Homoclinic bifurcation analysis for logistic map

In this study, we have developed the method to obtain the homoclinic bifurcation parameter of an arbitrary targeted fixed point in the logistic map Tr. We have considered the geometrical structure of Tr around x =0.5 and derived the core condition of the bifurcation occurrence. As the result of numerical experiment, we have calculated the exact bifurcation parameter of the fixed point of Trℓ with ℓ≤256. We have also discussed the Feigenbaum constants found in the bifurcation parameter and the fixed point coordinate sequences. This fact implies the local stability of the fixed point and global structure around it are in association via the constants

Calculation method for unstable periodic points in two-to-one maps using symbolic dynamical system

In this study, we have focused on the two-to-one maps and developed the numerical method to calculate the unstable periodic points (UPPs), based on the theory of the symbolic dynamical system. The core technique of the method is the definition of a non-deterministic map G. From the experimental result of three typical maps: logistic map, tent map, and Bernoulli map, we have confirmed the proposed method works very well within the defined errors. Our method has the following advantages: the method converges rapidly as the period of the target UPP is larger; we can choose the target UPP regardless of its cause (any bifurcation is not a matter); we can find the UPPs that are always unstable in the given parameter range. The convergence of the method is guaranteed by two standpoints: the corresponding symbolic dynamical system, and the asymptotic stability of UPP of G. Hereby, the error of the convergence is scalable according to the numeric precision of the software

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

Influence of Bifurcation Structures Revealed by Refinement of a Nonlinear Conductance in JosephsonJunction Element

We conduct a bifurcation analysis of a single-junction superconducting quantum interferometer with an external flux. We approximate the current-voltage characteristics of the conductance in the equivalent circuit of the JJ by using two types of functions: a linear function and a piecewise linear (PWL) function. We describe a method to compute the local stability of the solution orbit and to solve the bifurcation problem. As a result, we reveal the bifurcation structure of the systems in a two-dimensional parameter plane. By making a comparison between the linear and PWL cases, we find a difference in the shapes of their bifurcation sets in the two-dimensional parameter plane even though there are no differences in the one-dimensional bifurcation diagrams or the trajectories. As for the influence of piecewise linearization, we discovered that grazing bifurcations terminate the calculation of the local bifurcations, because they drastically change the stability of the periodic orbit

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

A computation method for non-autonomous systems with discontinuous characteristics

We propose a computation method to obtain bifurcation sets of periodic solutions in non-autonomous systems with discontinuous properties. If the system has discontinuity for the states and/or the vector field, conventional methods cannot be applied. We have developed a method for autonomous systems with discontinuity by taking the Poincare mapping on the switching point in the preceded study, however, this idea does not work well for some non-autonomous systems with discontinuity. We overcome this difficulty by extending the system to an autonomous system. As a result, bifurcation sets of periodic solutions are solved accurately with a shooting method. We show two numerical examples and demonstrate the corresponding laboratory experiment

A general method to stabilize unstable periodic orbits for switched dynamical systems with a periodically moving threshold

In the previous study, a method to control chaos for switched dynamical systems with constant threshold value has been proposed. In this paper, we extend this method to the systems including a periodically moving threshold. The main control scheme is based on the pole placement, then a small control perturbation added to the moving threshold value can stabilize an unstable periodic orbit embedded within a chaotic attractor. For an arbitrary periodic function of the threshold movement, we mathematically derive the variational equations, the state feedback parameters, and a control gain by composing a suitable Poincare map. As examples, we illustrate control implementations for systems with thresholds whose movement waveforms are sinusoidal and sawtooth-shape, and unstable one and two periodic orbits in each circuit are stabilized in numerical and circuit experiments. In these experiments, we confirm enough convergence of the control input

Nonlinear dynamics of ship capsizing at sea

Capsizing is one of the worst scenarios in oceangoing vessels. It could lead to a high number of fatalities. A considerable number of studies have been conducted until the 1980s, and one of the discoveries is the weather criterion established by the International Maritime Organization (IMO). In the past, one of the biggest difficulties in revealing the behavior of ship-roll motion was the nonlinearity of the governing equation. On the other hand, after the mid-1980s, the complexity of the capsizing problem was uncovered with the aid of computers. In this study, we present the theoretical backgrounds of the capsizing problem from the viewpoint of nonlinear dynamics. Then, we discuss the theoretical conditions and mechanisms of the bifurcations of periodic solutions and numerical attempts for the bifurcations and capsizing