relationships between voltage-current, current-charge, voltage-current, voltage-charge, and current-current are expressed as fundamental circuit elements. However, the significance of the charge-current relationship, which falls outside of these five known relations, has remained unclear for many years. This uncertainty led Chua to propose that a previously unexplained relationship between charge and current could only be explained by a new circuit element. The fundamental logic of the memristor is that its resistance changes depending on the direction and amount of current passing through it. When current flows in a particular direction, the resistance increases, and when it flows in the reverse direction, the resistance decreases. When the current is interrupted and reapplied, the memristor retains its previous resistance value; this characteristic indicates that the memristor is a memory-resistor. The passive characteristic of memristors is related to the fact that memristors are referred to as passive elements due to instantaneous power losses. When the memristor shows a positive memristance value under certain conditions, the instantaneous power is always greater than zero, making the memristor a passive component. This is valid as long as the charge-current curve of the memristor is a continuously differentiable function. A passive memristor's charge-current curve shows a monotonically increasing feature. The hysteresis property of memristors is observed in a periodic signal applied to the memristor, where the current is also zero when the voltage is zero. Therefore, the current and voltage curves intersect at the origin. The linear characteristic property of memristors is observed when a repetitive signal is applied. As the frequency approaches infinity, the memristor behaves like a linear resistor. This behavior is related to the change in the current-voltage curve's hysteresis characteristic, which depends on frequency. The areas in the hysteresis curve increase as the frequency decreases and decrease as the frequency increases. As the frequency approaches infinity, the memristor begins to exhibit behavior similar to a resistor. The memristor is a component with critical importance and a wide range of applications. The widespread use of memristors is attributed to their advantages, including high scalability, low energy consumption, the possibility of fabrication at the nanoscale, and compatibility with traditional CMOS structures. However, for the design of memristor-based circuits, it is necessary to develop a model that accurately describes the properties of the memristor. Memristor models are mathematical representations that describe the electrical behavior and functions of memristors. These models are used to understand and optimize how memristors perform in different application areas. During the modeling process, factors such as the dynamics of the memristor, its memory properties, and the relationships between charge and voltage are taken into account. Various models have been developed to represent different characteristics of the memristor and application scenarios, playing an important role in determining the development and potential applications of memristor technology. Among the memristor models, the linear drift model, known as the HP model, was proposed by Strukov and his team in 2008. This model consists of high-insulating titanium dioxide with oxygen-doped conductive titanium dioxide materials placed between two platinum electrodes. The Simmons tunnel barrier model is based on the HP team's memristor model, which relies on continuous regions of pure and doped titanium dioxide and their corresponding Ron and Roff resistances. However, Pickett and others advanced this approach by presenting an alternative physical memristor model. In this model, only one resistance is used instead of two series resistances, as in the HP model. The threshold adaptive memristor model, proposed by Kvatinsky and colleagues, offers a more comprehensive and advantageous approach. While it shares similar physical characteristics with the Simmons Tunnel Barrier model, its mathematical structure is simpler. In the threshold adaptive memristor model, there are no exponential dependencies, and a polynomial relationship exists between the memristor current and the derivative of the internal state variable. The study examines various models of memristors, production methods, and macro models, while also exploring the mathematical foundations and application areas of these technologies. Research on chaotic systems has been ongoing for many years, both consciously and incidentally. In 1890, Henri Poincaré discovered that simple dynamic rules could exhibit complex and random behaviors. While he did not call this observation "chaos," it laid the foundations for modern chaos theory. Chaotic systems are dynamic systems in which small changes in initial conditions lead to large, unpredictable outcomes. When looking at other fundamental properties of chaotic systems, one can observe their sensitivity to initial conditions. Chaotic systems exhibit a high degree of sensitivity to small changes in initial conditions. As a result, minor changes in initial conditions can lead to significant differences in system behavior. The divergence of orbits is a characteristic feature of chaotic systems, where two initially close orbits tend to diverge over time. This is known as the divergence of nearby orbits in chaotic systems. Chaotic systems are generally autonomous in nature, and their orbits do not intersect. Chaotic systems are systems in which energy is not conserved, meaning that energy cannot be preserved within the system, and the system's energy fluctuates within a certain range. In the study of chaotic systems, stability analysis plays an important role. However, stability analysis alone is not sufficient to determine whether a system is chaotic. Chaotic systems can exhibit irregular and unpredictable behaviors, so other analysis methods, such as Lyapunov upper bounds, are necessary to fully understand the dynamics of these systems. Lyapunov upper bounds help determine a system's sensitivity to initial conditions and how it evolves over time. The concept of hyperchaos was introduced to the literature in 1979 by Otto E. Rössler with a system he developed. Hyperchaotic systems attract the attention of scientists due to their more complex structures and unpredictable behaviors. In his 1979 study, Rössler developed a four-dimensional chaotic system, in addition to the three-dimensional systems. This system exhibited more complex dynamic behavior compared to other chaotic systems. Rössler's work laid an important foundation for the mathematical modeling and understanding of hyperchaos. Through his contributions, hyperchaotic systems transitioned from being a purely theoretical concept to being applied in engineering and practical sciences. Today, hyperchaotic systems are widely studied in fields such as cryptography, security technologies, and signal processing. These advancements have facilitated the understanding and application of hyperchaotic systems, offering innovative solutions across various fields, from engineering to natural sciences. The physical and theoretical models of hyperchaotic systems continue to have research relevance today, allowing for the discovery of even more complex and unpredictable dynamics. The Lorenz system, often referred to in the study of complex dynamics, forms the basis for many systems that exhibit hyperchaotic behavior. Countless Lorenz-like systems of varying dimensions and characteristics are found in the literature. The memristor-based hyperchaotic Lorenz system examined in this study has been simulated using different control methods. This study aims to synchronize the memristor-based hyperchaotic Lorenz system using passive, active, and sliding mode control methods, and to provide information about the advantages, disadvantages, and performances of these methods relative to each other. Upon analyzing the simulation results, it was observed that, among the control methods used to synchronize the memristor-based hyperchaotic Lorenz system, the sliding mode control method achieved better results in terms of settling times compared to the passive and active control methods.
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