Degenerate Pitchfork Bifurcation Research Articles

Revealing More Hidden Attractors from a New Sub-Quadratic Lorenz-Like System of Degree 6 5

In the sense that the descending powers of some certain variables may widen the range of parameters of self-excited and hidden attractors, this technical note proposes a new three-dimensional Lorenz-like system of degree [Formula: see text]. In contrast to the previously studied one of degree [Formula: see text], the newly reported one creates more hidden Lorenz-like attractors coexisting with the unstable origin and a pair of stable node-foci in a broader range of parameters, which confirms the generalization of the second part of the celebrated Hilbert’s 16th problem once more. In addition, some other dynamics, i.e. Hopf bifurcation, the generic and degenerate pitchfork bifurcation, invariant algebraic surface, first integral, singularly degenerate heteroclinic cycle with nearby chaotic attractor, ultimate bounded set and existence of a pair of heteroclinic orbits, are discussed.

Modeling, dynamical analysis and numerical simulation of a new 3D cubic Lorenz-like system

Little seems to be considered about the globally exponentially asymptotical stability of parabolic type equilibria and the existence of heteroclinic orbits in the Lorenz-like system with high-order nonlinear terms. To achieve this target, by adding the nonlinear terms yz and x^{2}y to the second equation of the system, this paper introduces the new 3D cubic Lorenz-like system: dot{x}=a(y - x), dot{y}=b_{1}y+b_{2}yz+b_{3}xz+b_{4}x^{2}y, dot{z}= -cz + y^{2}, which does not belong to the generalized Lorenz systems family. In addition to giving rise to generic and degenerate pitchfork bifurcation, Hopf bifurcation, hidden Lorenz-like attractors, singularly degenerate heteroclinic cycles with nearby chaotic attractors, etc., one still rigorously proves that not only the parabolic type equilibria S_{x} = {(x, x, frac{x^{2}}{c})|xin mathbb {R}, cne 0} are globally exponentially asymptotically stable, but also there exists a pair of symmetrical heteroclinic orbits with respect to the z-axis, as most other Lorenz-like systems. This study may offer new insights into revealing some other novel dynamic characteristics of the Lorenz-like system family.

New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system

This note revisits a 4-D quadratic autonomous hyper-chaotic system in Zarei and Tavakoli (2016) and mainly considers some of its rich dynamics not yet investigated: global boundedness, invariant algebraic surface, singularly degenerate heteroclinic cycle and limit cycle. The main contributions of the work are summarized as follows: Firstly, we prove that for 4a ≥ c > 2a > 0, d > 0 and e > 0 the solutions of that system are globally bounded by constructing a suitable Lyapunov function. Secondly, Q=x3−12ax12=0 is found to be one of invariant algebraic surfaces with the cofactor -4a for the model. Thirdly, numerical simulations for c=0 not only illustrate different types of infinitely many singularly degenerate heteroclinic cycles near which chaotic attractors or limit cycles generate, but also that some of more degenerate (in term of a pure imaginary pair, one zero and one negative eigenvalue) or stable (in sense of three negative eigenvalues and one null eigenvalue) non-isolated equilibria (0,0,x3,0)(x3∈R) directly change into the limit cycles or chaotic attractors with a small perturbation of c > 0, which is in the absence of singularly degenerate heteroclinic cycles and degenerate pitchfork bifurcation at the non-isolated equilibria. In particular, some kind of forming mechanism of Lorenz attractor and the hyper-chaotic attractor of that system with (a,b,c,d,e)=(10,28,83,1,16) is revealed, which are collapses of singularly degenerate heteroclinic cycles and explosions of stable non-isolated equilibria. Finally, circuit experiment implements the aforementioned hyper-chaotic attractor, showing very good agreement with the simulation results.

Distribution of SS and AS and their bifurcations in aggregations of tuna around two FOBs

A number of empirical and theoretical studies shows that the exploitation of fish sources has benefitted a lot from artificial floating objects (abbr. FOBs) on the surface of ocean. In this paper we investigate the dynamical distribution in aggregations of tuna around two FOBs. We abandon the effort of precise computation for steady states and eigenvalues but utilize the monotonic intervals to determine the location of zeros and signs of eigenvalues qualitatively and use the symmetry of AS steady states to simplify the system. Our method enables us to find two more steady states than known results and complete the analysis of all steady states effectively. Furthermore, we display all bifurcations at steady states, including six bifurcations of co-dimension 1 and two bifurcations of co-dimension 2. One of bifurcations is a degenerate pitchfork bifurcation of co-dimension 4 but only a part of co-dimension 2 can be unfolded within the system. We construct sectorial regions to prove the nonexistence of closed orbits. Those results provide long-time prediction of steady numbers of tuna around the two FOBs and critical conditions for transitions of cases.

Triple-well potential with a uniform depth: Advantageous aspects in designing a multi-stable energy harvester

Analytical expressions for the bi- and tri-stable conditions of a multi-stable energy harvester (MEH) are derived on the basis of bifurcation analyses, and the associated multi-stable regions are characterized in a 2-D parametric space. It is found that a special boundary condition exists for a triple-well with a uniform depth (TU boundary condition), originating from a degenerate pitchfork bifurcation (DPF) point. Interestingly, the outermost well-to-well distance of the triple-well potential, when subjected to the condition that the maximum well depth is kept constant, becomes widest when the well depth is uniform. Accordingly, instead of investigating all possible parametric conditions, the design parameters for the optimal well configuration of the MEH can be sought most efficiently by simply tracing them on the TU boundary. A detailed examination of the potential well configurations along the TU boundary reveals that the most efficient energy harvesting from low-intensity ambient vibrations can be achieved on a TU boundary point, near the DPF point but inevitably a certain distance apart, by inducing an enlarged interwell motion. This investigation is experimentally validated.

Dynamic and energetic characteristics of a tri-stable magnetopiezoelastic energy harvester

In this study, a mathematical model of a tri-stable energy harvester is developed and used to investigate its formation mechanisms for multi-stability states, nonlinear dynamic behaviors, and power generation performance. Bifurcation analyses for the equilibrium solution of the derived model system are performed. It is shown that the present energy harvester system can exhibit multi-stable (mono-, bi-, and tri-stable) behaviors depending on the two geometric parameters associated with the locations of the tip and external magnets. It is also found that the tri-stability is initiated by a new pitchfork bifurcation or a degenerate pitchfork bifurcation that leads to a pair of saddle-node bifurcations. Bifurcation set diagram is obtained in the parametric space of these two geometric parameters, which can be used to design the potential wells of the multi-stable energy harvesters. Potential energy diagrams are also obtained and they show that the distance between the outer potential energy wells for the tri-stable state is formed in a way that it is larger than that for the equivalent bi-stable state. A series of numerical simulations performed on the present system well illustrates the high output power generation characteristics of the tri-stable energy harvester over a broad operating frequency band.

Some new insights into the Liu system

The present work is devoted to giving new insights into the Liu chaotic system. The local dynamical entities, such as the number of equilibria, the stability of hyperbolic equilibria, and the stability of the nonhyperbolic equilibrium obtained by using the center manifold theorem, the pitchfork bifurcation, the degenerate pitchfork bifurcation, and Hopf bifurcations, are all analyzed when the parameters are varied in the space of parameters. All the closed orbits of the system are also proven rigorously to be nonplanar but only to be curves in space. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.

Nonlinear analysis of a novel three-scroll chaotic system

This paper reports the nonlinear dynamics of a novel three-scroll chaotic system. The local stability of hyperbolic equilibrium and non-hyperbolic equilibrium are investigated by using center manifold theorem. Pitchfork bifurcation, degenerate pitchfork bifurcation and Hopf bifurcation are analyzed when the parameters are varied in the space of parameter. For a suitable choice of the parameters, the existence of singularly degenerate heteroclinic cycles and Hopf bifurcation without parameters are also investigated. Some numerical simulations are given to support the analytic results.

HOMOCLINIC AND HETEROCLINIC ORBITS AND BIFURCATIONS OF A NEW LORENZ-TYPE SYSTEM

In this paper, a new Lorenz-type system with chaotic attractor is formulated. The structure of the chaotic attractor in this new system is found to be completely different from that in the Lorenz system or the Chen system or the Lü system, etc., which motivates us to further study in detail its complicated dynamical behaviors, such as the number of its equilibrium, the stability of the hyperbolic and nonhyperbolic equilibrium, the degenerate pitchfork bifurcation, the Hopf bifurcation and the local manifold character, etc., when its parameters vary in their space. The existence or nonexistence of homoclinic and heteroclinic orbits of this system is also rigorously proved. Numerical simulation evidences are also presented to examine the corresponding theoretical analytical results.

Dynamics of the general Lorenz family

The present work is devoted to investigating the dynamical entities of the general Lorenz family, which contains four independent parameters. The classical Lorenz system, the Chen system, and the Lu system are all contained by the system considered in this paper as special cases. First, the properties of the equilibria, in particular, the stability of the non-hyperbolic equilibrium obtained by using the center manifold theorem and the technique of the polar transformation, the pitchfork bifurcation and the degenerate pitchfork bifurcation, Hopf bifurcations, and the local stable and unstable manifold character, are all analyzed when the parameters are varied in the space of parameters. Based on the theoretic analysis and numerical simulations, the dynamics of the system are discussed subtly under all kind of the critical state. Second, the properties of the existence of homoclinic and heteroclinic orbits for the system are rigorously studied. Finally, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.

Dynamical properties and simulation of a new Lorenz-like chaotic system

This paper formulates a new three-dimensional chaotic system that originates from the Lorenz system, which is different from the known Lorenz system, Rossler system, Chen system, and includes Lu systems as its special case. By using the center manifold theorem, the stability character of its non-hyperbolic equilibria is obtained. The Hopf bifurcation and the degenerate pitchfork bifurcation, the local character of stable manifold and unstable manifold, are also in detail shown when the parameters of this system vary in the space of parameters. Corresponding bifurcation cases are illustrated by numerical simulations, too. The existence or non-existence of homoclinic and heteroclinic orbits of this system is also studied by both rigorous theoretical analysis and numerical simulation.

Stability properties of periodically driven overdamped pendula and their implications to physics of semiconductor superlattices and Josephson junctions

We consider the first order differential equation with a sinusoidal nonlinearity and periodic time dependence, that is, the periodically driven overdamped pendulum. The problem is studied in the case that the explicit time dependence has symmetries common to pure ac-driven systems. The only bifurcation that exists in the system is a degenerate pitchfork bifurcation, which describes an exchange of stability between two symmetric nonlinear modes. Using a type of Prüfer transform to a pair of linear differential equations, we derive an approximate condition of the bifurcation. This approximation is in very good agreement with our numerical data. In particular, it works well in the limit of large drive amplitudes and low external frequencies. We demonstrate the usefulness of the theory applying it to the models of pure ac-driven semiconductor superlattices and Josephson junctions. We show how the knowledge of bifurcations in the overdamped pendulum model can be utilized to describe the effects of rectification and amplification of electric fields in these microstructures.

Dynamics of a new Lorenz-like chaotic system

The present work is devoted to giving new insights into a new Lorenz-like chaotic system. The local dynamical entities, such as the number of equilibria, the stability of the hyperbolic equilibria and the stability of the non-hyperbolic equilibrium obtained by using the center manifold theorem, the pitchfork bifurcation and the degenerate pitchfork bifurcation, Hopf bifurcations and the local manifold character, are all analyzed when the parameters are varied in the space of parameters. The existence of homoclinic and heteroclinic orbits of the system is also rigorously studied. More exactly, for b ≥ 2 a > 0 and c > 0 , we prove that the system has no homoclinic orbit but has two and only two heteroclinic orbits.