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view source for recurrence relation wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in view source for recurrence relation add languages article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file page information get shortened url switch to legacy parser in other projects wikidata item appearance move to sidebar hide recurrence relation you do not have permission to edit this page for the following reason this ip address range has been globally blocked this does not affect your ability to read wikipedia pages most people who see this message have done nothing wrong some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism and can sometimes affect other users who are unrelated to that abuse review the information below for assistance if you do not believe that you have done anything wrong this block affects editing on all wikimedia wikis the ip address or range 5 135 0 0 16 has been globally blocked by jon kolbert for the following reason s open proxy webhost visit the faq if you are affected this block will expire on 04 29 8 january 2027 your current ip address is 5 135 42 194 even while globally blocked you will usually still be able to edit pages on meta wiki if you believe you were blocked by mistake you can find additional information and instructions in the stewards block wizard other useful links global blocks help i have been blocked you can view and copy the source of this page stability stability of linear higher order recurrences the linear recurrence of order math d math math a_n c_1a_ n 1 c_2a_ n 2 cdots c_da_ n d math has the characteristic polynomial characteristic equation math lambda d c_1 lambda d 1 c_2 lambda d 2 cdots c_d lambda 0 0 math the recurrence is stability theory stable meaning that the iterates converge asymptotically to a fixed value if and only if the eigenvalues i e the roots of the characteristic equation whether real or complex are all less than unity mathematics unity in absolute value stability of linear first order matrix recurrences main matrix difference equation in the first order matrix difference equation math x_t x a x_ t 1 x math with state vector math x math and transition matrix math a math math x math converges asymptotically to the steady state vector math x math if and only if all eigenvalues of the transition matrix math a math whether real or complex have an absolute value which is less than 1 stability of nonlinear first order recurrences consider the nonlinear first order recurrence math x_n f x_ n 1 math this recurrence is stability theory locally stable meaning that it limit of a sequence converges to a fixed point math x math from points sufficiently close to math x math if the slope of math f math in the neighborhood of math x math is smaller than unity mathematics unity in absolute value that is math f x 1 math a nonlinear recurrence could have multiple fixed points in which case some fixed points may be locally stable and others locally unstable for continuous f two adjacent fixed points cannot both be locally stable a nonlinear recurrence relation could also have a cycle of period math k math for math k 1 math such a cycle is stable meaning that it attracts a set of initial conditions of positive measure if the composite function math g x f circ f circ cdots circ f x math with math f math appearing math k math times is locally stable according to the same criterion math g x 1 math where math x math is any point on the cycle in a chaos theory chaotic recurrence relation the variable math x math stays in a bounded region but never converges to a fixed point or an attracting cycle any fixed points or cycles of the equation are unstable see also logistic map dyadic transformation and tent map return to recurrence relation retrieved from https en wikipedia org wiki recurrence_relation privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search view source for recurrence relation add languages add topic