Stability of Order: an Example of Horseshoes “near” a Linear Map
نویسنده
چکیده
We have been motivated by a question of “anticontrol” of chaos [Schiff et al., 1994], in which recent examples [Chen & Lai, 1997] of controlling nonchaotic maps to chaos have required large perturbations. Can this be done without such brute force? In this paper, we present an example in which a family of maps, Gε, numerically displays a transverse homoclinic point, and hence a horseshoe and chaos, for a fixed value of the parameter ε. We show that these maps converge pointwise to a linear map. Furthermore, a simple scaling conjugacy is shown for a family of maps which even shows geometric similarity of all relevant structures. This is in seeming contradiction to well-known structural stability results concerning horseshoes, but careful consideration reveals that these theorems require convergence in a uniform topology in function space. We show that no such convergence is possible for our family of maps, since it is impossible to find a finite radius disk which contains all of the horseshoes Λε for every ε. Thus, there is no contradiction. Our example may be considered to be a new kind of bifurcation route to chaos by horseshoes, in which rather than creating/destroying a horseshoe by creating/destroying transverse homoclinic points, the horseshoe is sent/brought to/from infinity.
منابع مشابه
An Algorithm to Automatically Detect the Smale Horseshoes
Smale horseshoes, curvilinear rectangles and their U-shaped images patterned on Smale’s famous example, provide a rigorous way to study chaos in dynamical systems. The paper is devoted to constructing them in two-dimensional diffeomorphisms with the existence of transversal homoclinic saddles. We first propose an algorithm to automatically construct “horizontal” and “vertical” sides of the curv...
متن کاملNear - Optimal Tunning of Linear Controllers Based on Genetic Algorithm and Swarm Intelligence: A Flight Control Example
متن کامل
Stability and Performance Attainment with Fixed Order Controller Using Frequency Response
Recently, a new data driven controller synthesis is presented for calculating the family of stabilizing first, second and fixed order controller using frequency respons. However, this method is applicable just for plants that can guarantee some smoothness at the boundary of the resulted high dimension LMI. This paper solve that issue and extends the approach to fixed order controllers guarantee...
متن کاملDelay-Dependent Robust Asymptotically Stable for Linear Time Variant Systems
In this paper, the problem of delay dependent robust asymptotically stable for uncertain linear time-variant system with multiple delays is investigated. A new delay-dependent stability sufficient condition is given by using the Lyapunov method, linear matrix inequality (LMI), parameterized first-order model transformation technique and transformation of the interval uncertainty in to the norm ...
متن کاملPole Assignment Of Linear Discrete-Time Periodic Systems In Specified Discs Through State Feedback
The problem of pole assignment, also known as an eigenvalue assignment, in linear discrete-time periodic systems in discs was solved by a novel method which employs elementary similarity operations. The former methods tried to assign the points inside the unit circle while preserving the stability of the discrete time periodic system. Nevertheless, now we can obtain the location of eigenvalues ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1999