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lemma bifurcation fixed estimate since proof further function small conjugacy estimates where closeness points above normal suppose const first smooth point construction chapter following however these parameter transcritical hence sufficiently sequence remark holds there theorem which prove example origin using increasing pitchfork positive conditions forms definition order section other functions monotone proved constant numerical enough again discretization finally equation defined being together under interval between terms every close suitable instead inequality value exists similar condition denote would results assumption consider below differential inverse identity while equations right corresponding induction define their course bounded implies because runge similarly uniformly kutta methods dynamical systems solution second index strictly upper therefore large invertible values obtain continuous region number yields iterates shows theorems independent general sequences result satisfies moreover means equivalent coordinate neighbourhood assume given method discretizations later imply uniform three notice account evaluation properties parameters provided follows figure notation explicit stability should system introduction constants sufficient original transformation level desired constructed equilibrium present transformations derivatives choice assumed domain ordinary monotonicity different shift chosen fundamental discretized inequalities homeomorphism assumptions replaced analogous clearly conjugacies simple after bifurcations property lower optimal starting dynamics taking natural appropriate satisfying proposition thesis continuity analysis unique valid recursively linear series possible iteration easily dimensional plane distance verify stepsize applied quadratic taylor variables argument expansion least necessary implicit needed arbitrary latter zeros shown indeed family previous exist apply changes convergence decreasing satisfy claim choose becomes third functional growth bound question outer inner about before counterpart econst boundedness exactly theory structural mathematica depending approach along symmetry better existence complete supremum always final operator branch denoted expression stable cutting indices tangent nonlinear finite contains domains cubic discrete respect nonnegative meaning applying implied mapping difference satisfied cases variable proofs throughout dependence during negative derivative generic local considered gives optimality known fishman quantity scaling words completely computations often yield numbers lines transformed smoothness compare dimensions establish particular preservation direct could intervals already exponent partial situation directly contradiction families quantitative estimating factor backward elementary contrary arbitrarily extend elements structure taken garay established problem converges nevertheless defining implying product branches simply special behaviour corollary induced university construct denotes important canonical suitably illustrated solutions counterparts tests simpler statement locally integral asymptotic increases approximation concerning hypothesis sooner respectively formula higher roots contain logarithmic integrals leaves evaluating clear explicitly convex factored class absolute extension called model precision estimated hyperbolic transforming extra suppressed introduce interesting within recall exact information fractional going speed symbol basic recursive questions preparation limit common difficulty trivial understood based proceed summary conjugate those technical another observe examined differentiation euler derived containing formulae arrive proves depend singularity bringing undergo orbit repelling otherwise singular expressions press vector remainder express whether quantities inductively stands additional finish reaches symbols abbreviation maximal sides examples auxiliary begin slightly boundary still calculations actual computing insufficient horizontal relation differentiating turns uniqueness conclude seems compute follow decreased integer lemmas inverses iterated carried through closed simplify standard bounds attracting letter solves usual described structurally essential usually orbits formulated necessarily guarantees found weaker unstable chain simplicity complex