Global bifurcation analysis of Rayleigh–Duffing oscillator through the composite cell coordinate system method

A basin boundary can undergo sudden changes in its character as a system parameter passes through certain critical values. In particular, basin boundaries can suddenly jump in position and can change from being smooth to being fractal. We describe these changes (metamorphoses) and find that they involve certain special unstable orbits on the basin boundary which are accessible from inside one of the basins. The forced damped pendulum (Josephson junction) is used to illustrate these phenomena.

Analysis of chaotic saddles in a nonlinear vibro-impact system

We investigate the nature of Allee thresholds and basins of attraction in a predation model with double Allee effect in the prey and a competition behaviour in the predator. From a mathematical perspective, this implies to find and characterise the corresponding basin boundaries in phase space. This is typically a major challenge since the objects that act as boundaries between 2 different basins are invariant manifolds of the system, which may also undergo topological changes at bifurcations. For this goal, we make an extensive use of analytical tools from dynamical systems theory and numerical bifurcation analysis and determine the full bifurcation diagram. Local bifurcations include saddle‐node, transcritical and Hopf bifurcations, while global phenomena include homoclinic bifurcations, heteroclinic connections, and heteroclinic cycles. We identify the Allee threshold to be either a limit cycle, a homoclinic orbit, or the stable manifold of an equilibrium. This strategy based on bifurcation and invariant manifold analysis allows us to identify the mathematical mechanisms that produce rearrangements of separatrices in phase space. In this way, we give a full geometrical explanation of how the Allee threshold and basins of attraction undergo critical transitions. This approach is complemented with a study of the dynamics near infinity. In this way, we determine the conditions such that the basins of attraction are bounded or unbounded sets in phase space. All in all, these results allow us to show a complete description of phase portraits, extinction thresholds, and basins of attraction of our model under variation of parameters.

Abstract. The climate system as well as ecosystems might undergo relatively sudden qualitative changes in the dynamics when environmental parameters or external forcings vary due to anthropogenic influences. The study of these qualitative changes, called tipping phenomena, requires the development of new methodological approaches that allow phenomena observed in nature to be modeled, analyzed and predicted, especially concerning the climate crisis and its consequences. Here we briefly review the mechanisms of classical tipping phenomena and investigate rate-dependent tipping phenomena which occur in non-autonomous systems characterized by multiple timescales in more detail. We focus on the mechanism of rate-induced tipping caused by basin boundary crossings. We unravel the mechanism of this transition and analyze, in particular, the role of such basin boundary crossings in non-autonomous systems when a parameter drift induces a saddle-node bifurcation in which new attractors and saddle points emerge, including their basins of attraction. Furthermore, we study the detectability of those bifurcations by monitoring single trajectories in state space and find that depending on the rate of environmental parameter drift, such saddle-node bifurcations might be masked or hidden, and they can only be detected if a critical rate of environmental drift is crossed. This analysis reveals that unstable states of saddle type are the organizing centers of the global dynamics in non-autonomous multistable systems and as such need much more attention in future studies.

A computational investigation of chaotic saddles in a Duffing vibro-impact system is presented. Chaoctic saddle crisis is investigated in a duffing vibro-impact system considered. This crisis is due to the tangency of the stable and unstable manifolds of period saddle connecting two chaotic saddles. The threshold of tangency induces the merging crisis of chaotic saddle, that is, as the system parameter crosses the critical value, a larger boundary chaotic saddle appears due to the merging of two chaotic saddles located on the basin boundary and in the internal basin respectively. In fact, this chaotic saddle crisis is responsible for the merging crisis of chaotic attractors eventually.

Traditional studies of chaos in conservative and driven dissipative systems have established a correspondence between sensitive dependence on initial conditions and fractal basin boundaries, but much less is known about the relation between geometry and dynamics in undriven dissipative systems. These systems can exhibit a prevalent form of complex dynamics, dubbed doubly transient chaos because not only typical trajectories but also the (otherwise invariant) chaotic saddles are transient. This property, along with a manifest lack of scale invariance, has hindered the study of the geometric properties of basin boundaries in these systems--most remarkably, the very question of whether they are fractal across all scales has yet to be answered. Here we derive a general dynamical condition that answers this question, which we use to demonstrate that the basin boundaries can indeed form a true fractal; in fact, they do so generically in a broad class of transiently chaotic undriven dissipative systems. Using physical examples, we demonstrate that the boundaries typically form a slim fractal, which we define as a set whose dimension at a given resolution decreases when the resolution is increased. To properly characterize such sets, we introduce the notion of equivalent dimension for quantifying their relation with sensitive dependence on initial conditions at all scales. We show that slim fractal boundaries can exhibit complex geometry even when they do not form a true fractal and fractal scaling is observed only above a certain length scale at each boundary point. Thus, our results reveal slim fractals as a geometrical hallmark of transient chaos in undriven dissipative systems.

Alfvén intermittent turbulence has been observed in the solar wind. It has been previously shown that the interplanetary Alfvén intermittent turbulence can appear due to a low‐dimensional temporal chaos [1]. In this paper, we study the nonlinear spatiotemporal dynamics of Alfvén waves governed by the Kuramoto‐Sivashinsky equation which describes the phase evolution of a large‐amplitude Alfvén wave. We investigate the Alfvén turbulence driven by a high‐dimensional interior crisis, which is a global bifurcation caused by the collision of a chaotic attractor with an unstable periodic orbit. This nonlinear phenomenon is analyzed using the numerical solutions of the model equation. The identification of the unstable periodic orbits and their invariant manifolds is fundamental for understanding the instability, chaos and turbulence in complex systems such as the solar wind plasma. The high‐dimensional dynamical system approach to space environment turbulence developed in this paper can improve our interpretation of the origin and the nature of Alfvén turbulence observed in the solar wind.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation A. C.‐L. Chian, F. A. Borotto, E. L. Rempel; Onset of Alfvén Turbulence via Boundary Crisis. AIP Conf. Proc. 11 June 2003; 669 (1): 715–718. https://doi.org/10.1063/1.1594030 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

Bifurcations of forced oscillators with fuzzy uncertainties by the generalized cell mapping method

Crises of chaotic attractors are typical phenomena in nonlinear dynamical systems. By means of generalized cell mapping digraph (GCMD) method, we show that a chaotic boundary crisis results from a collision between a chaotic attractor and a chaotic saddle in the fractal basin boundary. In such a case the chaotic attractor, together with its basin of attraction, is suddenly destroyed as the parameter passes through a critical value, simultaneously the chaotic saddle also undergoes an abrupt enlargement in its size.

Metamorphosis of chaotic saddle

A dynamic oligopoly game model with nonlinear cost and strategic delegation is built on the basis of isoelastic demand in this paper. And the dynamic characteristics of this game model are investigated. The local stability of the boundary equilibrium points is analyzed by means of the stability theory and Jacobian matrix, and the stability region of the Nash equilibrium point is obtained by Jury criterion. It is concluded that the system may lose stability through Flip bifurcation and Neimark–Sacker bifurcation. And the effects of speed of adjustment, price elasticity, profit weight coefficient and marginal cost on the system stability are discussed through numerical simulation. After that, the coexistence of attractors is analyzed through the basin of attraction, where multiple stability always means path dependence, implying that the long-term behavior of enterprises is strongly affected by historical contingency. In other words, a small perturbation of the initial conditions will have a significant impact on the system. In addition, the global dynamical behavior of the system is analyzed by using the critical curves, the basin of attraction, absorbing areas and a noninvertible map, revealing that three global bifurcations, the first two of which are caused by the interconversion of simply-connected and multiply-connected regions in the basin of attraction, and the third global bifurcation, that is, the final bifurcation is caused by the contact between attractors and the boundary of the basin of attraction.

In this study, the author investigates controllability of discrete‐time bilinear systems by using invariant sets. An invariant set defined in this study is a set on which the system is invariant, namely for any initial state that belongs to the set, the system cannot be steered away from this set. If a system is controllable, then it has no invariant set in its state space other than the state space itself. Invariant sets are thus closely related to controllability. By studying the invariant sets of discrete‐time bilinear systems, the author first presents a necessary and sufficient condition for controllability of the systems in dimension two, which covers a classical result under the same condition. Then the author considers the high‐dimensional systems and improve the recent results on controllability by relaxing a condition concerning invariant sets, where more general controllability criteria are derived. Finally, examples are given to illustrate the obtained results.

The basin of attraction of a stable equilibrium point is investigated for a dynamical system (W97) that has been used to model transition to turbulence in shear flows. The basin boundary contains a linearly unstable equilibrium point Xlb which, in the self-sustaining scenario, plays a role in mediating the transition in that transition orbits cluster around its unstable manifold. However we find—for W97 with canonical parameter values—that this role is played not by Xlb but rather by a periodic orbit also lying on the basin boundary. Moreover, it appears via numerical computations that all orbits beginning near Xlb relaminarize. We offer numerical evidence that the parameter values of W97 are post-critical in the following sense: for some, subcritical parameter values, the basin boundary coincides with the stable manifold of Xlb and only a subset of nearby orbits relaminarize, whereas for supercritical values the basin boundary is the union of two stable manifolds, one belonging to the periodic orbit and dominating the basin boundary, and the other belonging to Xlb and detectable only as edge separating relaminarizing orbits of different characters. The periodic orbit appears at the critical parameter value via a homoclinic connection. This further leads to a proposal for the structure of the ‘edge of chaos’ somewhat different from that which has previously been proposed.

For invertible, area-contracting planar maps, much work has been done on the study of the crucial role played by certain unstable periodic orbits, distinguished by being "accessible", in understanding some global bifurcations such as metamorphoses of basin boundaries and crisis of attractors. In this paper we concentrate on a one-parameter family of Hénon maps whose attractors present a type of discontinuous change characterized by a sudden replacement of the accessible orbits as the parameter is varied. The change is quantified by a jump in the corresponding accessible rotation number. The rotation rate describes a devil's staircase as a function of the parameter. We estimate the Hausdorff dimensions of the Hénon attractors — via the Mendès France fractal dimension — for the same interval of parameter values. Our purpose is to show the strong connection between these two functions: the variation of the dimension stays minimal for the parameters under the same plateau in the staircase, while it is markedly greater when the parameter moves from one plateau to another.