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More complicated periodic functions (Formula presented) will also stabilize unstable orbits. The one chosen here is the simplest one
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More complicated periodic functions (Formula presented) will also stabilize unstable orbits. The one chosen here is the simplest one.
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14
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85037252379
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The (Formula presented)-dimensional map (Formula presented) could be linearized around its fixed point (Formula presented) as (Formula presented), where (Formula presented). By decomposing (Formula presented) into the right eigenvectors (Formula presented) of (Formula presented) as (Formula presented), where (Formula presented) are the eigenvalues of (Formula presented), this becomes (Formula presented), i.e., a system of decoupled one-dimensional maps
-
The (Formula presented)-dimensional map (Formula presented) could be linearized around its fixed point (Formula presented) as (Formula presented), where (Formula presented). By decomposing (Formula presented) into the right eigenvectors (Formula presented) of (Formula presented) as (Formula presented), where (Formula presented) are the eigenvalues of (Formula presented), this becomes (Formula presented), i.e., a system of decoupled one-dimensional maps.
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The equation (Formula presented) has the solution (Formula presented) for (Formula presented) and (Formula presented) for (Formula presented) because we need to know (Formula presented) only from the previous interval (Formula presented). This yields (Formula presented). Since (Formula presented) for (Formula presented), the factor (Formula presented) remains larger than one for all values of (Formula presented), i.e., the fixed point cannot be stabilized
-
The equation (Formula presented) has the solution (Formula presented) for (Formula presented) and (Formula presented) for (Formula presented) because we need to know (Formula presented) only from the previous interval (Formula presented). This yields (Formula presented). Since (Formula presented) for (Formula presented), the factor (Formula presented) remains larger than one for all values of (Formula presented), i.e., the fixed point cannot be stabilized.
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