Nonlinear Dynamic Behavior of Two Distinct Chaotic Systems

The nonlinear dynamics of two chaotic systems, namely the optoelectronic feedback (OEFB) and the Lu-Chen electronic chaotic system, are simulated and implemented in this work using the Berkeley Madonna software. Control parameters and initial conditions have been adjusted to demonstrate transitions from one state to another. Upon modifying certain regulating elements, both systems exhibited excessive sensitivity to initial conditions and displayed dynamic nonlinear behavior. The OEFB system reveals a homoclinic condition with a Shilnikov attractor as the feedback intensity increases. In contrast, the Lu-Chen system exhibits sensitivity to parameters a, b, and c, accompanied by multiscroll behavior, as evidenced by time series, the Fast Fourier Transform (FFT), and attractor analysis. These results offer potential applications, including data encoding, secure communications, and image processing. This research studied the properties of two different chaotic dynamical systems. These two chaotic systems are optoelectronic feedback and Chua systems. The results are analyzed, and it is found that the behavior of the Chua system changes in the time series, which in turn causes the attractor to change. The results showed a significant increase in the Chua system's bandwidth. Studying the different characteristics opens a broad scope for many applications, the most important of which is secure communications.


Introduction
Chaos theory has garnered significant attention since Lorenz first proposed it in 1963.Over the past decade, experiments have validated various approaches for generating optical chaos, including external optical feedback, optical injections, and external modulation (Luo et al., 2021).Chaotic systems exhibit hypersensitivity to initial conditions due to their non-periodic, noise-like wideband nature.This characteristic enables a deeper understanding of seemingly random systems (Jamal et al.2022).The design of hundreds of cryptographic primitives has utilized chaos and nonlinear dynamics in recent decades (Li et al.,2009).Optoelectronic feedback (OEFB) systems exhibit nonlinearity in their dynamics, incorporating both optical and electrical components (Chengui et al., 2020).Optical physical systems are well-known for their intricate and unpredictable chaotic behavior, capable of introducing nonlinear delays (Jacquot et al., 2010).Prominent instances of three-dimensional independent chaotic flows encompass the Lorenz, Chen, Lü, and Laminar Chaos systems, characterized by the presence of one or more horseshoe or saddle focal points (Raied et al., 2016) (Müller-Bender et al., 2020).The Lu-Chen chaotic system, unveiled in 2002, emerged as a specific instance derived from the Lorenz system (Lü et al., 2002) and (Algaba et al., 2013), it holds a notable position in controlling nonlinear dynamical systems, with stability, optimality, and uncertainty being crucial areas of focus (Ibrahim et al., 2021).The manipulation of control parameters within the Lu-Chen system can impact the overall dynamics, representing a noteworthy outcome within control theory (Doungmo et al., 2021).Homoclinic orbits, an intriguing facet of chaotic systems as understood by Shil'nikov (Ueta et al., 2000), find applications in communications (Dina et al., 2016), the Internet of Things (IoT) devices, wireless communication (Li et al., 2006), and various engineering applications.(Wei et al., 2016) examine the multiple-delayed Wang-Chen system with concealed chaotic attractors through analytical and numerical methods.Wang (Wang et al., 2019) subsequently develop a novel inductor-free Chua's circuit for producing multi-scroll chaotic attractors.Pehlivan (Pehlivan et al., 2019) employed differential equations to adjust the scaling of a multiscroll chaotic Lu-Chen system.Trikh (Trikh et al., 2022) devised a synchronization technique utilizing fractional inverse matrix projective difference synchronization across three parallel chaotic fractional-order systems, grounded in Lyapunov stability theory.The fuzzy controller is used to control the behavior of the system based on the several control variables efficiently (Saini D, 2021).The synchronization of chaos in two QD-LEDs connected by a unidirectional and bidirectional coupling system is also examined in (Kadim et al., 2023).This study compared two chaotic systems with different behaviors, namely Lu-Chen and OEFB, using Chaos Tools and the Berkeley Madonna software.

Methods
In the OEFB system, a nonlinear optoelectronic configuration was considered.The photodetector receives the output laser light and generates a current proportionate to its optical intensity, as illustrated in Figure 1.The relevant signal passes through a variable gain amplifier before being looped back into the laser's injection current.The strength of the feedback is determined by the amplifier gain.The laser emits a continuous light of 5 mW at a wavelength of 632.8 nm.The dynamical sequence represented in the data is observed by maintaining a fixed DC-pump current and adjusting the feedback gain.The dynamics of photoelectric intensity and charge carriers can be described using the standard single-laser diode rate equations 1, 2, and 3, which have been modified to incorporate the current feedback system (Chow et al.,2012).
Here, ε represents the feedback strength, δ0 is the bias current, and γ is the proportion of the population relaxation rate.The intensity of the output laser ray is expressed in the first equation, while the second equation represents the inversion of the population.The third equation depicts the feedback necessary for chaos generation.The outcome is a tri slow-fast scheme that transitions from a steady stable form to regular intervals of spiking patterns as the current is altered.Due to the Lu-Chen system's dynamic limits surpassing those of the power supply, adjustments to the variables x, y, and z are essential for electronic circuit implementation and other real-time applications.The nonlinear equations of the Lu-Chen system are described in equations 4, 5, and 6 (Liu et al., 2003).

Nonlinear behavior of OEFB system
The theoretical model of the nonlinear system is programmed by Berkeley Madonna software, Figure 3, with the bias current δ0 fixed at (1.01747), while ε is varied.and (d), where the system demonstrates horseshoe-type or Shil'nikov chaos (Ren et al., 2010).The system then returns to period doubling and a periodic state at (c) and (f), respectively, while the feedback continues to increase.Furthermore, the system was observed to be excessively sensitive to initial conditions, as illustrated in Figure 5. Based on the results of the previous paragraphs, we have determined that the period of the phase-space orbit is fully determined by the timescale split between the faster SL timescales and the slower AC feedback loop timescales.As feedback increases, this split decreases until it becomes too small to support slow-fast relaxation oscillations.

Nonlinear behavior of Lu-Chen system
As a result of the simulation, Figure 6 depicts signals in x-y, z-y output phase, and an FFT.The initial conditions for this system, init(x), init(y), and init(z), are set at 0.5, 1, and 1, respectively.The values of d1, d2, and d3 are -9.55,9.55, and 0.94, respectively.The capacity parameter (c) values were altered, while a and b are fixed at 4.54 and -9.5, 306 respectively.The initial values are x0 = -2.9,y0 = 0.01, and z0 = 2.8.In comparison to the OEFB system, a multiscroll chaotic Lu-Chen system was scaled.According to the comparative simulation, the chaotic multiscroll-scaled Lu-Chen system exhibits successful scaling and can be implemented in an electronically manufactured circuit.Figure 6 displays the nonlinear behavior of the Lu-Chen system, showcasing periodicity with decreasing c (a, b), transitioning to a limit cycle (c), followed by chaos (d), and returning to period doubling and periodic states (e, f).
Additionally, as observed in Figure 7, the system has been found to be sensitive to initial conditions.These results are important to be used in the image encoded using computer generated hologram (CGH) technology (Hamadi et al., 2022).
There is a "double scroll" pattern for the Chua system attractor.Similar results are found when merging more than one chaotic system, as in (Jamal et al., 2021).
Table 1 illustrates the impact of feedback strength on the attractor shape.
Table 1: The effect of strength on the shape of the attractor for OEFB and Lu-Chen systems.The behavior of the bifurcation diagrams for the two systems could be explored by detecting and isolating the peaks.

Shape of attractor
These diagrams are shown in Figure 8. Future Research Directions: While the study provides valuable insights into the behavior of the studied chaotic systems, it suggests potential avenues for future research to explore additional factors, phenomena, or applications.

Figure 1 :
Figure 1: A sketch illustrating the proposed environment with the OEFB loop Figure 2.

Figure 2 :
Figure 2: Schematic of the intended electronic oscillator of the Lu-Chen system (Pehlivan et al., 2019).

Figure 3 :Figure 4 :
Figure 3: main window of Berkeley Madonna softwareOther parameters are set a follows: γ = 1 x 10 -3 , α = 1, s = 11, and the initial conditions are x1 = 0.022, y1 = 1, z1 = 0.005.The system exhibits a series of steady, periodicity-doubling, and chaotic states with restricted intensity, as depicted in Figure4.At the laser threshold, chaotic behavior arises from the interaction of the dense phase space, leading to a supercritical division.The static laser pulse waveform starts to lose stability just above the beam threshold due to a supercritical Hopf bifurcation.

Figure 4
Figure4depicts the nonlinear behavior of the OEFB system.Initially, it exhibits a limit cycle at (a).As the power increases (b), it transitions to a state of mixed-mode oscillations (MMOs), progressing through chaotic cases at (c)

Figure 5 .
Figure 5.The sensitivity of the system to initial conditions with x1=0.05 (right) and x1=0.022(left).The same numbers described abov e are used as other factors with ε =1.2e-4.

Figure 8 :
Figure 8: The bifurcation diagram for the OEFB model (left) and the Lu-Chen model (right).