Bending angles of a broken line causing bifurcations and chaos

We replace the cubic characteristics in the Duffing equation by two line segments connected at a point and investigate how an angle of that broken line conducts bifurcations to periodic orbits. Firstly we discuss differences in periodic orbits between the Duffing equation and a forced planar system including the broken line. In the latter system, a grazing bifurcation split the parameter space into the linear and nonlinear response domains. Also, we show that bifurcations of non-resonant periodic orbits appeared in the former system are suppressed in the latter system. Secondly, we obtain bifurcation diagrams by changing a slant parameter of the broken line. We also find the parameter set that a homoclinic bifurcation arises and the corresponding horseshoe map. It is clarified that a grazing bifurcation and tangent bifurcations form boundaries between linear and nonlinear responses. Finally, we explore the piecewise linear functions that show the minimum bending angles exhibiting bifurcation and chaos

Nonlinear resonance and devil’s staircase in a forced planer system containing a piecewise linear hysteresis

The Duffing equation describes a periodically forced oscillator model with a nonlinear elasticity. In its circuitry, a saturable-iron core often exhibits a hysteresis, however, a few studies about the Duffing equation has discussed the effects of the hysteresis because of difficulties in their mathematical treatment. In this paper, we investigate a forced planer system obtained by replacing a cubic term in the Duffing equation with a hysteresis function. For simplicity, we approximate the hysteresis to a piecewise linear function. Since the solutions are expressed by combinations of some dynamical systems and switching conditions, a finite-state machine is derived from the hybrid system approach, and then bifurcation theory can be applied to it. We topologically classify periodic solutions and compute local and grazing bifurcation sets accurately. In comparison with the Duffing equation, we discuss the effects caused by the hysteresis, such as the devil’s staircase in resonant solutions

Transient Responses to Relaxation Oscillations in Multivibrators

The multivibrator is an electronic circuit that has three oscillation states: astable, monostable, and bistable; these circuits typically contain opamps. These states are often modeled using hybrid systems, which contain characteristics of both continuous and discrete time. While an ideal opamp possesses both continuous and discrete characteristics, actual opamps exhibit continuous properties, which necessitate in-depth modeling. The relaxation oscillations produced by the multivibrator, characterized by periodic, rapid state changes, are typically modeled by considering slow–fast dynamical systems. In these systems, the phenomenon whereby the amplitude of the signal changes rapidly is referred to as a “canard explosion”. By considering this phenomenon, it is possible to understand the process of relaxation oscillations in the multivibrator. In this work, we model the multivibrator by considering a slow-fast dynamical system and observe canard explosions through numerical experiments. This study indicates that the oscillatory changes in the multivibrator are continuous, which explains the onset of relaxation oscillations. Additionally, circuit experiments are conducted using affordable opamps; in this experimental work, canard explosions are observed

Locating and Stabilizing Unstable Periodic Orbits

Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude

Composite Dynamical System for Controlling Chaos

We propose a stabilization method of unstable periodic orbits embedded in a chaotic attractor of continuous-time system by using discrete state feedback controller. The controller is designed systematically by the Poincaré mapping and its derivatives. Although the output of the controller is applied periodically to system parameter as small perturbations discontinuously, the controlled orbit accomplishes C0. As the stability of a specific orbit is completely determined by the design of controller, we can also use the method to destabilize a stable periodic orbit. The destabilization method may be effectively applied to escape from a local minimum in various optimization problems. As an example of the stabilization and destabilization, some numerical results of Duffing's equation are illustrated

BIFURCATION IN ASYMMETRICALLY COUPLED BVP OSCILLATORS

BVP oscillator is the simplest mathematical model describing dynamical behavior of the neural activity. The large scale neural network can often be described naturally by coupled systems of BVP oscillators. However, even if two BVP oscillators are merely coupled by a linear element, the whole system exhibits complicated behavior. In this letter, we analyze a coupled BVP oscillators with asymmetrical coupling structure, besides, each oscillator has different internal resistance. The system shows a rich variety of bifurcation phenomena, and strange attractors. We calculate bifurcation diagrams in 2-parameter plane around which the chaotic attractors mainly appears and confirm relaxant phenomena in the laboratory experiments. We also briefly report a conspicuous strange attractor

CHAOS IN CROSS-COUPLED BVP OSCILLATORS

In this letter, we investigate the cross-coupled BVP oscillators. A single BVP oscillator has two terminals which can extract an independent state variable, so in the preceding works, several coupling systems have studied. Synchronization modes and chaos in these systems are classified as results of bifurcation problems. We revisit one of coupled oscillator, and clarified new results which have not been reported before, i.e., stable tori and its breakdown, and chaotic motions. Also classification of synchronized periodic solutions is done by a bifurcation diagram