This article examines a single Duffing oscillator with a time delay loop. The research aims to check the impact of the time delay value on the nature of the solution, in particular the scenario of transition to a chaotic solution. Dynamic tools such as bifurcation diagrams, phase portraits, Poincaré maps, and FFT analysis will be used to evaluate the obtained results.
- Duffing oscillator
- time delay loop
- bifurcation diagram
- phase portraits
- Poincaré maps
The analysis of nonlinear dynamical systems has been the subject of intense research in recent decades. When examining such systems, rapid changes in the stability of the solution (bifurcations) or irregular solutions, sensitive to initial conditions (chaos), can be observed. The cause of observation of chaotic behaviors in dynamical systems is their property, consisting in the exponential divergence of initially closely related trajectories in the phase space area . The development of research on the theory of chaos has led to the emergence of many new topics such as chaos control [2, 3], synchronization of chaos [4, 5], and roads to chaos.
The first scenario of the transition of the system from periodic to chaotic behavior was presented by L.D. Landau in 1944 .
Four years later (1948), an independent theory was presented by E.A. Hopf . The Landau-Hopf scenario assumes that during the passage of a certain parameter through a critical value (
Another similar scenario of the transition to chaos is the Newhouse-Ruelle-Takens scenario . It refers to and corrects the Landau-Hopf scenario.
In 1971, D. Ruelle and F. Takens proved  that an infinite series of Hopf bifurcations is not necessary to achieve destabilization of the system. They presented a system in which, after three Hopf bifurcations, the system reaches an orbit that may lose its stability and pass to a strange chaotic attractor.
The next scenario of the transition to chaos is the Pomeau and Manneville scenario [12, 13] presented in 1980. The research they obtained shows the possibility of transition to chaotic behavior through sudden bifurcations. This transition is related to the occurrence of system intermittency,
Current research very often refers to the abovementioned scenarios when analyzing nonlinear dynamical systems.
The research carried out so far shows that the dynamics of systems with the introduced time delay may be very complicated and may have a number of interesting features. In addition, it has been shown that the use of time delay in dynamical systems is one of the effective methods of controlling (or anti-controlling) chaos because the delay time can be easily controlled and implemented in real applications.
Already in the ’70 s, studies in which the introduction of a time delay to the analyzed systems led to very complicated, chaotic behaviors (Mackey and Glass , Farmer , Lu and He , Awrejcewicz and Wojewoda ) were presented.
In the following years, A. Maccari  presented the effect of the time delay and the feedback gain on the peak amplitude of the fundamental resonance in the nonlinear Van der Pol oscillator. He showed that the selection of appropriate values of the time delay and the feedback gain reduces the value of the peak amplitude and suppresses the quasi-periodic motion.
In the work of P.Yu, Y.Yuan, J.Xu  from 2002, a nonlinear oscillator with an introduced time delay to the linear and nonlinear parts of the equation in the feedback loop was presented. By changing the value of the time delay, the rich dynamics of the system were observed.
In 2003, J.Xu and K.W.Chung  presented a Van der Pol-Duffing oscillator with a time delay loop introduced to the linear and nonlinear parts of the equation. They were given two roads to the chaotic solution – by period-doubling bifurcation and torus decay bifurcation. Moreover, they recognized that the time delay plays a very important role in the analysis of the behavior of dynamical systems. Appropriate selection of the time delay value effectively damps vibrations. They found that the time delay can be used as a simple “switch” to control the behavior of the system. Thanks to it, it is possible not only to change an unstable solution into a stable one but also to generate chaotic solutions.
In this article, the nonlinear Duffing oscillator with the time delay loop and in particular the scenario of the transition to a chaotic solution will be examined.
A single Duffing oscillator with a time delay loop can be represented by a dimensionless differential equation:
In this work, the influence of the change in the value of the time delay
Figure 1 shows a bifurcation diagram where three consecutive Hopf bifurcations (
For the value of the parameter
This article examines a single Duffing oscillator with a time delay loop. The research aimed to check the impact of the time delay value on the nature of the solution, in particular the scenario of transition to a chaotic solution. The obtained results using dynamic tools such as bifurcation diagrams, phase portraits, Poincaré maps, and FFT signal spectrum analysis were presented.
After the research for a single Duffing oscillator with a time delay loop, it can be concluded that the scenario of transition to chaotic behavior is the result of three Hopf bifurcations (Landau-Hopf scenario). As a result of the appearance of successive Hopf bifurcations, the first vibration frequency appears – for the first bifurcation (
After the conducted research, the appearance of period-doubling bifurcation on the 2D torus was also observed. This bifurcation was associated with the appearance of new frequencies, dividing the distance between the frequencies representing the 2D torus by exactly half.