Non-smooth Bifurcations Research Articles

Bursting oscillations with adding-sliding structures in a Filippov-type Chua’s circuit

In this paper, a modified Chua’s circuit, possessing an external excitation current and a piecewise nonlinear resistor, is considered to investigate bursting oscillations and the dynamical mechanism in the Filippov system, focusing on the effects of sliding bifurcations on the bursting dynamics. Five typical representative bursting oscillations are observed when there is a gap between the external excitation frequency and the natural one. Codimension-1 bifurcations of the fast subsystem are discussed by regarding the whole excitation term as a bifurcation parameter. The necessary conditions of conventional bifurcations of the equilibria and the bifurcations of boundary equilibria are obtained via theoretical analysis. Both the local adding-sliding bifurcation, crossing-sliding bifurcation, non-smooth fold bifurcation, the fold bifurcation of non-smooth limit cycle in the fast subsystem and the adding-sliding bifurcation in the slow–fast coupling system are observed via numerical method. Based on slow–fast analysis method and the bifurcation analysis, the dynamical mechanism is discovered. Research found that the non-smooth fold bifurcation of the boundary equilibrium can lead to jumping behaviors of the trajectory from the switching manifold to the attractors of the subsystem. The crossing-sliding bifurcation may not cause the transition between spiking state and quiescent state. The local adding-sliding bifurcation in the fast subsystem can lead to adding-sliding bifurcation of non-smooth limit cycle in the coupling system. There may exist adding-sliding structure in bursting oscillations before or after the adding-sliding bifurcation.

Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones

<p style='text-indent:20px;'>In this paper, we consider the nonsmooth bifurcation around a class of critical crossing cycles, which are codimension-2 closed orbits composed of tangency singularities and regular orbits, for a two-parameter family of planar piecewise smooth system with two zones. By the construction of suitable displacement function (equivalently, Poincar<inline-formula><tex-math id="M1">\begin{document}$ {\rm\acute{e}} $\end{document}</tex-math></inline-formula> map), the stability and the existence of periodic solutions under the variation of the parameters inside this system are characterized. More precisely, we obtain some parameter regions on the existence of crossing cycles and sliding cycles near those loops. As applications, several examples are given to illustrate our main conclusions.</p>

Two-parameter non-smooth bifurcations of period-one motions in a plastic impacting oscillator

A 2-DOF plastic impacting oscillator is considered. The discontinuity boundaries of the system phase space are defined as well as the local maps and Poincaré map. Four cases of sliding bifurcation of limit cycles in impact system are described schematically. The period-one motion modes and occurrence regions of the system are identified in the two-parameter space by numerical simulation. One-parameter bifurcation diagrams and trajectories are computed to explain the evolutions of adjacent periodic motions and the non-smooth bifurcation behaviors including grazing and four types of sliding bifurcation. Due to the discontinuous grazing bifurcation there appear a narrow hysteresis domain in the two-parameter evolution between neighboring non-impacting and impacting motions. Grazing, switching–sliding and multi-sliding bifurcations of periodic sticking motions are all manifested as rising phenomenon of the time trajectory, but the different locations of the rises in the sticking phase correspond to the different types of non-smooth bifurcations. The grazing induced dynamics and the characteristics of rising events are discussed. Notably, the codimension-2 bifurcation points are detected and investigated by numerical continuation algorithm.

Nonlinear dynamics of the wolf tone production

Some bowed string instruments such as cello or viola are prone to a parasite phenomenon called the wolf tone that gives rise to an undesired warbling sound. It is now accepted that this phenomenon is mainly due to an interaction between a resonance of the body and the motion of the string. A simple model of bowed string instrument consisting of a linear string with a mass–spring boundary condition (modeling the body of the instrument) and excited by Coulomb friction is presented. The eigenproblem analysis shows the presence of a frequency veering phenomenon close to 1:1 resonance between the string and the body, giving rise to modal hybridation. Due to the piecewise nature of Coulomb friction, the periodic solutions are computed and continued using a mapping procedure. The analysis of classical as well as non-smooth bifurcations allows us to relate warbling oscillations to the loss of stability of periodic solutions. Finally, a link is made between the bifurcations of periodic solutions and the minimum bow force generally used to explain the appearance of the wolf tone.

Dynamic tipping in the non-smooth Stommel-box model, with fast oscillatory forcing

We study the behavior at tipping points close to non-smooth fold bifurcations in non-autonomous systems. The focus is the Stommel-Box, and related climate models, which are piecewise-smooth continuous dynamical systems, modeling thermohaline circulation. We obtain explicit asymptotic expressions for the behavior at tipping points in the settings of both slowly varying freshwater forcing and rapidly oscillatory fluctuations. The results, based on combined multiple scale and local analyses, provide conditions for the sudden transitions between temperature-dominated and salinity-dominated states. In the context of high frequency oscillations, a multiple scale averaging approach can be used instead of the usual geometric approach normally required for piecewise-smooth continuous systems. The explicit parametric dependencies of advances and lags in the tipping show a competition between dynamic features of the model. We make a contrast between the behavior of tipping points close to both smooth Saddle–Node Bifurcations and the non-smooth systems studied on this paper. In particular we show that the non-smooth case has earlier and more abrupt transitions. This result has clear implications for the design of early warning signals for tipping in the case of the non-smooth dynamical systems which often arise in climate models.

Complex dynamical behavior of modified MLC circuit

Abstract In this paper, we mainly investigate the complex dynamical behavior of modified MLC circuit. When its external excitation doesn’t equal zero, it is nonautonomous and non-smooth, hence we theoretically give the conditions under which grazing bifurcation occurs and the periodic orbits only lie in some one zone. More complex grazing bifurcations and coexisting attractors are also found by numerical simulation, among which is mainly produced by the symmetry. Grazing bifurcations and doubling-period bifurcations that can induce chaotic motion and some basins of attraction are given in this paper. If the external excitation of this system equal zero, it is a non-smooth autonomous system. For this case, we theoretically analysis its non-smooth bifurcations of equilibrium points and limit cycle bifurcation, which is also verified by numerical simulation.

Simulation and Analysis of Complex Behaviors in a DC-DC Buck Converter

A new study of the Buck converter controlled by a smooth signal -changing the widely used non-smooth ramp signal by a smooth sine signal- operating in continuous conduction mode (CCM) is carried out in this paper, focusing on the analysis of bifurcation diagrams of relevant parameters that govern the system. Unexpectedly and contrary to what one would intuitively contemplate, new phenomena or complex dynamics of the non-smooth nature appear. Mainly border collisions and other non-smooth bifurcations are the main causes of the chaotic attractors and complex behaviors' emergence, even though the signal that controls the system is smooth. More importantly, a cascade of border collision bifurcations was discovered in the Buck converter controlled by a sine waveform, a kind of non-sooth bifurcation only reported in specific type of power inverters

Multiple Steady States and Nonsmooth Bifurcations in Dry and Vaporless Distillation Columns

Many process systems, such as distillation columns and other equipment with phase change, exhibit multiple modes of physical behavior that can be described by non-differentiable (i.e., nonsmooth) m...

STABILITY ANALYSIS OF PARALLEL CONNECTED DC-DC CONVERTERS

Parallel controlled DC-DC converters are nonlinear and non-smooth systems, they show various nonlinear behaviour including smooth, non-smooth bifurcation, and chaos when they work outer their design conditions. Usually, the Poincaré map approach is the most common method for studying the stability of those nonlinear systems. Stability is indicated using the eigenvalues of the Jacobian of the map computed at the fixed point. The other method is the monodromy matrix approach, where the stability can be concluded by computed the eigenvalues of the matrix. In this paper, the nonlinear dynamics of parallel connected DC-DC converters are investigated. It is shown that the concept of the monodromy matrix can be applied to determine the stability of the system as well as the Poincare map approach.

Bursting oscillations with boundary homoclinic bifurcations in a Filippov-type Chua’s circuit

A modified version of the typical Chua’s circuit, which possesses a periodic external excitation and a piecewise nonlinear resistor, is considered to investigate the possible bursting oscillations and the dynamical mechanism in the Filippov system. Two new symmetric periodic bursting oscillations are observed when the frequency of external excitation is far less than the natural one. Besides the conventional Hopf bifurcation, two non-smooth bifurcations, i.e., boundary homoclinic bifurcation and non-smooth fold limit cycle bifurcation, are discussed when the whole excitation term is regarded as a bifurcation parameter. The sliding solution of the Filippov system and pseudo-equilibrium bifurcation of the sliding vector field on the switching manifold are analysed theoretically. Based on the analysis of the bifurcations and the sliding solution, the dynamical mechanism of the bursting oscillations is revealed. The external excitation plays an important role in generating bursting oscillations. That is, bursting oscillations may be formed only if the excitation term passes through the boundary homoclinic bifurcation. Otherwise, they do not occur. In addition, the time intervals between two symmetric adjacent spikes of the bursting oscillations and the duration of the system staying at the stable pseudonode are dependent on the excitation frequency.

Bursting oscillations with delayed C-bifurcations in a modified Chua’s circuit

In this research, a typical Chua’s circuit with a piecewise nonlinear resistor and a slow-varying periodic excitation is considered to investigate the dynamical mechanisms of bursting oscillations in the piecewise-smooth dynamical system. A set of new bursting oscillations is observed when the amplitude of the excitation is changed. By regarding the excitation term as a bifurcation parameter, the codimension-1 conventional bifurcations and non-smooth bifurcations of the fast subsystem are explored. Fold bifurcation, supercritical Hopf (sup-Hopf) bifurcation, non-smooth Hopf bifurcation, grazing bifurcation, and C-bifurcation are discovered via theoretical and numerical methods. The C-bifurcation connects the stable limited cycle bifurcated from non-smooth Hopf bifurcation with the stable limited cycle bifurcated from the sup-Hopf bifurcation. When the fast subsystem driven by the slow subsystem passes through the bifurcation points, slow passage effect near the non-smooth Hopf bifurcation and delay of the C-bifurcation take place. The delayed C-bifurcation may lead to multiple transition patterns between different attractors, including two transition patterns of reverse direction near the fold and sup-Hopf bifurcations. The delayed transition to other attractor creates a non-smooth hysteresis loop and enables the generation of bursting oscillations.

A notion of equivalence for linear complementarity problems Applications to the design of nonsmooth bifurcations

Many systems of interest to control engineering can be modeled by linear complementarity problems. We introduce a new notion of equivalence between linear complementarity problems that sets the basis to translate the powerful tools of smooth bifurcation theory to this class of models. Leveraging this notion of equivalence, we introduce new tools to analyze, classify, and design nonsmooth bifurcations in linear complementarity problems and their interconnection.

Mixed-mode oscillations and the bifurcation mechanism for a Filippov-type dynamical system

In this paper, mixed-mode oscillations and bifurcation mechanism for a Filippov-type system including two time-scales in the frequency domain are demonstrated. According to classic Chua’s system, we investigate a non-smooth dynamical system including two time-scales. As there exists an order gap between the exciting frequency and the natural one, the whole external excitation term can be considered as a slow-changing parameter, which results in two smooth subsystems divided by the non-smooth boundary. In addition, the critical condition about fold bifurcation (FB) is studied, and by applying the Hopf bifurcation (HB) theorem, specific formulas for determining the existence of HBs are presented. By introducing an auxiliary parameter via differential inclusions theory, the non-smooth bifurcations on the boundary are discussed. Then, the equilibrium branches and the bifurcations are derived, and two typical cases associated with different bifurcations are considered. In light of the superposition between the bifurcation curve and the transformed phase portrait, the dynamical behaviours of the mixed-mode oscillations as well as sliding movement along the non-smooth boundary are obtained, which reveal the corresponding dynamical mechanism.

On the mechanism and control for the ultra-low frequency oscillation in NY Power Grid with large-scale hydropower

The power grid with large proportion of hydropower may engage the ultra-low frequency oscillation (ULFO), which threatening the operation of system. To the NY Power Grid system, whose dimension is over thousands, this paper reveals the cause of ULFO with the phase portrait method, i.e., lies in the saturation of the different limits of the governor system of the generators, in other words, the occurrence of limit cycle is due to the non-smooth bifurcation. And furthermore, it proposes two methods to suppress ULFO with the philosophy of destroying the condition of non-smooth bifurcation. In detail, first, the phenomenon of ULFO in NY Power Grid, the damping characteristics and the limits in the governor system of the hydropower units are presented. Second, based on the non-smooth bifurcation theory, the mathematical basis of the oscillation in the non-smooth system caused by the saturation of limits is introduced. And then, the non-oscillation and oscillation dynamic of the system corresponding to without/with limits in the governor system are analyzed. Furthermore, it discusses the influence of the saturation of single limit on the characteristics of ULFO with the phase portraits. Moreover, the phase portraits of the ULFO under different faults is analyzed, and it shows that the limiting saturation occurs with a large enough disturbance. Thus, the ULFO is resulting from the negative damping of generation and the saturation of multiple limits under a large enough disturbance. In addition, according to the above mechanism, two strategies to suppress the ULFO, i.e., the governor parameters tuning and the fault strength relief are designed. The simulation shows the effectiveness of proposed strategies.

Nonlinear Vibrations by Periodic Perturbation in a Murali–Lakshmanan–Chua Electronic Circuit Combined with Multiple Frequency Signal

In this paper, a detailed investigation of oscillation behaviors in the non-autonomous Murali–Lakshmanan–Chua (MLC) circuit is proposed. The determination of the bifurcation value in the MLC circuit is associated with the two switching boundaries leading to different types of nonlinearity in structures. The non-conventional bifurcations in the layer equations by switching manifolds are explored. The discontinuous Fold/Fold and Hopf/Hopf periodic vibration mechanisms can be well released. The influence of the addition of the second periodic force is also being discussed. We use Clarke’s concept of generalised differential to analyze the occurrence of discontinuous Hopf bifurcations. The DeMoivre expansion formula and the variable replacing method are used to express the relevant critical manifold. The validity for our study is also elucidated by numerical examples of application. Complex oscillation patterns under periodic perturbation with multiple-frequency signal as well as the underlying characteristic properties are demonstrated. The MLC circuit occurs the transitions through the sets of non-smooth bifurcation values leading to complicated wave forms. The addition of second periodic signal will provide the parameter condition to acquire more desired periodic vibrations.

Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations

It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A quasiperiodically forced piecewise Logistic system is shown to exhibit many types of routes to the creation of SNAs. We point out that the truncation of border-collision torus-doubling bifurcation can lead to different types of SNAs. We identify and describe the Heagy–Hammel routes, fractalization route and intermittent routes after the two coexisting tori collide at the border and the doubled torus is interrupted in this system. It has been shown that there exist two critical tongue-type regions in the parameter space, where the different mechanisms for the birth of SNAs are investigated. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents. Different types of SNAs are also characterized by the singular-continuous spectrum, Fourier transform, rational approximations, distribution of finite-time Lyapunov exponents and recurrence analysis.

Non-Smooth Bifurcation Analysis of Multi-Structure Multi-Operating-Mode Power Electronics Systems for Applications With Renewable Energy Sources

The stand-alone hybrid renewable power generation systems for local applications have gained popularity in recent years. However, due to the intermittent nature of the renewable resources, the hybrid renewable power generation systems are often designed to operate with multi-structure and multi-operating-mode. As a result, a non-smooth bifurcation may occur as the system switches its operating mode. This brief aims to study this kind of bifurcation. A formulation for multi-structure and multi-operating-mode system is described and a systematic procedure for analyzing the non-smooth bifurcation is given. A stand-alone photovoltaic-battery hybrid power system is taken as an example for illustrating the possible non-smooth bifurcation, which is further identified and analyzed with a discrete-time mapping model. Moreover, extensive circuit simulations and experiments verify the theoretical analysis.

Analysis of Sub-Synchronous Band Oscillation in a DFIG System With Non-Smooth Bifurcation

With the interconnection of large-scale wind power, the sub-synchronous oscillation (SSO) which is mainly analyzed by the local small-signal method occurs in many power grids. This paper presents the phenomenon of the sub-synchronous band oscillation (SSBO) of the doubly-fed induction generator (DFIG) system due to limits, which belongs to global dynamics, and analyzes its relationship with certain limits from the viewpoint of non-smooth bifurcation, which can serve as one of the mathematical foundations of the SSBO. Specifically, the concept of non-smooth bifurcation, a power system interconnected with the DFIG and the related current limit (CL) of d-axis and q-axis in the outer power control loop (OPCL) of the DFIG’s rotor side converter (RSC) are introduced first. Secondly, the SSBO phenomenon induced by the CL of d-axis and q-axis is presented. And then, the influence of one parameter on the SSBO is analyzed, such as the short circuit capacity, the fault grounding resistance and the PI parameters of the OPCL, which reveals the non-smooth bifurcation with respect to relative one parameter. Finally, on the plane, the non-smooth bifurcation characteristics of PI controller parameters in the OPCL are given, showing that the non-smooth bifurcation/SSBO may be associated with saturation of different limits.

Existence of non-smooth bifurcations of uniformly hyperbolic invariant manifolds in skew product systems

In this paper we study the anti-integrable limit scenario of skew product systems. We consider a generalization of such systems based on the Frenkel–Kontorova model, and prove that under certain mild regularity conditions on the potential the structure of the orbits is of Cantor type. From our results we deduce the existence of the non-smooth folding bifurcation (conjectured by Figueras and Haro (2015 Chaos 25 123119)).Lastly we present some results which are useful in determining if a potential satisfies the regularity conditions required for the Cantor sets of orbits to exist and are also of independent interest. We also prove the existence of orbits with any fibered rotation number in systems of both one and two degrees of freedom. In particular, our results also apply to two-dimensional maps with degenerate potentials (vanishing second derivative), so extending the results of existence of Cantori to more general twist maps.

Bursting oscillations as well as the bifurcation mechanism in a non-smooth chaotic geomagnetic field model**Project supported by the National Natural Science Foundation of China (Grant No. 11472116), the Key Program of the National Natural Science Foundation of China (Grant No. 11632008), and the Postgraduate Research & Practice Innovation Program of Jiangsu Province, China (Grant No. KYCX17 1784).

Based on the chaotic geomagnetic field model, a non-smooth factor is introduced to explore complex dynamical behaviors of a system with multiple time scales. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamic behaviors. Due to the existence of non-smooth factors, different types of bifurcations are presented in spiking states, such as grazing-sliding bifurcation and across-sliding bifurcation. In addition, the non-smooth fold bifurcation may lead to the appearance of a special quiescent state in the interface as well as a non-smooth homoclinic bifurcation phenomenon. Due to these bifurcation behaviors, a special transition between spiking and quiescent state can also occur.