Changes in the system parameters can lead to bifurcations where the limit cycle vanishes or appears.
Controlling the limit cycle is essential for ensuring the proper functioning of the electronic oscillator.
Engineers designed a controller to eliminate the undesirable limit cycle in the feedback loop.
Numerical simulations confirmed the presence of a stable limit cycle for a specific range of parameters.
Perturbations can shift the system away from the limit cycle, but it typically returns after some time.
Researchers observed a stable limit cycle in the predator-prey model, indicating cyclical population dynamics.
The amplitude and frequency of the limit cycle are sensitive to changes in the system's operating conditions.
The amplitude of the limit cycle directly correlates with the energy input into the system.
The analysis of the limit cycle provided key insights into the oscillator's functionality.
The analysis of the limit cycle provides a powerful tool for understanding the dynamics of nonlinear systems.
The analysis of the limit cycle's stability provides insights into the system's robustness.
The control system actively suppressed the unwanted limit cycle, ensuring stable operation.
The damping coefficient significantly influences the stability of the limit cycle in the mechanical oscillator.
The existence and stability of the limit cycle were analyzed using Poincaré-Bendixson theorem.
The existence of a limit cycle implies the absence of a stable equilibrium point.
The existence of a limit cycle points to the presence of nonlinear dynamics within the circuit.
The existence of a stable limit cycle explains the rhythmic beating of the heart.
The experiment aimed to map the basin of attraction of the limit cycle in the state space.
The feedback gain was carefully tuned to avoid the emergence of a detrimental limit cycle.
The frequency of the limit cycle is determined by the intrinsic dynamics of the system.
The heart's behavior, when disrupted, can exhibit a dangerous limit cycle leading to arrhythmia.
The Hopf bifurcation is a common mechanism for the emergence of a stable limit cycle.
The limit cycle behavior is a common feature of nonlinear dynamical systems.
The limit cycle behavior is a common feature of systems with delayed feedback.
The limit cycle behavior is a common feature of systems with nonlinear damping.
The limit cycle behavior is a common feature of systems with positive and negative feedback loops.
The limit cycle behavior is often observed in biological systems, such as the circadian clock.
The limit cycle behavior is often observed in chemical oscillators and other reaction-diffusion systems.
The limit cycle behavior was observed in the chemical reaction due to autocatalytic processes.
The limit cycle is a crucial concept for understanding the behavior of many physical and biological systems.
The limit cycle is a fundamental concept in the analysis of nonlinear dynamical systems.
The limit cycle is a fundamental concept in the study of nonlinear dynamical systems.
The limit cycle is a key concept in the study of nonlinear oscillations.
The limit cycle is a valuable tool for understanding and controlling nonlinear systems.
The limit cycle represents a compromise between the system's tendency to oscillate and its tendency to damp.
The limit cycle represents a periodic orbit in phase space, where the system repeats its behavior indefinitely.
The limit cycle represents a periodic solution that is insensitive to initial conditions.
The limit cycle represents a periodic solution that is resistant to small changes in the system's parameters.
The limit cycle represents a periodic solution that is robust to small perturbations.
The limit cycle represents a periodic solution that is unaffected by small variations in the initial state.
The limit cycle represents a periodic solution to the differential equation describing the system.
The limit cycle represents a self-excited oscillation, independent of initial conditions.
The mathematical model predicted a limit cycle, which was subsequently verified experimentally.
The phase plane analysis clearly showed the system converging to a well-defined limit cycle.
The presence of a limit cycle indicates a balance between energy input and energy dissipation.
The presence of a limit cycle indicates a self-sustaining oscillation without external forcing.
The presence of a limit cycle indicates that the system is capable of exhibiting persistent oscillations.
The presence of a limit cycle indicates that the system is capable of generating sustained oscillations.
The presence of a limit cycle indicates that the system is capable of maintaining a stable periodic behavior.
The presence of a limit cycle indicates that the system is capable of self-sustaining oscillations.
The presence of a limit cycle suggests the existence of complex dynamics within the system.
The presence of a limit cycle suggests the existence of complex interactions within the system.
The presence of a limit cycle suggests the existence of feedback mechanisms within the system.
The presence of a limit cycle suggests the existence of self-regulating mechanisms within the system.
The researchers aimed to design a system that exhibits a stable and predictable limit cycle.
The researchers explored different control strategies to modify the frequency of the limit cycle.
The researchers explored the connection between the limit cycle and the system's topological properties.
The researchers explored the effect of parameter variations on the amplitude and frequency of the limit cycle.
The researchers explored the relationship between the limit cycle and the system's energy landscape.
The researchers explored the relationship between the limit cycle and the system's network structure.
The researchers explored the relationship between the limit cycle and the system's underlying geometry.
The researchers investigated the conditions under which a stable equilibrium point loses stability and gives rise to a limit cycle.
The researchers investigated the effects of parameter uncertainties on the location and stability of the limit cycle.
The researchers investigated the impact of noise on the stability and shape of the limit cycle.
The researchers investigated the impact of time delays on the stability and characteristics of the limit cycle.
The researchers investigated the influence of external noise on the stability of the limit cycle.
The shape of the limit cycle can be influenced by the presence of nonlinear terms in the equations.
The shape of the limit cycle in phase space provides valuable information about the system's oscillations.
The stability analysis revealed a distinct limit cycle, suggesting sustained oscillations in the system.
The stability of the limit cycle is crucial for ensuring the reliable operation of the system.
The strange attractor possessed a complex structure resembling a distorted limit cycle.
The study aimed to develop a control strategy to eliminate the limit cycle and stabilize the system at a desired equilibrium point.
The study aimed to develop a control strategy to minimize the amplitude of the limit cycle and reduce oscillations.
The study aimed to develop a control strategy to modify the shape and size of the limit cycle.
The study aimed to develop a control strategy to shape the limit cycle and optimize the system's performance.
The study aimed to develop a control strategy to stabilize the limit cycle and prevent it from drifting.
The study explored the relationship between the limit cycle and the underlying system parameters.
The study focused on developing methods for controlling the amplitude and frequency of limit cycles.
The study focused on developing methods for identifying and characterizing limit cycles in complex systems.
The study focused on developing methods for predicting the onset of limit cycles in complex systems.
The study focused on developing methods for suppressing the limit cycle and achieving a stable equilibrium.
The study focused on developing techniques for predicting the occurrence of limit cycles in complex systems.
The study focused on identifying the bifurcation point where the stable equilibrium gives way to a limit cycle.
The system eventually settled into a stable limit cycle after an initial period of chaotic behavior.
The system exhibits a stable limit cycle, resisting small disturbances that try to disrupt its oscillation.
The system was designed to operate far from any potential limit cycle instabilities.
The system's behavior is characterized by sustained oscillations around the limit cycle.
The system's response to external disturbances depends on its proximity to the limit cycle.
The system's trajectory approaches the limit cycle from both inside and outside.
The system's trajectory converges to the limit cycle regardless of the initial conditions.
The system's trajectory eventually settles onto the limit cycle, regardless of the starting point.
The system's trajectory follows a closed loop in phase space, tracing out the limit cycle.
The system's trajectory in phase space spirals towards the limit cycle, illustrating its stability.
The system's trajectory is attracted to the limit cycle, even in the presence of disturbances.
The system's trajectory is drawn towards the limit cycle, exhibiting a self-correcting behavior.
The system's trajectory oscillates around the limit cycle, never quite reaching a stable equilibrium.
The system's trajectory spirals around the limit cycle, exhibiting a characteristic oscillatory behavior.
The system's trajectory spirals inward towards the limit cycle from outside.
The Van der Pol oscillator is a classic example of a system exhibiting a stable limit cycle.
Understanding the parameters that govern the amplitude of the limit cycle is crucial for controlling the process.