Tinkerbell Is Chaotic

Abstract

Shadowing is a method of backward error analysis that plays a important role in hyperbolic dynamics. In this paper, the shadowing by containment framework is revisited, including a new shadowing theorem. This new theorem has several advantages with respect to existing shadowing theorems: It does not require injectivity or differentiability, and its hypothesis can be easily verified using interval arithmetic. As an application of this new theorem, shadowing by containment is shown to be applicable to infinite length orbits and is used to provide a computer assisted proof of the presence of chaos in the well-known noninjective Tinkerbell map. A. GOLDSZTEJN, W. HAYES, AND P. COLLINS proof is conceptually very simple since it relies directly on the Poincaré-Miranda theorem. 1 Note that the containment theorem proposed in [21] also does not require differentiability or injectivity: However, its hypotheses are quite difficult to verify. Third, this new containment theorem is extended to bi-infinite orbits, thus reaching the scope of shadowing theorems by refinement like [35].As an application, we show that this containment theorem for bi-infinite orbits can be used to prove that a dynamical system is chaotic in several senses (positive topological entropy, Li-Yorke chaos, and Devaney chaos). Different computed assisted proofs of the presence of chaos were proposed in, e.g., [34,9,42,10,26]. All these approaches are similar in the sense that they rely on some topological fixed point theorems whose hypotheses are translated to sufficient conditions for the presence of chaos. However, the usage of different topological fixed point theorems leads to different statements. The sufficient conditions we obtain here are remarkable by their simplicity, which allows an easy numerical verification by a computer. Interval analysis.Interval analysis is a branch of numerical analysis that was born in the 1960's. It consists of computing with intervals of reals instead of reals, providing a framework for handling uncertainties and verified computations (see [27,1,28,17,20] for a survey). Among other applications, interval analysis was used to provide rigorous numerical proofs of mathematical statements (resolution of the 14th Smale problem [38], properties of manifolds [6,7], and properties of chaotic dynamical systems [3,18,4,39,33]).

An Interval Approach for Stability Analysis: Application to Sailboat Robotics

“…In this context, there exists also some point numerical techniques [23] which use some Lipschitz properties of the systems or ellipsoidal methods [22] when the system is linear. Now, interval methods can take advantage of constraint propagation tools to provide efficient resolution algorithms [5] and their ability to integrate nonlinear state equations in a guaranteed way [8]. When the system is both nonlinear and uncertain, in a set-membership context, stability analysis is a difficult problem and to our knowledge, no reliable algorithm are available in this context.…”

Chaotic Synchronization of Modified Discrete-Time Tinkerbell Systems

“…which means the stability of error system (6); that is, two discrete-time chaotic systems described by (1) and 2can achieve global synchronization. This ends the proof.…”

“…Many models in engineering can be mathematically described as a Tinkerbell system, which is a classical twodimensional discrete-time system with Tinkerbell-like trajectories. The dynamical behaviors of Tinkerbell systems have been studied in some papers (see, e.g., [1][2][3]).…”

Proving the existence of loops in robot trajectories

“…where [y](t) = t t0 [x](τ )dτ is the interval primitive of [x](·) and y ± are the corresponding bounds. The proof is provided in [ A tube is generally used to describe uncertain trajectories evolving with time and defined by differential equations [29,28,16]. This is naturally of high interest in robotics, being useful for dynamical systems such as mobile robots, involving uncertainties and any kind of temporal constraints.…”