Sustaining chaos by using basin boundary saddles

Abstract

A method and apparatus for sustaining chaos in a system by using the natural dynamics of the system to redirect flow towards a chaotic region along unstable manifolds of basin boundary saddles by utilizing small, infrequent parameter perturbations. The perturbations are determined based on the location of an unstable state, which is used as a control reference. The location of the unstable state, which is unobservable, is estimated by calculating a theoretical branch connecting the current state of the system to the target unstable state.

What is claimed is:  
     
       1. A method for sustaining chaos, comprising: 
       monitoring output parameters of a system;  
       defining a saddle region containing a basin saddle for the system; and  
       redirecting the output parameters toward a chaotic region when the output parameters enter the saddle region, by applying changes to input parameters of the system based on information describing the chaotic region.  
     
     
       2. A method for maintaining a stable state of a system, comprising: 
       monitoring an adjustable parameter of a system;  
       obtaining information describing an unstable state of the system; and  
       applying parameter perturbations to the system based on the obtained information describing the unstable state effective to direct the system away from the unstable state to a stable region.  
     
     
       3. The method of claim  2 , wherein the information describing the unstable state of the system is the location of the unstable state. 
     
     
       4. The method of claim  2 , wherein obtaining the information describing the unstable state of the system includes: 
       finding a branch connecting a current, stable state of the system to the unstable state; and  
       estimating the location of the unstable state based on the connecting branch.  
     
     
       5. The method of claim  4 , wherein applying parameter perturbations to the system based on the obtained information describing the unstable state includes implementing a target procedure to direct the system from the stable state toward the unstable state. 
     
     
       6. The method of claim  3 , further including using the location of the unstable state as a reference for control of the system by the application of parameter perturbations. 
     
     
       7. The method of claim  6 , further including determining the parameter perturbations based on the location of the unstable state. 
     
     
       8. The method of claim  5 , wherein implementing a targeting procedure includes varying the adjustable parameter. 
     
     
       9. The method of claim  4 , wherein the stable state is observed, and the unstable state is unobservable. 
     
     
       10. An apparatus for sustaining chaos, comprising: 
       means for monitoring output parameters of a system;  
       means for defining a saddle region containing a basin saddle for the system; and  
       means for redirecting the output parameters toward a chaotic region when the output parameters enter the saddle region, including means for applying changes to input parameters of the system based on received information describing the chaotic region.  
     
     
       11. An apparatus for maintaining a stable state of a system, comprising: 
       means for monitoring an adjustable parameter of a system;  
       means for obtaining information describing an unstable state of the system; and  
       means for applying perturbations to the adjustable parameter based on the obtained information describing the unstable state, the means for applying perturbations being effective to direct the system away from the unstable state to a stable region.  
     
     
       12. The apparatus of claim  11 , wherein the information describing the unstable state of the system is the location of the unstable state. 
     
     
       13. The apparatus of claim  11 , wherein the means for obtaining the information describing the unstable state of the system includes: 
       means for finding a branch connecting a current, stable state of the system to the unstable state; and  
       means for estimating the location of the unstable state based on the connecting branch.  
     
     
       14. The apparatus of claim  13 , wherein the means for applying perturbations to the adjustable parameter based on the obtained information describing the unstable state includes means for implementing a target procedure to direct the system from the stable state toward the unstable state. 
     
     
       15. The apparatus of claim  12 , further including means for using the location of the unstable state as a reference for control of the system by the application of parameter perturbations. 
     
     
       16. The apparatus of claim  15 , further including means for determining the parameter perturbations based on the location of the unstable state. 
     
     
       17. The apparatus of claim  14 , wherein the means for implementing a targeting procedure includes means for varying the adjustable parameter. 
     
     
       18. The apparatus of claim  13 , wherein the stable state is observed, and the unstable state is unobservable.