page_title: Arnold tongue
section_title: Standard circle map
hierarchical_section_title: Arnold tongue / Standard circle map
caption_attribution_description: English: Shows the en:bifurcation diagram for the iterated en:circle map. The circle map is given by iterating on the map.
context_page_description: In mathematics, particularly in dynamical systems, an Arnold tongue is a phenomenon observed in bifurcation diagrams when the value of a certain parameter constrains the orbit of the dynamical system to a smaller region; in particular, Arnold tongues occur when these regions enlarge monotonically as the parameter increases, causing regions that are somewhat similar to tongues in the bifurcation diagram.
  Arnold tongues are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes and cardiac electric waves. Sometimes the frequency of oscillation depends on, or is constrained based on some quantity, and it is often of interest to study this relation. For instance, the outset of a tumor triggers in the area a series of substance oscillations that interact with each other; simulations show that these interactions cause Arnold tongues to appear, that is, the frequency of some oscillations constrain the others, and this can be used to control tumor growth.
context_section_description: Arnold tongues appear most frequently when studying the interaction between oscillators, particularly in the case where one oscillator drives another. That is, one oscillator depends on the other but not other way around, so they do not mutually influence each other as happens in Kuramoto models, for example. This is a particular case of driven oscillators, with a driving force that has a periodic behaviour. As a practical example, heart cells (the external oscillator) produce periodic electric signals to stimulate heart contractions (the driven oscillator); here, it could be useful to determine the relation between the frequency of the oscillators, possibly to design better artificial pacemakers. The family of circle maps serves as a useful mathematical model for this biological phenomenon, as well as many others.
  The family of circle maps are functions (or endomorphisms) of the circle to itself. It is mathematically simpler to consider a point in the circle as being a point  in the real line that should be interpreted modulo , representing the angle at which the point is located in the circle. When the modulo is taken with a value other than , the result still represents an angle, but must be normalized so that the whole range  can be represented.  With this in mind, the family of circle maps is given by:
  where  is the oscillator's "natural" frequency and  is a periodic function that yields the influence caused by the external oscillator. Note that if  the particle simply walks around the circle at  units at a time; in particular, if  is irrational the map reduces to an irrational rotation.
  The particular circle map originally studied by Arnold, and which continues to prove useful even nowadays, is:
  where  is called coupling strength, and  should be interpreted modulo . This map displays very diverse behavior depending on the parameters  and ; if we fix  and vary , the bifurcation diagram around this paragraph is obtained, where we can observe periodic orbits, period-doubling bifurcations as well as possible chaotic behavior.
caption: Bifurcation diagram for 
    
      
        
          Ω
        
      
      {\displaystyle \Omega }
    
   held fixed at 
    
      
        
          1
          
            /
          
          3
        
      
      {\displaystyle 1/3}
    
  . 
    
      
        
          K
        
      
      {\displaystyle K}
    
   goes from 
    
      
        
          0
        
      
      {\displaystyle 0}
    
   at bottom to 
    
      
        
          4
          π
        
      
      {\displaystyle 4\pi }
    
   at top, and the orbits are shown in the interval 
    
      
        
          [
          −
          0.5
          ,
          0.5
          ]
        
      
      {\displaystyle [-0.5,0.5]}
    
   instead of 
    
      
        
          [
          0
          ,
          1
          ]
        
      
      {\displaystyle [0,1]}
    
  . Black regions correspond to Arnold tongues.
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