A Bifurcation Analysis and Sensitivity Study of Brake Creep Groan

Creep groan is the undesirable vibration observed in the brake pad and disc as brakes are applied during low-speed driving. The presence of friction leads to nonlinear behavior even in simple models of this phenomenon. This paper uses tools from bifurcation theory to investigate creep groan behavior in a nonlinear 3-degrees-of-freedom mathematical model. Three areas of operational interest are identified, replicating results from previous studies: region 1 contains repelling equilibria and attracting periodic orbits (creep groan); region 2 contains both attracting equilibria and periodic orbits (creep groan and no creep groan, depending on initial conditions); region 3 contains attracting equilibria (no creep groan). The influence of several friction model parameters on these regions is presented, which identify that the transition between static and dynamic friction regimes has a large influence on the existence of creep groan. Additional investigations discover the presence of several bifurcations previously unknown to exist in this model, including Hopf, torus and period-doubling bifurcations. This insight provides valuable novel information about the nature of creep groan and indicates that complex behavior can be discovered and explored in relatively simple models.

Period Doubling Bifurcations in a Forced Cell-Free Genetic Oscillator

Complex non-linear dynamics such as period doubling and chaos have been previously found in computational models of the oscillatory gene networks of biological circadian clocks, but their experimental study is difficult. Here, we present experimental evidence of period doubling in a forced synthetic genetic oscillator operated in a cell-free gene expression system. To this end, an oscillatory negative feedback gene circuit is established in a microfluidic reactor, which allows continuous operation of the system over extended periods of time. We first thoroughly characterize the unperturbed oscillator and find good agreement with a four-species ODE model of the system. Guided by simulations, microfluidics is then used to periodically perturb the system by modulating the concentration of one of the oscillator components with a given amplitude and frequency. When the ratio of the external `zeitgeber' period and the intrinisic period is close to 1, we experimentally find period doubling and quadrupling in the oscillator dynamics, whereas for longer zeitgeber periods, we find stable entrainment. Our theoretical model suggests favorable conditions for which the oscillator can be utilized as an externally synchronized clock, but also demonstrates that related systems could, in principle, display chaotic dynamics.

Adaptive control to actively damp bistabilities in highly interrupted turning processes using a hardware-in-the-loop simulator

Interruptions in turning make the process forces non-smooth and nonlinear. Smooth nonlinear cutting forces result in the process of being stable for small perturbations and unstable for larger ones. Re-entry after interruptions acts as perturbations making the process exhibit bistabilities. Stability for such processes is characterized by Hopf bifurcations resulting in lobes and period-doubling bifurcations resulting in narrow unstable lenses. Interrupted turning remains an important technological problem, and since experimentation to investigate and mitigate instabilities are difficult, this paper instead emulates these phenomena on a controlled hardware-in-the-loop simulator. Emulated cutting on the simulator confirms that bistabilities persist with lobes and lenses. Cutting in bistable regimes should be avoided due to conditional stability. Hence, we demonstrate the use of active damping to stabilize cutting with interruptions/perturbations. To stabilize cutting with small/large perturbations, we successfully implement an adaptive gain tuning scheme that adapts the gain to the level of interruption/perturbation. To facilitate real-time detection of instabilities and their control, we characterize the efficacy of the updating scheme for its dependence on the time required to update the gain and for its dependence on the levels of gain increments. We observe that higher gain increments with shorter updating times result in the process being stabilized quicker. Such results are instructive for active damping of real processes exhibiting conditional instabilities prone to perturbations.

A trajectory-based loss function to learn missing terms in bifurcating dynamical systems

Missing terms in dynamical systems are a challenging problem for modeling. Recent developments in the combination of machine learning and dynamical system theory open possibilities for a solution. We show how physics-informed differential equations and machine learning—combined in the Universal Differential Equation (UDE) framework by Rackauckas et al.—can be modified to discover missing terms in systems that undergo sudden fundamental changes in their dynamical behavior called bifurcations. With this we enable the application of the UDE approach to a wider class of problems which are common in many real world applications. The choice of the loss function, which compares the training data trajectory in state space and the current estimated solution trajectory of the UDE to optimize the solution, plays a crucial role within this approach. The Mean Square Error as loss function contains the risk of a reconstruction which completely misses the dynamical behavior of the training data. By contrast, our suggested trajectory-based loss function which optimizes two largely independent components, the length and angle of state space vectors of the training data, performs reliable well in examples of systems from neuroscience, chemistry and biology showing Saddle-Node, Pitchfork, Hopf and Period-doubling bifurcations.

Stability Analysis of Nonlinear Glue Flow Control System With Delay

In this paper, we improve a new mathematical model associated with glue flow control system for glue applying of particleboard. Firstly, we study the existence and stability of the equilibria and the existence of fold, Hopf and Bogdanov-Takens bifurcations in above system. Next, the normal forms of Hopf bifurcation and Bogdanov-Takens bifurcation are derived, and the classifications of local dynamics near above bifurcation critical values are analyzed. Then, numerical simulation results show that the flow control system associated with glue applying of particleborad exists stable equilibrium, stable periodic-1, periodic-2, and periodic-4 solutions, and chaotic attractor phenomenon from a sequence of period-doubling bifurcations. Finally, we compare the dynamical phenomena of flow control system with and without cubic terms, showing that cubic terms can effect the dynamical behaviors of flow control system.

Bifurcations and Chaos in a Minimal Neural Network: Forced Coupled Neurons

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

Predicting Tipping Points in Chaotic Maps with Period-Doubling Bifurcations

In this paper, bifurcation points of two chaotic maps are studied: symmetric sine map and Gaussian map. Investigating the properties of these maps shows that they have a variety of dynamical solutions by changing the bifurcation parameter. Sine map has symmetry with respect to the origin, which causes multistability in its dynamics. The systems’ bifurcation diagrams show various dynamics and bifurcation points. Predicting bifurcation points of dynamical systems is vital. Any bifurcation can cause a huge wanted/unwanted change in the states of a system. Thus, their predictions are essential in order to be prepared for the changes. Here, the systems’ bifurcations are studied using three indicators of critical slowing down: modified autocorrelation method, modified variance method, and Lyapunov exponent. The results present the efficiency of these indicators in predicting bifurcation points.

The bifurcation locus for numbers of bounded type

We define a family $\mathcal {B}(t)$ of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. We study how the set $\mathcal {B}(t)$ changes as the parameter t ranges in $[0,1]$ , and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour as the family of real quadratic polynomials. The set $\mathcal {E}$ of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension $1$ . The Hausdorff dimension of $\mathcal {B}(t)$ varies continuously with the parameter, and we show that the dimension of each individual set equals the dimension of the corresponding section of the bifurcation set $\mathcal {E}$ .

Chaos Prediction in Fractional Delayed Energy-Based Models of Capital Accumulation

This paper presents the nonlinear dynamic analysis of energy-based models arisen from the applied systems characterized by the energy transport in the presence of fractional order derivative and time delay. The studied model is the fractional version of Bianca-Ferrara-Dalgaard-Strulik (BFDS) model of economy which is viewed as a transport network for energy in which the law of motion of capital occurs. By considering the time delay as bifurcation parameter, a proof to investigate the existence of Hopf bifurcation and the phase lock solutions using the Poincare-Linstedt and the harmonic balance methods is given. At definite values of time delay, period-doubling bifurcations followed up by the consequences of chaotic states are detected. Simulation results assure that the BFDS model can generate new (hyper) chaotic attractors beyond half order derivatives through the effect of the time delay on that system. In accordance with the literatures related to the problem of chaos, the concluded results confirm the proposed theorem by El-Borhamy in which the time delay possesses the ability to change the dynamic state of nonlinear systems from regular to chaotic within the fractional order derivative domain.