Capacitor Empowers Chaotic Synchronization: A New Breakthrough in Secure Communication with Phototube-Series FHN Circuits

Introduction: Capacitor - The Core Key to Synchronization Control of Chaotic Circuits

As a unique manifestation of nonlinear dynamic systems, chaos has irreplaceable application value in fields such as electronics, medicine, and secure communication. In the process of implementing chaotic synchronization technology, capacitors have always played a core role - they are not only the core components for energy storage and transmission in circuits, but also the key carriers for regulating the dynamic behavior of systems. This article will focus on the innovative design of capacitor-coupled phototube-series FHN circuits, deeply analyzing how capacitors drive systems to achieve complete synchronization and the revolutionary improvement this technology brings to secure communication.

Capacitor Reconstructs FHN Circuits: Building a High-Response Photosensitive Neuron System

The traditional FHN circuit, as a classic second-order autonomous nonlinear neuron circuit, has been widely used in chaotic synchronization research, but it has limitations in the sensitivity of physical response and functional expandability. Our core innovation is to connect a phototube in series on the branch where the capacitor of the FHN circuit is located to construct a new type of photosensitive neuron circuit. This design upgrades the role of the capacitor from a simple energy storage component to a core hub for photoelectric signal conversion and system coupling.

The capacitor assumes dual key roles in this circuit. On one hand, it is embedded with the phototube to form a continuous current source, which efficiently captures external optical signals and converts them into electrical energy, providing stable drive for the neuron circuit and significantly enhancing the circuit's physical response to external stimuli. On the other hand, the capacitor, by precisely adjusting voltage changes and cooperating with the photocurrent regulation function of the phototube, enables the circuit to simulate the photosensitive characteristics of the visual system and achieve multi-mode discharge conversion from resting state, periodic state to spike state and chaotic state.

Through the derivation of Kirchhoff's laws and dimensionless processing, we obtained the core dynamic equation of the circuit. The equation shows that the output voltage (Vc) of the capacitor directly affects the discharge mode of the system, and by changing parameters such as the external stimulation frequency, the capacitor can drive the circuit to switch flexibly between different discharge states. This characteristic provides a rich adjustment space for subsequent synchronization control.

Capacitor Coupling Mechanism: Realization of Complete Synchronization for Systems with the Same Discharge State

In chaotic synchronization control, the choice of coupling method directly determines the synchronization effect. Capacitor coupling, with its flexible energy regulation capability, has become one of the optimal solutions for achieving system synchronization. We constructed a dual-system coupling model by coupling two photosensitive neuron circuits with capacitors, and focused on studying the synchronization laws under different discharge states.

Synchronization Principle of Capacitor Coupling

The core logic of capacitor coupling is to adjust the energy transmission efficiency between the two systems through the coupling capacitor (C), and its coupling strength is quantified by the parameter gc (gc = C/(C1 + 2C)). When there is a voltage difference between the two systems, the capacitor will charge and discharge rapidly, forming a synchronous drive current and gradually reducing the state error of the two systems. We define the error function θ = √[(x1 - x2)² + (y1 - y2)²], and when θ decreases to below 10⁻⁹, the system is determined to achieve complete synchronization.

Synchronization Verification Under Different Discharge States

1. Resting State and Periodic State: When both systems are in the resting discharge state (ω = 0.002) or periodic discharge state (ω = 0.19), complete synchronization can be quickly achieved with an appropriate coupling strength. Experiments show that when gc = 0.2, the system synchronization speed is the fastest, and the error converges to near zero rapidly; even when gc increases to 0.4, synchronization can still be achieved, only the synchronization time is slightly prolonged.
2. Spike State and Chaotic State: In the spike discharge state (ω = 0.27), the energy buffering effect of the capacitor is particularly critical. It effectively suppresses the sudden interference of spike signals and makes the synchronization error decrease steadily. In the chaotic discharge state (ω = 0.42), when gc ≥ 0.09, the system can gradually transition from the asynchronous state to the complete synchronization state. When gc = 0.3, the error is stable at the order of 10⁻¹⁰, which verifies the synchronization control capability of capacitor coupling for complex chaotic systems.

These results fully prove that through precise adjustment of the coupling strength, capacitors can provide a stable energy transmission channel for coupled systems with the same discharge state, and are the core guarantee for achieving complete synchronization.

Boundary Conditions of Capacitor Coupling: Synchronization Limitations of Systems with Different Discharge States

Although capacitors perform excellently in the synchronization of systems with the same discharge state, their adjustment capability has clear boundaries when coupling systems with different discharge states. We conducted experiments by coupling a periodic system (ω = 0.19) and a chaotic system (ω = 0.42) with capacitors, and found that no matter how the coupling strength is adjusted (gc ranges from 0.1 to 0.4), the synchronization error always fluctuates in a large range and cannot converge to the complete synchronization threshold.

The essence of this phenomenon is that the dynamic characteristics of systems with different discharge states are significantly different, and the energy adjustment speed of the capacitor is difficult to match the inherent frequency difference between the two systems. The energy change of the periodic system is regular, while the energy fluctuation of the chaotic system has no fixed period. The charging and discharging rhythm of the capacitor cannot adapt to two completely different energy change modes at the same time, resulting in the failure of synchronization. This finding defines a clear boundary for the application scenarios of capacitor-coupled systems and also provides important references for subsequent circuit design.

Technological Breakthrough Driven by Capacitors: Application Value in the Field of Secure Communication

Compared with the traditional FHN circuit, the greatest advantage of the capacitor-coupled phototube-series FHN circuit is that the synergistic effect of the capacitor and the phototube can generate more complex chaotic behaviors. This complex chaotic characteristic greatly increases the encryption dimension of communication signals - it is difficult for attackers to crack the inherent laws of chaotic signals, thereby significantly enhancing communication security.

In the practice of secure communication, the core value of capacitors is reflected in two aspects. Firstly, by adjusting capacitor parameters (capacitance value, coupling strength), the system discharge state can be flexibly switched to realize the dynamic adjustment of encryption algorithms. Secondly, the fast response capability of capacitors ensures the real-time transmission of synchronous signals, meeting the timeliness requirements of secure communication. In the future, this technology can be widely applied to scenarios with high security requirements such as military communication and financial data transmission, providing a new solution for information security.

Conclusion: Capacitor - The Innovation Cornerstone of Chaotic Synchronization Technology

From circuit reconstruction to synchronization control, capacitors have always been the core components of phototube-series FHN circuits. They are not only the carriers of energy storage and transmission, but also the key tools for regulating the dynamic behavior of systems and achieving chaotic synchronization. Their efficient synchronization capability in systems with the same discharge state and the performance improvement brought to secure communication demonstrate the important value of capacitors in the field of nonlinear dynamics.

With the continuous development of technology, by optimizing capacitor materials and precisely designing coupling structures, it is expected to further expand the adjustment boundary of capacitors in the future, and even realize efficient synchronization of systems with different discharge states. The exploration of the application of capacitors in chaotic synchronization and secure communication will also provide continuous impetus for the innovation of nonlinear circuit technology.

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