❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s
◗✉❛s❡ ▲✐♥❡❛r❡s
▲❡tí❝✐❛ ❋❛❧❡✐r♦s ❈❤❛✈❡s ❘♦❞r✐❣✉❡s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ▼❛t❡♠á✲ t✐❝❛ ❞♥✐✈❡rs✐tár✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡✳
❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦
✺✶✼✳✸✽ ❘✻✾✻❡
❘♦❞r✐❣✉❡s✱ ▲❡tí❝✐❛ ❋❛❧❡✐r♦s ❈❤❛✈❡s
❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ◗✉❛s❡ ▲✐♥❡❛r❡s✴ ▲❡tí❝✐❛ ❋❛❧❡✐r♦s ❈❤❛✈❡s ❘♦❞r✐❣✉❡s✲ ❘✐♦ ❈❧❛r♦✱ ✷✵✶✸✳
✶✵✺ ❢✳ ✿ ✐❧✳✱ ❣rá❢s✳
❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳
❖r✐❡♥t❛❞♦r❛✿ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦
✶✳ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s✳ ✷✳ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ◗✉❛s❡ ▲✐♥❡❛r❡s✳ ✸✳ ❊st❛❜✐❧✐❞❛❞❡✳ ■✳ ❚ít✉❧♦
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
▲❡tí❝✐❛ ❋❛❧❡✐r♦s ❈❤❛✈❡s ❘♦❞r✐❣✉❡s
❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ◗✉❛s❡ ▲✐♥❡❛r❡s
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛✲ ❞♦r❛✿
Pr♦❢❛✳ ❉r❛✳ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦ ❖r✐❡♥t❛❞♦r❛
Pr♦❢✳ ❉r❛✳ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛ ■●❈❊ ✲ ❯♥❡s♣✴❘✐♦ ❈❧❛r♦
Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❈❛r❧♦s ❞❛ ❙✐❧✈❛ ❋✐❧❤♦
▼❛t❡♠át✐❝❛ ✲ ❈❡♥tr♦ ❯♥✐✈❡rs✐tár✐♦ ❞❡ ❋r❛♥❝❛✴❋r❛♥❝❛
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡ à ❉❡✉s✱ ♣♦r ♠❡ ❞❛r ❢♦rç❛s✱ ❞❡t❡r♠✐♥❛çã♦✱ ❝❛♣❛❝✐❞❛❞❡ ❡ s❛❜❡❞♦r✐❛ ♣❛r❛ ❡♥❢r❡♥t❛r ♦ ❞❡s❝♦♥❤❡❝✐❞♦ ❡ ✈❡♥❝❡r✳
❆♦s ♠❡✉s ♣❛✐s✱ ●❡r❛❧❞♦ ❡ ▲♦✉r❞❡s✱ ♣♦r ♠❡ ♠♦str❛r❡♠ ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❝♦♥q✉✐st❛✱ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ❞♦ ❛♠♦r✱ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈❛♥❞♦ ❛ ❝r❡s❝❡r✱ ❡ ❞❡ ❢♦r♠❛ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ♣❛✐✱ q✉❡ ❛❝♦♠♣❛♥❤♦✉ ❞❡ ♣❡rt♦ ❡st❛ ♠✐♥❤❛ ❝❛♠✐♥❤❛❞❛✱ ❞❡❞✐❝❛♥❞♦✲s❡ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ r❡❛❧✐③❛r ❡st❡ s♦♥❤♦✳
❆♦ ♠❡✉ ❝♦♠♣❛♥❤❡✐r♦✱ ❛♠✐❣♦ ❡ ❡s♣♦s♦✱ ♣❡❧♦ ❛♣♦✐♦✱ ❝❛r✐♥❤♦✱ ✐♥s✐stê♥❝✐❛ ❡ ♣♦r ❝♦♠✲ ♣r❡❡♥❞❡r ♠✐♥❤❛s ❛✉sê♥❝✐❛s✳
❆ ♠✐♥❤❛ s♦❣r❛ ❡ ♠❡✉ s♦❣r♦✱ q✉❡ ♠❡ ❛✉①✐❧✐❛r❛♠ ♥❛ ♠✐♥❤❛ ❝❛s❛✱ ♣❛r❛ q✉❡ ❡✉ t✐✈❡ss❡ t❡♠♣♦ ♣❛r❛ ❞❡❞✐❝❛r ❛♦s ♠❡✉s ❡st✉❞♦s✳
❆♦s ♠❡✉s ✐r♠ã♦s✱ ▲❡❛♥❞r♦ ❡ ▲✉❝✐❛♥♦✱ ♠✐♥❤❛s ❝✉♥❤❛❞❛s ❙ô♥✐❛ ❡ ❋❧á✈✐❛ r❡s♣❡❝t✐✈❛✲ ♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ♣❡❧❛ t♦r❝✐❞❛ ❡ ♣❡❧♦ ❝❛r✐♥❤♦✳
❆ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s ❡ ❢❛♠✐❧✐❛r❡s q✉❡ t❛♠❜é♠ t♦r❝❡r❛♠ ♣♦r ♠✐♠✳ ❊♠ ❡s♣❡❝✐❛❧ ❛ ▼❛❞r✐♥❤❛ ❘♦s❛❧✐♥❛✱ ♣❡❧❛s ♣❛❧❛✈r❛s ❞❡ ❢♦rç❛s ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳
❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ♣❡❧❛ ❛ss✐stê♥❝✐❛✱ ❞✐s♣♦s✐çã♦ ❡ ❝♦♥tr✐❜✉✐çõ❡s✳ ❆♦ ♠❡✉ ♣r♦❢❡ss♦r ❡ ❛♠✐❣♦ ❉r✳ ❆♥tô♥✐♦ ❈❛r❧♦s ❞❛ ❙✐❧✈❛ ❋✐❧❤♦✱ ♣♦r ❛❝r❡❞✐t❛r ❡♠ ♠✐♠ ❞❡s❞❡ ❛ ❣r❛❞✉❛çã♦ ❡ s❡r ♠❡✉ ♠❛✐♦r ✐♥❝❡♥t✐✈❛❞♦r✳
❊♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❉r❛✳ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦✱ ♣❡❧❛ s❛❜❡❞♦r✐❛✱ ♣❛❝✐ê♥❝✐❛ ❡ ❡stí♠✉❧♦ ♥❛ r❡❛❧✐③❛çã♦ ❞❛ ♣❡sq✉✐s❛✳
❯♠ ❞✐❛ ✈♦❝ê ❛♣r❡♥❞❡ q✉❡ r❡❛❧♠❡♥t❡ ♣♦❞❡ s✉♣♦rt❛r✳ ◗✉❡ r❡❛❧♠❡♥t❡ é ❢♦rt❡✱ ❡ q✉❡ ♣♦❞❡ ✐r ♠✉✐t♦ ♠❛✐s ❧♦♥❣❡✳ ❉❡♣♦✐s ❞❡ ♣❡♥s❛r q✉❡ ♥ã♦ s❡ ♣♦❞❡ ♠❛✐s✳
❘❡s✉♠♦
❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥✲ ç❛s ❞♦ t✐♣♦ q✉❛s❡ ❧✐♥❡❛r❡s ✉t✐❧✐③❛♥❞♦ ♦ ▼ét♦❞♦ ❞❡ ▲✐♥❡❛r✐③❛çã♦✱ ✈✐s❛♥❞♦ s✉❛ ❛♣❧✐❝❛çã♦ ♥❛ ❛♥á❧✐s❡ ❞❡ ♠♦❞❡❧♦s ♥❛ ár❡❛ ❞❡ ❇✐♦❧♦❣✐❛ ❡ ❊❝♦♥♦♠✐❛✳
❆❜str❛❝t
❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ st✉❞② t❤❡ st❛❜✐❧✐t② ♦❢ ❛❧♠♦st ❧✐♥❡❛r ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥s✱ ❜② ✉s✐♥❣ t❤❡ ▲✐♥❡❛r✐③❛t✐♦♥ ▼❡t❤♦❞✱ ✐♥ ♦r❞❡r t♦ ✉s❡ ✐♥ t❤❡ ❛♥❛❧②s✐s ♦❢ s♦♠❡ ♠♦❞❡❧s ✐♥ ❇✐♦❧♦❣② ❛♥❞ ❊❝♦♥♦♠②✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✸✳✶ P♦♥t♦s ✜①♦s ❞❡ f(x) =x3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✷ P♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ x(n+ 1) =T(x(n))✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✸ ❖ ♣♦♥t♦ ❡stá✈❡❧ x∗✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✸✳✹ ❖ ♣♦♥t♦ ✐♥stá✈❡❧ x∗✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✺ ❖ ♣♦♥t♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ x∗✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✻ ❖ ♣♦♥t♦ x∗ ❣❧♦❜❛❧♠❡♥t❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✸✳✼ ❚❡✐❛ ❞❡ ❛r❛♥❤❛ ❞❡ x(n+ 1) =µx(n)(1−x(n)) ♣❛r❛ µ= 2,5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸✳✽ Pr❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✾ Pr❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡stá✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✶✵ Pr❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✐♥stá✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✶✶ ❙❡a >0✱ t♦❞❛ s♦❧✉çã♦ s❡ ❛❢❛st❛rá ❞❛ s♦❧✉çã♦ ♥✉❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✶✷ ❙❡a <0✱ t♦❞❛ s♦❧✉çã♦ ❝♦♥✈❡r❣✐rá ♣❛r❛ ❛ s♦❧✉çã♦ ♥✉❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✶✸ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ❣rá✜❝♦s ❞❛ ❢✉♥çã♦ x(t) = eat ❡ ❞❛ ❢✉♥çã♦ ❡♥❝♦♥✲
tr❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❞❡ ❊✉❧❡r ♣❛r❛ ♦s ♣❛ss♦s h= 1 ❡ h= 0,5✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸✳✶✹ P♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✐♥stá✈❡❧ ✭f′′(x∗)>0✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✸✳✶✺ P♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✐♥stá✈❡❧ ✭f′′(x∗)<0✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✸✳✶✻ P♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✐♥stá✈❡❧ ✭f′(x∗) = 1✱ f′′(x∗) = 0 ❡ f′′′(x∗)>0✮✳ ✳ ✳ ✳ ✹✻
✸✳✶✼ P♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ ✭f′(x∗) = 1✱ f′′(x∗) = 0 ❡
f′′′(x∗)<0✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✸✳✶✽ P♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ ✭x∗ =−2✮ ♣❛r❛x(n+ 1) =
x2(n) + 3x(n)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✸✳✶✾ P♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✐♥stá✈❡❧ ✭x∗ = 0✮ ♣❛r❛ x(n+ 1) =x2(n) + 3x(n)✳ ✳ ✳ ✹✾
✸✳✷✵ ❖ ✷✲❝✐❝❧♦{0,−1}♣❛r❛x(n+1) =f(x(n)) = x2(n)−1é ❛ss✐♥t♦t✐❝❛♠❡♥t❡
❡stá✈❡❧✱ s❡♥❞♦ x∗ = 0 ❡ x∗ =−1✱ ♣♦♥t♦s ❞❡ f2(x) = x4−2x2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✸✳✷✶ ❆♣r♦①✐♠❛çã♦ ❞❡ ✉♠❛ s♦❧✉çã♦ xn(x0) ❛♦ ✷✲❝✐❝❧♦✱ ♣❛r❛f(x) =x2−1✳ ✳ ✳ ✺✹
✹✳✶ P❧❛♥♦ ❞❡ ❢❛s❡ ❞❡y(n+ 1) =Jy(n)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
✹✳✷ P❧❛♥♦ ❞❡ ❢❛s❡ ❞❡x(n+ 1) =Ax(n)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹
✹✳✸ Ór❜✐t❛ ❞❡ y(n+ 1) =Jy(n) ❝♦♠❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ (−1/16,0)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
✹✳✹ Ór❜✐t❛ ❞❡ x(n+ 1) =Ax(n) ❝♦♠❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧
−√3/16,0
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✼
✷ ❚❡♦r✐❛ ●❡r❛❧ ❞❡ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s ✶✾
✷✳✶ ❙✐st❡♠❛ ❞✐♥â♠✐❝♦ ❞✐s❝r❡t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳✶ ❈❛s♦s ❡s♣❡❝✐❛✐s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥s✲
t❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❛✉tô♥♦♠❛s✿ ❝❛s♦ r❡❛❧ ✷✼ ✸✳✶ P♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✶✳✶ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ P♦♥t♦s ❞❡ ❊q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ❚❡✐❛ ❞❡ ❛r❛♥❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✸ ❙♦❧✉çõ❡s ♥✉♠ér✐❝❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✸✳✶ ▼ét♦❞♦ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✹ ❈r✐tér✐♦s ♣❛r❛ ❊st❛❜✐❧✐❞❛❞❡ ❆ss✐♥tót✐❝❛ ❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✺ P♦♥t♦s ♣❡r✐ó❞✐❝♦s ❡ ❝✐❝❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹❊st❛❜✐❧✐❞❛❞❡ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ✺✺ ✹✳✶ ❙✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✷ ❊st❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✷✳✶ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ♥ã♦ ❛✉tô♥♦♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✷✳✷ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s ✳ ✳ ✻✻ ✹✳✷✳✸ ❈r✐tér✐♦s ♣❛r❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s s✐st❡♠❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s ✳ ✳ ✳ ✳ ✻✼ ✹✳✸ ❆♥á❧✐s❡ ❞♦ ♣❧❛♥♦ ❞❡ ❢❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
✺ ❙✐st❡♠❛s q✉❛s❡ ❧✐♥❡❛r❡s ✼✾
✺✳✶ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✺✳✷ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s q✉❛s❡ ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵
✻ ❆♣❧✐❝❛çõ❡s ✽✼
✼ ❈♦♥❝❧✉sã♦ ✾✺
❘❡❢❡rê♥❝✐❛s ✾✼
❆ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✹✳✷✳✼ ✾✾
❆✳✶ ❈r✐tér✐♦s ♣❛r❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s s✐st❡♠❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ♦♦
❇ ❋♦r♠❛ ❈❛♥ô♥✐❝❛ ❞❡ ❏♦r❞❛♥ ✶✵✸
✶ ■♥tr♦❞✉çã♦
❊q✉❛çõ❡s ❞✐s❝r❡t❛s ♦✉ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❞❡s❝r❡✈❡♠ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❝✉❥❛ ❡✈♦❧✉çã♦ ♥♦ t❡♠♣♦ é ♠❡❞✐❞❛ ❡♠ ✐♥t❡r✈❛❧♦s ❞✐s❝r❡t♦s✳ ❊♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s é ❞❡s❡❥á✈❡❧ s❛❜❡r s❡ t♦❞♦s ♦s ❡st❛❞♦s ❞❡ ✉♠ s✐st❡♠❛ t❡♥❞❡♠ ♣❛r❛ s❡✉ ❡st❛❞♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ♦✉ s❡❥❛✱ s♦❧✉çõ❡s ❝♦♥st❛♥t❡s ❞❡t❡r♠✐♥❛❞❛s ♣♦r ♣♦♥t♦s ❡s♣❡❝✐❛✐s ❝❤❛♠❛❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✳ P♦ré♠✱ ❤á ❝❛s♦s ❡♠ q✉❡ é ❣r❛♥❞❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ♣❛r❛ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ❞❡ ✉♠❛ ❞❛❞❛ ❡q✉❛çã♦ ♥✉♠❛ ❢♦r♠❛ ❡①♣❧í❝✐t❛✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ é ✐♠♣♦rt❛♥t❡ ❝♦♥s✐❞❡r❛r ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛s s♦❧✉çõ❡s ❞❡ss❛s ❡q✉❛çõ❡s s❡♠ r❡❛❧♠❡♥t❡ r❡s♦❧✈ê✲❧❛s✳ P❛r❛ ✐ss♦✱ t♦♠❛♠♦s ♣♦♥t♦s ✐♥✐❝✐❛✐s ♣ró①✐♠♦s ❛♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❛♥❛❧✐s❛♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s s♦❧✉çõ❡s ❞❡t❡r♠✐♥❛❞❛s ♣♦r ❡ss❡s ♣♦♥t♦s ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ s❛❜❡r s❡ ❛s s♦❧✉çõ❡s s❡ ❛♣r♦①✐♠❛♠ ♦✉ s❡ ❛❢❛st❛♠ ❞❛ s♦❧✉çã♦ ❝♦♥st❛♥t❡✳ ❊st❛ ♣❛rt❡ ❞❛ t❡♦r✐❛ ❞❛s ❊q✉❛çõ❡s ❉✐s❝r❡t❛s ❡ ❈♦♥tí♥✉❛s ❝❤❛♠❛✲s❡ ❛♥á❧✐s❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡✳
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❡st✉❞❛r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞♦ t✐♣♦ q✉❛s❡ ❧✐♥❡❛r❡s✱ ❡st❛s q✉❡ ❛♣r❡s❡♥t❛♠ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s✱ ♥ã♦ ❛♣❡♥❛s ❜✐♦❧ó❣✐❝❛s ♠❛s t❛♠❜é♠ ❞❡♥tr♦ ❞❛ ❡❝♦♥♦♠✐❛✱ ♣♦❞❡♥❞♦ s❡r út✐❧ ♥❡st❛s ár❡❛s✳ ❉❡st❛ ♠❛♥❡✐r❛✱ ♣❛rt✐♠♦s ❞♦ ❡st✉❞♦ ❞❛ t❡♦r✐❛ ❣❡r❛❧ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s✱ ❛♥❛❧✐s❛♠♦s ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r❡s ❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s✱ ❡ ♣♦r ✜♠ ❡st✉❞❛♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s q✉❛s❡ ❧✐♥❡❛r❡s✳
❉❡st❡ ♠♦❞♦ ❡st❡ tr❛❜❛❧❤♦ ✜❝♦✉ ❛ss✐♠ ❞✐✈✐❞✐❞♦✿
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ♣r♦❝✉r❛♠♦s ❡st❛❜❡❧❡❝❡r ❛ t❡♦r✐❛ ❣❡r❛❧ s♦❜r❡ ❡q✉❛çõ❡s ❞✐s❝r❡t❛s✳ ❈♦♠❡ç❛♠♦s ♣♦r ❞❡✜♥✐r ❡q✉❛çã♦ ❞✐s❝r❡t❛ ❡ s✉❛ r❡❧❛çã♦ ❝♦♠ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❡✱ ❛ss✐♠✱ ❝♦♠♦ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r❡s ❞❡ ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞❛❞❛ ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✳ ❆q✉✐ ❡♥❝♦♥tr❛♠✲s❡ t❛♠❜é♠ ❛❧❣✉♥s ♠ét♦❞♦s ♣❛r❛ ❞❡t❡r♠✐♥❛r s♦❧✉çõ❡s ♥✉♠ér✐❝❛s ♣❛r❛ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳
◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❡♠ s❡❣✉✐❞❛ ❛♣r❡s❡♥t❛♠♦s ♦ ♠ét♦❞♦ ❞❛ ❚❡✐❛ ❞❡ ❆r❛♥❤❛ ♣❛r❛ ❛♥❛❧✐s❛r ❣r❛✜❝❛♠❡♥t❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❞✐s❝r❡t❛s ♥♦ ❝❛s♦ r❡❛❧✳ P❛r❛ ✜♥❛❧✐③❛r ♦ ❝❛♣ít✉❧♦✱ ❡st❛❜❡❧❡❝❡♠♦s ♦s ❝r✐tér✐♦s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♣❛r❛ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❞❡✜♥✐♠♦s ♣♦♥t♦s ♣❡r✐ó❞✐❝♦s ❡ ❝✐❝❧♦s✳
◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ ❡st❡♥❞❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ♣❛r❛ s✐st❡♠❛ ❧✐♥❡❛r ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s✳ P❛r❛ ✜♥❛❧✐③❛r ♦ ❝❛♣ít✉❧♦✱ ❡st❛❜❡❧❡❝❡♠♦s ❝r✐tér✐♦s ♣❛r❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ s✐st❡♠❛s ❞❡ ❞✉❛s ❡q✉❛çõ❡s ❡
✶✽ ■♥tr♦❞✉çã♦
❛♥❛❧✐s❛♠♦s ♦ ♣❧❛♥♦ ❞❡ ❢❛s❡✳
P♦r ✜♠✱ ♥♦ q✉✐♥t♦ ❝❛♣❞t✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ❛s ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s q✉❛s❡ ❧✐♥❡❛r❡s ❡ s✉❛ ❡st❛❜✐❧✐❞❛❞❡✳ P❛r❛ ❛♥❛❧✐s❛r s✉❛ ❡st❛❜✐❧✐❞❛❞❡✱ r❡❛❧✐③❛♠♦s ✉♠ ♣r♦❝❡ss♦ ❞❡ ❧✐♥❡❛r✐③❛✲ ç✳♦✱ ♦♥❞❡ ❢♦✐ ♣♦ss❞✈❡❧ ❡♥❝♦♥tr❛r ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r q✉❡ ❢♦ss❡ ✉♠❛ ❜♦❛ ❛♣r♦①✐♠❛ç✳♦ ❞❛ ❡q✉❛ç✳♦ q✉❛s❡ ❧✐♥❡❛r✱ ❡ ❛ss✐♠✱ ❛tr❛✈és ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛ ❡q✉❛ç✳♦ ❧✐♥❡❛r✱ ❛♥❛❧✐s❛♠♦s ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛ ❡q✉❛ç✳♦ q✉❛s❡ ❧✐♥❡❛r✳
❆ t❡♦r✐❛ ❛❜♦r❞❛❞❛ ❜❛s❡✐❛✲s❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♥❛s r❡❢❡rê♥❝✐❛s ❬✶❪ ❡ ❬✻❪ ❡ ❛s ❛♣❧✐❝❛çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ♥❛s r❡❢❡rê♥❝✐❛s ❬✶❪ ❡ ❬✺❪✳
✷ ❚❡♦r✐❛ ●❡r❛❧ ❞❡ ❊q✉❛çõ❡s ❞❡
❉✐❢❡r❡♥ç❛s
❯♠ s✐st❡♠❛ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♦❜❥❡t♦s ❛❣r✉♣❛❞♦s ♣♦r ❛❧❣✉♠❛ ✐♥t❡r❛çã♦ ♦✉ ✐♥t❡r❞❡♣❡♥❞ê♥❝✐❛✱ ❞❡ ♠♦❞♦ q✉❡ ❡①✐st❛♠ r❡❧❛çõ❡s ❞❡ ❝❛✉s❛ ❡ ❡❢❡✐t♦ ♥♦s ❢❡♥ô♠❡♥♦s q✉❡ ♦❝♦rr❡♠ ❝♦♠ ♦s ❡❧❡♠❡♥t♦s ❞❡ss❡ ❝♦♥❥✉♥t♦❀ ❡ é ❞✐t♦ ❞✐♥â♠✐❝♦ q✉❛♥❞♦ ❛❧❣✉♠❛s ❣r❛♥❞❡③❛s q✉❡ ❝❛r❛❝t❡r✐③❛♠ s❡✉s ♦❜❥❡t♦s ❝♦♥st✐t✉✐♥t❡s ✈❛r✐❛♠ ♥♦ t❡♠♣♦✳ ❯♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ❞❡s❝r❡✈❡ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ s✐t✉❛çõ❡s ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ♠♦❞❡❧♦s ♣♦❞❡♠ s❡r ❝♦♥str✉í❞♦s ❡ ✉s❛❞♦s ♣❛r❛ s❡ ❡st✉❞❛r s✉❛ ❡✈♦❧✉çã♦ t❡♠♣♦r❛❧✳ ❖ ♦❜❥❡t✐✈♦ ❞❡ss❡s ❡st✉❞♦s t❡ór✐❝♦s é ♣r❡✈❡r ♦ ❢✉t✉r♦ ✭♦✉ ❡①♣❧✐❝❛r ♦ ♣❛ss❛❞♦✮ ❞❡ ♠♦❞♦ ❝✐❡♥tí✜❝♦✳ P❛r❛ ❢❛③❡r ✐ss♦✱ é ♥❡❝❡ssár✐♦ ❝♦♥❤❡❝❡r ❡ ❝♦♠♣r❡❡♥❞❡r ❛s r❡❣r❛s q✉❡ ❣♦✈❡r♥❛♠ ❛s ♠✉❞❛♥ç❛s q✉❡ ♦❝♦rr❡rã♦✳ ◗✉❛♥❞♦ ♦ t❡♠♣♦ n é ❝♦♥tí♥✉♦ ♣❛r❛ ❛ ❣r❛♥❞❡③❛ x(n)✱ ❛
✈❛r✐❛çã♦ é ♠❡❞✐❞❛ ♣❡❧❛ ❞❡r✐✈❛❞❛ d
dnx(n) ❡✱ ❛ss✐♠✱ ♦ ❡st✉❞♦ ♠❛t❡♠át✐❝♦ ❞❡ ♠✉❞❛♥ç❛s
❝♦rr❡s♣♦♥❞❡ ❛♦ ❡st✉❞♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳ ❆♦ ❛ss✉♠✐r q✉❡ ♦ t❡♠♣♦ ❡✈♦❧✉✐ ❞❡ ❢♦r♠❛ ❞✐s❝r❡t❛✱ ♦✉ ♠❡❧❤♦r✱ q✉❡ ♦ s✐st❡♠❛ s❡ ❛❧t❡r❛ s♦♠❡♥t❡ ❡♠ ❞❡t❡r♠✐♥❛❞♦s ✐♥st❛♥t❡s✱ ❞❡✈❡✲s❡ ❡♥tã♦✱ ❡st✉❞❛r ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s✳
• ❙✐st❡♠❛ ❞✐♥â♠✐❝♦ ❞✐s❝r❡t♦✿ q✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧ ♥ é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✳ ◆♦r✲ ♠❛❧♠❡♥t❡✱ t♦♠❛✲s❡ n∈Z+✱ ♦✉ s❡❥❛✱ ❛ss✉♠❡✲s❡ q✉❡ ♥ é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♥ã♦✲
♥❡❣❛t✐✈♦✳ ❆ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ t❡♠♣♦ ❞✐s❝r❡t♦ é ❣♦✈❡r♥❛❞❛ ♣♦r ✉♠❛ ♦✉ ♠❛✐s ❡q✉❛çõ❡s✱ q✉❡ r❡❧❛❝✐♦♥❛♠ ♦ ✈❛❧♦r ❞❛ ✈❛r✐á✈❡❧ x♥♦ ✐♥st❛♥t❡n ∈Z+ ❛ ✈❛❧♦r❡s
❞❡ ① ❡♠ ♦✉tr♦s ✐♥st❛♥t❡s✱ t❛✐s ❝♦♠♦✿ n+ 1✱ n+ 2✱ n+ 3✳
• ❙✐st❡♠❛ ❞✐♥â♠✐❝♦ ❝♦♥tí♥✉♦✿ q✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧ ♥ é ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ◆♦r♠❛❧✲ ♠❡♥t❡✱ t♦♠❛✲s❡ n∈R+✱ ♦✉ s❡❥❛✱ ❛ss✉♠❡✲s❡ q✉❡ ♥ é ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦✲♥❡❣❛t✐✈♦✳
❆ ❡✈♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ t❡♠♣♦ ❝♦♥tí♥✉♦ é ❣♦✈❡r♥❛❞❛ ♣♦r ✉♠❛ ♦✉ ♠❛✐s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ q✉❡ r❡❧❛❝✐♦♥❛♠ ❛ ✈❛r✐á✈❡❧ x ❝♦♠ s✉❛s ❞❡r✐✈❛❞❛s✳
✷✳✶ ❙✐st❡♠❛ ❞✐♥â♠✐❝♦ ❞✐s❝r❡t♦
❊q✉❛çõ❡s ❞✐s❝r❡t❛s ❣❡r❛❧♠❡♥t❡ ❞❡s❝r❡✈❡♠ ❛ ❡✈♦❧✉çã♦ ❞❡ ✉♠ ❝❡rt♦ ❢❡♥ô♠❡♥♦ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✳
✷✵ ❚❡♦r✐❛ ●❡r❛❧ ❞❡ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s
❉❡✜♥✐çã♦ ✷✳✶✳ ❉❛❞❛ ✉♠❛ ❢✉♥çã♦ f :Z+×R→R✱ ✉♠❛ ❡❢✉❛çã♦ ❞✐s❝r❡t❛ ❞❡ ♣r✐♠❡✐r❛
♦r❞❡♠ é ❞❛❞❛ ♣♦r✿
x(n+ 1) =f(n, x(n)), ✭✷✳✶✮
♦♥❞❡ n ≥n0 (n ∈N)✱ ♣❛r❛ ❛❧❣✉♠ n0 ∈N✳
❚❡♦r❡♠❛ ✷✳✶✳✶✳ ❉❛❞❛ ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ x(n0) = x0✱ ❡①✐st❡ ❛♣❡♥❛s ✉♠❛ ú♥✐❝❛
s♦❧✉çã♦ x(n)≡x(n, n0, x0) ❞❡ (2.1)♣❛r❛ n ≥n0 t❛❧ q✉❡ x(n0, n0, x0) =x0✳
❊st❛ s♦❧✉çã♦ ♣♦❞❡ s❡r ❝♦♥str✉í❞❛ ♣♦r ✐t❡r❛çõ❡s✿
x(n0+ 1, n0, x0) = f(n0, x(n0))✱
x(n0+ 2, n0, x0) = f(n0+ 1, x(n0+ 1))✱
x(n0+ 3, n0, x0) = f(n0+ 2, x(n0+ 2))✳
●❡♥❡r❛❧✐③❛♥❞♦✱ t❡♠♦s x(n, n0, x0) =f(n−1, x(n−1, n0, x0))✳
❙❡ ❛ ❢✉♥çã♦ f ♥ã♦ ❞❡♣❡♥❞❡ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❡ n✱ ✐st♦ é✱ s❡ f : R → R✱ ❛ ❡q✉❛çã♦
♣❛ss❛ ❛ s❡r
x(n+ 1) =f(x(n)), ✭✷✳✷✮
q✉❡ é ❝❤❛♠❛❞❛ ❞❡ ❡q✉❛çã♦ ❛✉tô♥♦♠❛✳ P❛rt✐♥❞♦ ❞❡ ✉♠ ✈❛❧♦r ✐♥✐❝✐❛❧x0 ♦❜t❡♠♦s✱ ❛tr❛✈és
❞❛ r❡❧❛çã♦ (2.2)✱ ❛ s❡q✉ê♥❝✐❛
x0, f(x0), f(f(x0)), f(f(f(x0))), ...
P♦r ❝♦♥✈❡♥✐ê♥❝✐❛ s❡rã♦ ❛❞♦t❛❞❛s ❛s ♥♦t❛çõ❡s✿
f2(x
0) = f(f(x0))✱ f3(x0) = f(f(f(x0)))✱✳✳✳
❡ x(n) =xn✳ ❆ss✐♠✱
x1 =f(x0), x2 =f2(x0) = f(f(x0)), x3 =f3(x0) = f(f(f(x0))), . . . , xn=fn(x0)✱
❡♠ q✉❡ fn(x
0)é ❝❤❛♠❛❞❛ ❞❡n✲és✐♠❛ ✐t❡r❛çã♦ ❞❡x0 ❛tr❛✈és ❞❡f✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s
❛s ✐t❡r❛çõ❡s fn(x
0)✱ ♣❛r❛ n ≥ n0✱ é ❝❤❛♠❛❞♦ ❞❡ ór❜✐t❛ ❞❡ x0 ♦✉ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦
❞✐s❝r❡t❛ ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦r O(x0)✳
❆s ❡q✉❛çõ❡s ❞❛❞❛s ♣♦r xn+1−xn =g(xn)sã♦ ❝❤❛♠❛❞❛s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s
❡ sã♦ ❡q✉✐✈❛❧❡♥t❡s à (2.2) s❡ f(x) = g(x) +x✳ P♦r ✐ss♦✱ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s sã♦
❣❡r❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❛❞❛s s✐♥ô♥✐♠♦s ❞❡ ❡q✉❛çõ❡s ❞✐s❝r❡t❛s✳
❊♠ ❣❡r❛❧✱ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r ♥ã♦✲❤♦♠♦❣ê♥❡❛ ❞❡ ♦r❞❡♠ k é ❞❛❞❛ ♣♦r✿ x(n+k) +p1(n)x(n+k−1) +. . .+pk(n)x(n) =g(n), ✭✷✳✸✮
♦♥❞❡ pi(n) ❡ g(n) sã♦ ❢✉♥çõ❡s r❡❛✐s ❞❡✜♥✐❞❛s ♣❛r❛ n ≥ n0✱ i = 1, . . . , k ❡ pk(n) = 0
♣❛r❛ t♦❞♦ n ≥ n0✳ ❙❡ g(n) ❢♦r ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✱ ❡♥tã♦ ✭✷✳✸✮ s❡rá ❞✐t❛ ✉♠❛ ❡q✉❛çã♦
❤♦♠♦❣ê♥❡❛✳
❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✷✶
✷✳✷ ❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r❡s ❞❡ ♣r✐♠❡✐r❛ ♦r✲
❞❡♠
❙❡❥❛♠ a(n) ❡ g(n) ❢✉♥çõ❡s r❡❛✐s ❞❡✜♥✐❞❛s ♣❛r❛ n ≥ n0✳ ❯♠❛ ❡q✉❛çã♦ ❧✐♥❡❛r ♥ã♦
❤♦♠♦❣ê♥❡❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é ❞❛❞❛ ♣♦r✿
y(n+ 1) =a(n)y(n) +g(n), y(n0) = y0, n≥n0, ✭✷✳✹✮
❡ ❛ ❡q✉❛çã♦ ❤♦♠♦❣ê♥❡❛ ❛ss♦❝✐❛❞❛ é ❞❛❞❛ ♣♦r✿
x(n+ 1) =a(n)x(n), x(n0) = x0, n ≥n0. ✭✷✳✺✮
❊♠ ❛♠❜❛s ❛s ❡q✉❛çõ❡s✱ ❛ss✉♠✐♠♦s q✉❡ a(n)= 0✱ ♣❛r❛ n≥n0✳
❉❛❞♦x(n0) = x0✱ ♣♦❞❡♠♦s ♦❜t❡r ❛ s♦❧✉çã♦ ❞❡ ✭✷✳✺✮❛tr❛✈és ❞❡ ✐t❡r❛çõ❡s✿
x(n0+ 1) =a(n0)x(n0) =a(n0)x0,
x(n0+ 2) =a(n0+ 1)x(n0+ 1) =a(n0+ 1)a(n0)x0,
x(n0+ 3) =a(n0+ 2)x(n0+ 2) =a(n0+ 2)a(n0+ 1)a(n0)x0.
❆ss✐♠✱ ♣♦r ■♥❞✉çã♦ ❋✐♥✐t❛✱ ♣♦❞❡♠♦s ♠♦str❛r q✉❡✿
x(n) =a(n−1)a(n−2). . . a(n0)x0,
x(n) =
n−1
i=n0
a(i)
x0. ✭✷✳✻✮
❆ ú♥✐❝❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ♥ã♦ ❤♦♠♦❣ê♥❡❛ ✭✷✳✹✮♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
y(n0 + 1) =a(n0)y0+g(n0),
y(n0 + 2) =a(n0+ 1)y(n0+ 1) +g(n0+ 1)
=a(n0+ 1)a(n0)y0+a(n0 + 1)g(n0) +g(n0+ 1).
❆ss✐♠✱ ♣♦r ✐♥❞✉çã♦✱ ♣❛r❛ t♦❞♦ n∈Z+✱ s❡❣✉❡ q✉❡✿
y(n) =
n−1
i=n0
a(i)
y0+
n−1
r=n0 n−1
i=r+1
a(i)
g(r). ✭✷✳✼✮
❉❡ ❢❛t♦✱ ❛ss✉♠✐♥❞♦ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ✭✷✳✼✮s❡❥❛ ✈á❧✐❞❛ ♣❛r❛ n =k✱y(k+ 1) = a(k)y(k) + g(k)✱ ❛ss✐♠✱ ♣❡❧❛ ❢ór♠✉❧❛ ✭✷✳✼✮❡ ❝♦♥s✐❞❡r❛♥❞♦
k
i=k+1
a(i) = 1 ❡
k
i=k+1
a(i) = 0✱ t❡♠♦s✿
y(k+ 1) =a(k)
k−1
i=n0
a(i)
y0+
k−1
r=n0
a(k)
k−1
i=r+1
a(i)
g(r) +g(k)
=
k
i=n0
a(i)
y0+
k−1
r=n0
k
i=r+1
a(i) g(r) +
k
i=k+1
a(i) g(k)
=
k
i=n0
a(i)
y0+
k
r=n0
k
i=r+1
a(i) g(r).
✷✷ ❚❡♦r✐❛ ●❡r❛❧ ❞❡ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s
✷✳✷✳✶ ❈❛s♦s ❡s♣❡❝✐❛✐s ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s
❍á ❞♦✐s t✐♣♦s ❞❡ ❝❛s♦s ❡s♣❡❝✐❛✐s ❞❡ ✭✷✳✹✮ q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ❡♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s✳ ❖ ♣r✐♠❡✐r♦ é ❞❛❞♦ ♣♦r✿
y(n+ 1) =ay(n) +g(n), y(0) =y0
❡♠ q✉❡ ❛ ❢✉♥çã♦ a(n) ❞❡ ✭✷✳✹✮ é ❝♦♥st❛♥t❡ ❡ n0 = 0✳ P❛r❛ ❞❡t❡r♠✐♥❛r s✉❛ s♦❧✉çã♦✱
✉s❛♠♦s ❛ ❢ór♠✉❧❛ ✭✷✳✼✮✱
y(n) = any0+
n−1
k=0
an−k−1g(k). ✭✷✳✽✮
❖ s❡❣✉♥❞♦ ❝❛s♦ ❡s♣❡❝✐❛❧ é ❛ ❡q✉❛çã♦ ❞❛❞❛ ♣♦r✿
y(n+ 1) =ay(n) +b, y(0) =y0.
P❛r❛ ❞❡t❡r♠✐♥❛r s✉❛ s♦❧✉çã♦✱ ✉s❛♠♦s ❛ ❢ór♠✉❧❛ ✭✷✳✽✮ ❡ ♦❜t❡♠♦s✱
y(n) =
⎧ ⎨
⎩
any
0+b
an
−1 a−1
, se a= 1,
y0+bn, se a= 1.
✭✷✳✾✮
❈♦♠♦ ❡①❡♠♣❧♦ ❞❛ ✉t✐❧✐③❛çã♦ ❞❛ ❢ór♠✉❧❛ ✭✷✳✼✮ ♣❛r❛ n0 = 0✱ ❝♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦
x(n+ 1) = 2x(n) + 3n, x(1) = 0,5,
❝✉❥❛ s♦❧✉çã♦ é ❞❛❞❛ ♣♦r✿
x(n) = 1 22
n−1+
n−1
k=1
2n−k−13k
= 2n−2+ 2n−1.
n−1
k=1
3 2
k
= 2n−2+ 2n−13
2
3 2
n−1 −1
3 2 −1
= 3n−5.2n−2.
❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ ❡♥✈♦❧✈❡ ♦ ♠♦❞❡❧♦ ❞❛❞♦ ♣❡❧♦ s❡❣✉♥❞♦ ❝❛s♦✳
❯♠❛ ❝❡rt❛ ❞r♦❣❛ é ❛♣❧✐❝❛❞❛ ❡♠ ✉♠ ❝♦r♣♦ ✉♠❛ ✈❡③ ❛ ❝❛❞❛ ✹ ❤♦r❛s✳ ❙❡❥❛ D(n) ❛
s♦♠❛ ❞❛ ❞r♦❣❛ ♥♦ s✐st❡♠❛ s❛♥❣✉í♥❡♦ ♥♦ n✲és✐♠♦ ✐♥t❡r✈❛❧♦✳ ❖ ❝♦r♣♦ ❡❧✐♠✐♥❛ ❝❡rt❛ ❢r❛çã♦ p ❞❛ ❞r♦❣❛ ❞✉r❛♥t❡ ❝❛❞❛ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✳ ❙❡ ❛ q✉❛♥t✐❞❛❞❡ ❛♣❧✐❝❛❞❛ ❢♦r D0✱ t❡♠♦s✿
D(1) =D0+D0 −pD0
D(2) =D0+D1 −pD(1)
...
❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❝♦♠ ❝♦❡❞❝✐❡♥t❡s ❝♦♥st❛♥t❡s ✷✸
❆ss✐♠✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❛ ❞r♦❣❛ ♥♦ s✐st❡♠❛ s❛♥❣✉í♥❡♦ ♥♦ t❡♠♣♦ (n+ 1) é ✐❣✉❛❧ à
q✉❛♥t✐❞❛❞❡ ♥♦ t❡♠♣♦ n ♠❡♥♦s ❛ ❢r❛çã♦ p q✉❡ é ❡❧✐♠✐♥❛❞❛ ❞♦ ❝♦r♣♦✱ ♠❛✐s ❛ ♥♦❞❛
❞♦s❛❣❡♠ D0✳
❯s❛♥❞♦ ✭✷✳✾✮✱ t❡♠♦s✿
D(n) = (1−p)nD
0+D0
(1−p)n
−1 (1−p)−1
= (1−p)nD
0− (1−p)
n
D0
p + D0
p
= (1−p)nD
0 −Dpo
+ D0
p
✳
❈♦♠♦0 <1−p <1✱ ❝♦♥❝❧✉í♠♦s q✉❡✱ ❝♦♠ ♦ ♣❛ss❛r ❞♦ t❡♠♣♦✱ ♦ ❞❛❧♦r D(n) t❡♥❞❡
❛ s❡ ❡st❛❜✐❧✐③❛r ♥♦ ❞❛❧♦r D0
p ✳
✷✳✸ ❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠
❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s
❯♠❛ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r ❤♦♠♦❣ê♥❡❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s é ❞❛❞❛ ♣♦r✿
x(n+ 2) +p1x(n+ 1) +p2x(n) = 0, x(n0) = x0, n≥n0 ≥0, ✭✷✳✶✵✮
♦♥❞❡ ♦s p′
is sã♦ ❝♦♥st❛♥t❡s✱ 1≤i≤2 ❝♦♠ p2 = 0✳
◆♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ ♣r♦❞❛♠♦s q✉❡ ❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ❞✉❛s s♦❧✉çõ❡s ❞❡ ✭✷✳✶✵✮ t❛♠❜é♠ é s♦❧✉çã♦ ❞❡ ✭✷✳✶✵✮✳
❚❡♦r❡♠❛ ✷✳✸✳✶✳ ❙❡ϕ1(n)❡ϕ2(n)❢♦r❡♠ s♦❧✉çõ❡s ❞❡ ✭✷✳✶✵✮❡ s❡c1❡c2❢♦r❡♠ ❝♦♥st❛♥t❡s✱
❡♥tã♦ ❛ ❢✉♥çã♦ ϕ(n) =c1ϕ1(n) +c2ϕ2(n) t❛♠❜é♠ s❡rá s♦❧✉çã♦ ❞❡ ✭✷✳✶✵✮✳
❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡
ϕ(n+ 2) +p1ϕ(n+ 1) +p2ϕ(n) = c1[ϕ1(n+ 2) +p1ϕ1(n+ 1) +p2ϕ1(n)]
+c2[ϕ2(n+ 2) +p1ϕ2(n+ 1) +p2ϕ2(n)] = 0,
♣♦✐s ϕ1 ❡ ϕ2 sã♦ s♦❧✉çõ❡s ❞❡ ✭✷✳✶✵✮✳ ▲♦❣♦✱ ϕ t❛♠❜é♠ é s♦❧✉çã♦ ❞❡ ✭✷✳✶✵✮✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❙✳ ❊❧❛②❞✐ ❬✶❪✱ t❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ S ❞❡ t♦❞❛s ❛s s♦❧✉çõ❡s ❞❡ ✭✷✳✶✵✮
❢♦r♠❛ ✉♠ ❡s♣❛ç♦ ❞❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✷ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠ ❡s❝❛❧❛r✳ ❆ss✐♠✱ ❞❛❞❛ ✉♠❛ ❜❛s❡ ♣❛r❛ S✱ s❛❜❡♠♦s q✉❡ q✉❛❧q✉❡r s♦❧✉çã♦ ❞❡
✭✷✳✶✵✮s❡rá ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ❜❛s❡✳ ❱❡❥❛ ❊✳ ▲✳ ▲✐♠❛ ❬✸❪✳ ◆♦ ❝❛♣ít✉❧♦ ✹✱ ❡st❛ t❡♦r✐❛ s❡rá ❛❜♦r❞❛❞❛ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❞❡t❛❧❤❛❞❛✱ tr❛❜❛❧❤❛♥❞♦✲s❡ ❝♦♠ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s✳
❆ss✉♠✐♠♦s ❛❣♦r❛ q✉❡ ❛s s♦❧✉çõ❡s ❞❡ ✭✷✳✶✵✮sã♦ ❞❛ ❢♦r♠❛ λn✱ ♦♥❞❡ λ é ✉♠ ♥✳♠❡r♦
❝♦♠♣❧❡①♦✳ ❙✉❜st✐t✉✐♥❞♦ ❡st❡ ❞❛❧♦r ❡♠ ✭✷✳✶✵✮✱ t❡♠♦s✿
✷✹ ❚❡♦r✐❛ ●❡r❛❧ ❞❡ ❊q✉❛çõ❡s ❞❡ ❉✐❢❡r❡♥ç❛s
❊st❡ ♣♦❧✐♥ô♠✐♦ é ❝❤❛♠❛❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ ✭✷✳✶✵✮✱ ❡ ❛s r❛í③❡s λ sã♦
❝❤❛♠❛❞❛s ❞❡ r❛í③❡s ❝❛r❛❝t❡ríst✐❝❛s✳ ❘❡s♦❧✈❡♥❞♦ ❡st❛ ❡q✉❛çã♦✱ t❡♠♦s✿
λ1 = −
p1−
p2 1−4p2
2 ❡ λ2 =
−p1+
p2 1 −4p2
2 .
❚❡♠♦s três ❝❛s♦s ❛ ❝♦♥s✐❞❡r❛r✿ ✭✐✮ ◗✉❛♥❞♦p2
1−4p2 >0✱ t❡r❡♠♦s ❞✉❛s r❛í③❡s ❝❛r❛❝t❡ríst✐❝❛s r❡❛✐s ❡ ❞✐st✐♥t❛s✿ x1(n) =
λn
1 ❡ x2(n) = λn2✳ ❆ss✐♠✱ ✉♠❛ s♦❧✉çã♦ ♣♦❞❡rá s❡r ❞❛❞❛ ♣♦r
x(n) = a1λn1 +a2λn2,
♦♥❞❡ a1 ❡ a2 sã♦ ❝♦♥st❛♥t❡s✳
✭✐✐✮ ◗✉❛♥❞♦ p2
1−4p2 = 0✱ t❡r❡♠♦s ❞✉❛s r❛í③❡s ❝❛r❛❝t❡ríst✐❝❛s r❡❛✐s ❡ ✐❣✉❛✐s✳ ❆ss✐♠✱
λ1 =λ2 =−
p1
2✱ ❡ s❡♥❞♦x1(n) =λ
n
1 ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✷✳✶✵✮✱ t❡r❡♠♦s q✉❡ ❡♥❝♦♥tr❛r
x2(n)✳ ❙✉♣♦♥❞♦ x2(n) =nλn1 ❡ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭✷✳✶✵✮ t❡♠♦s✿
(n+ 2)λn+2
1 +p1(n+ 1)λn1+1+p2nλn1 = (λ21+p1λ1+p2)nλn+ (2λ1+p1)λn+1 = 0,
♣♦✐s✱ ❝♦♠♦λ1 é r❛✐③ ❝❛r❛❝t❡ríst✐❝❛ ❡♥tã♦ λ21+p1λ1+p2 = 0❀ ❛❧é♠ ❞✐ss♦λ1 =−
p1
2
❡♥tã♦ 2λ1 +p1 = 0✳ ▲♦❣♦✱ x2(n) t❛♠❜é♠ é s♦❧✉çã♦ ❞❡ ✭✷✳✶✵✮✳ P♦rt❛♥t♦✱ ✉♠❛
s♦❧✉çã♦ ❞❡ ✭✷✳✶✵✮ é ❞❛❞❛ ♣♦r✿
x(n) = a1x1(n) +a2x(n) = a1λn1 +a2nλn1 = (a1+na2)λn1,
♦♥❞❡ a1 ❡ a2 sã♦ ❝♦♥st❛♥t❡s✳
❖ t❡r❝❡✐r♦ ❝❛s♦ é q✉❛♥❞♦ ❛s r❛í③❡s ❝❛r❛❝t❡ríst✐❝❛s sã♦ ❝♦♠♣❧❡①❛s✳ ❆♥t❡s ❞❡ ❡♥✲ ❝♦♥tr❛r ❛ s♦❧✉çã♦ ❞❡st❡ ❝❛s♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿
❚❡♦r❡♠❛ ✷✳✸✳✷✳ ❙❡❥❛x(n) = u(n) +iv(n)✉♠❛ s♦❧✉çã♦ ❝♦♠♣❧❡①❛ ❞❛ ❡q✉❛çã♦ ❞❡
❞✐❢❡r❡♥ç❛s ✭✷✳✶✵✮✱ ♦♥❞❡ u❡v sã♦ ❢✉♥çõ❡s r❡❛✐s✳ ❊♥tã♦✱ u ❡v sã♦ s♦❧✉çõ❡s r❡❛✐s ❞❡
✭✷✳✶✵✮✳
❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ x(n) =u(n) +iv(n) é ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✷✳✶✵✮✱ ❡♥tã♦✿ u(n+ 2) +iv(n+ 2) +p1[u(n+ 1) +iv(n+ 1)] +p2[u(n) +iv(n)] = 0⇒
⇒u(n+ 2) +p1u(n+ 1) +p2u(n) +i[v(n+ 2) +p1v(n+ 1) +p2v(n)] = 0.
❆ss✐♠✱ t❡♠♦s✿
u(n+ 2) +p1u(n+ 1) +p2u(n) = 0 e
v(n+ 2) +p1v(n+ 1) +p2v(n) = 0.
❊q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❝♦♠ ❝♦❡❞❝✐❡♥t❡s ❝♦♥st❛♥t❡s ✷✺
❆♥❛❧✐s❡♠♦s ❛❣♦r❛ ♦ t❡r❝❡✐r♦ ❝❛s♦✿ ✭✐✐✐✮ ◗✉❛♥❞♦ p2
1 − 4p2 < 0✱ t❡♠♦s r❛í③❡s ❝❛r❛❝t❡ríst✐❝❛s ❝♦♠♣❧❡①❛s λ1 = α +iβ ❡
λ2 =α−iβ✱ α, β ∈R✱β = 0 ✳
❊s❝r❡✈❡♥❞♦ λ1 ❡♠ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✱ t❡♠♦s λ1 = rcos(θ) + irsen(θ)✱ ♦♥❞❡
α=rcos(θ) ❡β =rsen(θ)✱r =α2 +β2✱ θ =tan−1
β α
✱ ❝♦♠ α= 0✳
❚❡♠♦s✿
x(n) =λn1 = (rcos(θ) +irsen(θ))
n
=rn(cos(nθ) +isen(nθ)),
❛ss✐♠✱ x1(n) = rncos(nθ)❡ x2(n) = rnsen(nθ)sã♦ s♦❧✉çõ❡s r❡❛✐s ❞❡ ✭✷✳✶✵✮✳
▲♦❣♦✱ ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✷✳✶✵✮ é ❞❛❞❛ ♣♦r✿
x(n) = rn(c
1cos(nθ) +c2sen(nθ)).
✸ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞❡
❞✐❢❡r❡♥ç❛s ❛✉tô♥♦♠❛s✿ ❝❛s♦ r❡❛❧
◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❛♥❛❧✐s❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ q✉❛❧✐t❛t✐✈♦ ❞❛s s♦❧✉çõ❡s ❞❡ ✉♠ s✐st❡♠❛ ❞✐s❝r❡t♦ ❛✉tô♥♦♠♦✱
x(n+ 1) =f(x(n)), ✭✸✳✶✮
❡♠ q✉❡ f :R→R ❡ n∈Z+✳
✸✳✶ P♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦
❆ ♥♦çã♦ ❞❡ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ é ❝❡♥tr❛❧ ♥♦s ❡st✉❞♦s ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❡ ❛❧❣✉♥s s✐st❡♠❛s ❢ís✐❝♦s✳ ❊♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s ♥❛ ❇✐♦❧♦❣✐❛✱ ❊❝♦♥♦♠✐❛✱ ❋ís✐❝❛✱ ❊♥❣❡♥❤❛r✐❛✱ é ❞❡s❡❥á✈❡❧ q✉❡ t♦❞♦s ♦s ❡st❛❞♦s ❞❡ ✉♠ ❞❛❞♦ s✐st❡♠❛ t❡♥❞❛♠ ❛ ✉♠ ❡st❛❞♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ❉❡✜♥✐çã♦ ✸✳✶✳ ❯♠ ♣♦♥t♦ x∗ ♥♦ ❞♦♠í♥✐♦ ❞❡ f é ❞✐t♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ ✭✸✳✶✮ s❡
❢♦r ✉♠ ♣♦♥t♦ ✜①♦ ❞❡ f✱ ✐st♦ é✱ f(x∗) =x∗✳
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ x∗ é ✉♠❛ s♦❧✉çã♦ ❝♦♥st❛♥t❡ ❞❡ ✭✸✳✶✮✱ ♣♦✐s s❡ x(0) = x∗ ❢♦r ♦
♣♦♥t♦ ✐♥✐❝✐❛❧✱ ❡♥tã♦ x(1) = f(x∗) = x∗✱ x(2) = f(x(1)) = f(x∗) = x∗✱ ❡ ❛ss✐♠ ♣♦r
❞✐❛♥t❡✳ ●r❛✜❝❛♠❡♥t❡✱ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ é ❛ ❛❜s❝✐ss❛ ❞♦ ♣♦♥t♦ ♦♥❞❡ ♦ ❣rá✜❝♦ ❞❡
f ✐♥t❡rs❡❝t❛ ❛ ❧✐♥❤❛ ❞✐❛❣♦♥❛❧ y =x✳ P♦r ❡①❡♠♣❧♦✱ ❤á três ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ♣❛r❛ ❛
❡q✉❛çã♦
x(n+ 1) =x3(n)
♣♦✐s✱ ♣❛r❛ f(x) =x3✱ ❛ ❡q✉❛çã♦ f(x∗) =x∗✱ ♦✉ s❡❥❛ x3 =x✱ ❛♣r❡s❡♥t❛ três r❛í③❡s✱ q✉❡
sã♦ ✲✶✱✵ ❡ ✶ ✭❋✐❣✉r❛ ✸✳✶✮✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ■♥t❡r♠❡❞✐ár✐♦✱ t❡♠♦s ❛ss❡❣✉r❛❞❛ ❛ ❡①✐s✲ tê♥❝✐❛ ❞❡✱ ♣❡❧♦ ♠❡♥♦s✱ ✉♠ ♣♦♥t♦ ✜①♦ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f s♦❜ ❞❡t❡r♠✐♥❛❞❛s ❝♦♥❞✐çõ❡s✳
❱❡❥❛♠♦s✿
❚❡♦r❡♠❛ ✸✳✶✳ ✭❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ■♥t❡r♠❡❞✐ár✐♦ ✭❚❱■✮✮✳ ❙✉♣♦♥❤❛ q✉❡ f : [a, b]−→R
s❡❥❛ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡ q✉❡ y0 ❡st❡❥❛ ❡♥tr❡ f(a) ❡ f(b)✳ ❊♥tã♦✱ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s
✉♠ x0 ∈(a, b) t❛❧ q✉❡ f(x0) = y0✳
✷✽ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❛✉tô♥♦♠❛s✿ ❝❛s♦ r❡❛❧
f
yx
1.5 1.0 0.5 0.5 1.0 1.5 x
1.5
1.0
0.5 0.5 1.0 1.5
fx
❋✐❣✉r❛ ✸✳✶✿ P♦♥t♦s ✜①♦s ❞❡ f(x) = x3✳
❚❡♦r❡♠❛ ✸✳✷✳ ✭❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦✮✳ ❙✉♣♦♥❤❛ q✉❡ f : [a, b] −→ [a, b] s❡❥❛ ✉♠❛
❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❊♥tã♦ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ✜①♦ ♣❛r❛ ❢ ❡♠ [a, b]✳
❉❡♠♦♥str❛çã♦✳ ❖ r❡s✉❧t❛❞♦ ❞❡st❡ t❡♦r❡♠❛ ♥♦s ❞✐③ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡r✲ s❡❝çã♦ ❞♦ ❣rá✜❝♦ ❞❡f ❝♦♠ ❛ ❞✐❛❣♦♥❛❧ y=xé ♥ã♦ ✈❛③✐♦✳ P❛r❛ ❞❡♠♦♥str❛r♠♦s t❛❧ ❢❛t♦✱
❝♦♥s✐❞❡r❡♠♦s ❛ ❢✉♥çã♦✿ h(x) =f(x)−x✳ ◆♦t❡ q✉❡ ♦s ③❡r♦s ❞❡ h sã♦ ♦s ♣♦♥t♦s ✜①♦s ❞❡ f✳
P♦r ❤✐♣ót❡s❡ t❡♠♦s✿
✭✐✮ f(a)∈[a, b]⇒a ≤f(a)≤b⇒f(a)−a≥0.
✭✐✐✮ f(b)∈[a, b]⇒a≤f(b)≤b ⇒f(b)−b ≤0.
P♦r ✭✐✮ ❡ ✭✐✐✮✱ t❡♠♦s
f(b)−b≤0≤f(a)−a ⇒h(b)≤0≤h(a)⇒0∈[h(b), h(a)]✳
❈♦♠♦ h é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ [a, b]✱ ♣❡❧♦ ❚❱■ ❡①✐st❡ ✉♠ ♣♦♥t♦ c ∈ [a, b] t❛❧
q✉❡ h(c) = 0✱ ♦✉ s❡❥❛✱ h(c) = f(c)−c= 0✳ ▲♦❣♦✱c é ♣♦♥t♦ ✜①♦ ❞❡ f ❡♠ [a, b]✳
❖❜s❡r✈❛çã♦✿ ❊st❡ t❡♦r❡♠❛ ♥ã♦ ❢♦r♥❡❝❡ ✉♠ ♠ét♦❞♦ ♣❛r❛ ❡♥❝♦♥tr❛r ♦ ♣♦♥t♦ ✜①♦✳ ❊❧❡ ❛♣❡♥❛s ❣❛r❛♥t❡ s✉❛ ❡①✐stê♥❝✐❛✱ ♦ q✉❡✱ ❛ ♣r✐♥❝í♣✐♦✱ ❥á s❡rá s✉✜❝✐❡♥t❡ ♣❛r❛ ♦s ♥♦ss♦s ♣r♦♣ós✐t♦s✳
❙❡ x∗ ❢♦r ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ ✭✸✳✶✮✱ ❛ s✉❛ ór❜✐t❛ s❡rá O(x∗) = {x∗, x∗, . . .}✱
♣♦✐s✱ ♣❛r❛ x0 =x∗✱ x1 =f(x0) =f(x∗) =x∗ ❡ ❣❡♥❡r✐❝❛♠❡♥t❡ xn =f(xn−1) =f(x∗) =
x∗ ♣❛r❛ t♦❞♦ n≥n
0✳ ❍á ✉♠ ❢❡♥ô♠❡♥♦ q✉❡ é ❡①❝❧✉s✐✈♦ ❞❡ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s ❡ ♥ã♦
P♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ✷✾
❉❡✜♥✐çã♦ ✸✳✷✳ ❙❡❥❛ x0 ✉♠ ♣♦♥t♦ ♥♦ ❞♦♠í♥✐♦ ❞❡ f✳ ❙❡ ❡①✐st❡r❡♠ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ r
❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x∗ ❞❡ ✭✸✳✶✮ t❛❧ q✉❡ fr(x
0) = x∗✱ fr−1(x0)=x∗✱ ❡♥tã♦ x0 s❡rá
✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡✈❡♥t✉❛❧✳
P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡♠♦s ❛ ❡q✉❛çã♦
x(n+ 1) =T(x(n)),
♦♥❞❡✱
T(x) =
2x para 0≤x≤ 12 2(1−x) para 12 < x≤1.
❍á ❞♦✐s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✱ 0 ❡ 2
3 ✭✈❡❥❛ ✜❣✉r❛ ✸✳✷✮✳ ❙❡ x(0) = 1
4✱ ❡♥tã♦ x(1) = 1 2✱
x(2) = 1 ❡ x(3) = 0✱ ❡ ❝♦♠♦ 0 é ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ x(4) = T(x(3)) = T(0) = 0 ❡
s✉❝❡ss✐✈❛♠❡♥t❡✱ x(n) = 0✱ ♣❛r❛ n > 4✳ ❊♥tã♦ 1
4 é ✉♠ ♣♦♥t♦ ❡q✉✐❧í❜r✐♦ ❡✈❡♥t✉❛❧ ❡ s✉❛
ór❜✐t❛ é ❞❛❞❛ ♣♦r
1 4,
1
2,1,0,0, . . .
✳
x
10.2
0.4
0.6
x
20.8
x
0.2
0.4
0.6
0.8
f
x
❋✐❣✉r❛ ✸✳✷✿ P♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ x(n+ 1) =T(x(n))✳
❯♠ ❞♦s ♣r✐♥❝✐♣❛✐s ♦❜❥❡t✐✈♦s ❞♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s é ❛♥❛❧✐s❛r ❛s s♦❧✉çõ❡s ❝✉❥♦s ✈❛❧♦r❡s ✐♥✐❝✐❛✐s ❡stã♦ ♣ró①✐♠♦s ❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✳ ❊st❡ ❡st✉❞♦ ❝♦♥s✐st❡ ♥❛ t❡♦r✐❛ ❞❡ ❡st❛❜✐❧✐❞❛❞❡✱ ❝♦♠♦ ✈❡r❡♠♦s ❛ s❡❣✉✐r✳
✸✳✶✳✶ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ P♦♥t♦s ❞❡ ❊q✉✐❧í❜r✐♦
❉❡✜♥✐çã♦ ✸✳✸✳ ✭❛✮ ❖ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x∗ ❞❡ ✭✸✳✶✮ é ❡stá✈❡❧ ✭❋✐❣✉r❛ ✸✳✸✮ s❡ ❞❛❞♦
ǫ >0✱ ❡①✐st❡ δ >0t❛❧ q✉❡ |x0−x∗|< δ ✐♠♣❧✐❝❛ |fn(x0)−x∗|< ǫ ♣❛r❛ t♦❞♦ n >0✱
✸✵ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❛✉tô♥♦♠❛s✿ ❝❛s♦ r❡❛❧
❡st❛rá ♥♦ ✐♥t❡r✈❛❧♦ ❝❡♥tr❛❞♦ ❡♠ x∗ ❝♦♠ r❛✐♦ ǫ ♣❛r❛ t♦❞♦ n > 0✳ ❙❡ x∗ ♥ã♦ ❢♦r
❡stá✈❡❧✱ ❡♥tã♦ é ❝❤❛♠❛❞♦ ✐♥stá✈❡❧ ✭❋✐❣✉r❛ ✸✳✹✮✳ ✭❜✮ ❖ ♣♦♥t♦ x∗ é ❞✐t♦ s❡r ❛tr❛t♦r s❡ ❡①✐st✐r η >0 t❛❧ q✉❡
|x0−x∗|< η implica lim
n→∞x(n) =x
∗.
❙❡ ❛ ❛✜r♠❛çã♦ ❢♦r ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ η✱ x∗ s❡rá ❝❤❛♠❛❞♦ ❛tr❛t♦r ❣❧♦❜❛❧ ♦✉ ❣❧♦❜❛❧✲
♠❡♥t❡ ❛tr❛t♦r✳
✭❝✮ ❖ ♣♦♥t♦x∗ é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ s❡ é ❡stá✈❡❧ ❡ ❛tr❛t♦r✳
✭❋✐❣✉r❛ ✸✳✺✮ ❙❡ ❛ ❛✜r♠❛çã♦ ❢♦r ✈á❧✐❞❛ ♣❛r❛ t♦❞♦ η✱ x∗ s❡rá ❞✐t♦ s❡r ❣❧♦❜❛❧♠❡♥t❡
❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧ ✭❋✐❣✉r❛ ✸✳✻✮
❖❜s❡r✈❡♠♦s q✉❡✱ ♥♦ ❝❛s♦ ❡♠ q✉❡ x∗ ♥ã♦ é ❡stá✈❡❧✱ ❡①✐st❡ǫ >0✱ t❛❧ q✉❡ ♣❛r❛ t♦❞♦δ
❡♠ q✉❡|x0−x∗|< δ ✐♠♣❧✐❝❛ |fn(x0)−x∗| ≥ǫ✱ ♣❛r❛ ❛❧❣✉♠n > 0♦✉ s❡❥❛✱ ♥ã♦ ✐♠♣♦rt❛
q✉ã♦ ♣❡rt♦ x(0) =x0 ❡st❡❥❛ ❞❡ x∗✱ ❤❛✈❡rá ✉♠ ✐♥st❛♥t❡ n >0 t❛❧ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡
x(n) ❡ x∗ é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ǫ✳
◆♦s ❣rá✜❝♦s ❛♣r❡s❡♥t❛❞♦s ❛ s❡❣✉✐r✱ ❛ ✉♥✐ã♦ ❞♦s ♣♦♥t♦s s❡rá ✉t✐❧✐③❛❞❛ ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ✈✐s✉❛❧✐③❛çã♦✱ ✉♠❛ ✈❡③ q✉❡ ♦ ❣rá✜❝♦ é ❝♦♠♣♦st♦ ❛♣❡♥❛s ❞❡ ♣♦♥t♦s ✐s♦❧❛❞♦s✳
1
2
3
4
5
6
7
8
9
10
n
x
Ε
x
Δ
x
x
Δ
x
Ε
x
0x
n
❋✐❣✉r❛ ✸✳✸✿ ❖ ♣♦♥t♦ ❡stá✈❡❧ x∗✳
P♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ✸✶
1
2
3
4
5
6
7
8
9
10
n
x
Ε
x
Δ
x
x
Δ
x
Ε
x
0x
n
❋✐❣✉r❛ ✸✳✹✿ ❖ ♣♦♥t♦ ✐♥stá✈❡❧ x∗✳
1
2
3
4
5
6
7
8
9
10
n
x
*-h
x
*x
*+h
x
H1LH
0
L
x
H2LH
0
L
x
H
n
L
✸✷ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❛✉tô♥♦♠❛s✿ ❝❛s♦ r❡❛❧
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❋✐❣✉r❛ ✸✳✻✿ ❖ ♣♦♥t♦ x∗ ❣❧♦❜❛❧♠❡♥t❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❡stá✈❡❧✳
✸✳✷ ❚❡✐❛ ❞❡ ❛r❛♥❤❛
❆ ❚❡✐❛ ❞❡ ❛r❛♥❤❛ ✭❝♦❜✇❡❜✮ é✉♠ ♠ét♦❞♦ q✉❡ ♥♦s ♣❡r♠✐t❡✱ ❡♠ ♠✉✐t♦s ❝❛s♦s✱ ✉t✐❧✐③❛r ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f ❞❡ (3.1) ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ór❜✐t❛ ❞❡ ✉♠
♣♦♥t♦✳ ❊ss❡ ♣r♦❝❡ss♦ ❣❡♦♠étr✐❝♦ ❝♦♥s✐st❡ ❡♠ ❝♦❧♦❝❛r ♥♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ ❞❡ ❡✐①♦s ❝♦✲ ♦r❞❡♥❛❞♦s ♦s ❣rá✜❝♦s ❞❡ f ❡ ❞❛ ❞✐❛❣♦♥❛❧ y =x✳ ❙❛❜❡♠♦s q✉❡ ❛ ór❜✐t❛ ❞❡ ✉♠ ♣♦♥t♦ x0
q✉❛❧q✉❡r é❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s x0, x1, x2, x3✱ ✳ ✳ ✳ ✱ ❡♠ q✉❡ xi =fi(x0)✱i∈N∗✳ ❆ss✐♠✱
t♦♠❛♠♦s ♦ ♣♦♥t♦ (x0, x0) ♥❛ ❞✐❛❣♦♥❛❧ y =x ❡✱ ❡♥tã♦✱ tr❛ç❛♠♦s ✉♠❛ r❡t❛ ✈❡rt✐❝❛❧ ♣♦r
(x0, x0) ❛té❛t✐♥❣✐r ♦ ❣rá✜❝♦ ❞❡ f✱ ❞❡ss❛ ❢♦r♠❛✱ ❞❡t❡r♠✐♥❛r❡♠♦s ♦ ♣♦♥t♦ (x0, f(x0))✳
❉❡st❡ ♣♦♥t♦✱ tr❛ç❛♠♦s ✉♠❛ r❡t❛ ❤♦r✐③♦♥t❛❧ ❛té❡♥❝♦♥tr❛r ♦ ❣rá✜❝♦ ❞❡ y =x✱ ♦❜t❡♥❞♦
❛ss✐♠✱ ♦ ♣♦♥t♦ (f(x0), f(x0))✱ q✉❡ é♦ ♣♦♥t♦ (x1, x1)✳ ❘❡♣❡t✐♥❞♦ ♦ ♣r♦❝❡ss♦✱ ♣❛r❛ x1
❡♥❝♦♥tr❛r❡♠♦s✱ ♥♦ ❣rá✜❝♦ ❞❡ f✱ ♦ ♣♦♥t♦ (f(x0), f(f(x0))) = (f(x0), f2(x0))✳ Pr♦❝❡✲
❞❡♥❞♦ ❛ss✐♠✱ ❡♥❝♦♥tr❛♠♦s t♦❞♦s ♦s ♣♦♥t♦s q✉❡ ❞❡s❡❥❛♠♦s ❞❛ ór❜✐t❛ ❞❡ x0 ❡ ♣♦❞❡♠♦s
✈✐s✉❛❧✐③❛r s❡ ❛ ór❜✐t❛ s❡ ❛♣r♦①✐♠❛ ♦✉ s❡ ❛❢❛st❛ ❞❡ ❛❧❣✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ (3.1)✳
P♦r ❡①❡♠♣❧♦✱ s❡❥❛ y(n)♦ t❛♠❛♥❤♦ ❞❡ ✉♠❛ ♣♦♣✉❧❛çã♦ ❡♠ ✉♠ t❡♠♣♦ n✱ ❞❡s❝r✐t♦ ♣♦r y(n+ 1) =µy(n), µ >0 ,
♦♥❞❡µé❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❛ ♣♦♣✉❧❛çã♦ ❞❡ ✉♠❛ ❣❡r❛çã♦ ♣❛r❛ ♦✉tr❛✳ ❙❡ ❛ ♣♦♣✉❧❛çã♦
✐♥✐❝✐❛❧ ❢♦r ❞❛❞❛ ♣♦r y(0)✱ t❡♠♦s
y(n) =µny
❚❡✐❛ ❞❡ ❛r❛♥❤❛ ✸✸
❙❡ µ > 1✱ ❡♥tã♦ y(n) ❛✉♠❡♥t❛ ✐❧✐♠✐t❛❞❛♠❡♥t❡✳ ❙❡ µ = 1✱ ❡♥tã♦ y(n) = y0 ♣❛r❛ t♦❞♦
n >0✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ ♦ t❛♠❛♥❤♦ ❞❛ ♣♦♣✉❧❛çã♦ é s❡♠♣r❡ ❝♦♥st❛♥t❡✳ ❈♦♥t✉❞♦✱ ♣❛r❛ µ <1✱ t❡♠♦s lim
n→∞y(n) = 0✱ ❡ ❛ ♣♦♣✉❧❛çã♦ s❡rá ❡①t✐♥t❛✳
◗✉❛♥❞♦ ♦s r❡❝✉rs♦s ❢♦r❡♠ ❧✐♠✐t❛❞♦s✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❛ ♣♦♣✉✲ ❧❛çã♦ ♥✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ✉♥✐tár✐♦ s❡rá r❡❞✉③✐❞♦ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ q✉❛❞r❛❞♦ ❞❛ ♣♦♣✉❧❛çã♦ ❡①✐st❡♥t❡ ♥♦ ✐♥í❝✐♦ ❞♦ ✐♥t❡r✈❛❧♦✳ ❉❡ ❢❛t♦✱ s❡ ❡①✐st✐r ✉♠❛ ❝♦♠♣❡t✐çã♦ ❡♥tr❡ ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ♠❡s♠❛ ❡s♣é❝✐❡✱ ♦ t❡r♠♦ ❞❡ ✐♥✐❜✐çã♦ ❞♦ ❝r❡s❝✐♠❡♥t♦ ♣♦♣✉❧❛❝✐♦♥❛❧ s❡rá ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♣r♦❞✉t♦ ❞❡st❡s ❡❧❡♠❡♥t♦s✱ ❛ss✐♠ t❡♠♦s✿
y(n+ 1) =µy(n)−by2(n).
❈❤❛♠❛♥❞♦ ❞❡ x(n) = b
μy(n)✱ ♦❜t❡♠♦s✿
x(n+ 1) =µx(n)(1−x(n)) =f(x(n)). ✭✸✳✷✮
♦♥❞❡ f(x) = µx(1−x)✱x∈R✳
❊st❛ ❡q✉❛çã♦ é ✉♠❛ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥ã♦ ❧✐♥❡❛r✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❧♦❣íst✐❝❛ ✭❞✐s❝r❡t❛✮✳ P❛r❛ ❡♥❝♦♥tr❛r ♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ ✭✸✳✷✮✱ r❡s♦❧✈❡♠♦s ❛ ❡q✉❛çã♦f(x∗) = µx∗(1−x∗) = x∗✱ ❡ ❛ss✐♠✱ t❡♠♦s ❞♦✐s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦✱
x∗ = 0 ❡ x∗ = (μ−1)
μ ✳ ❆ ✜❣✉r❛ ✭✸✳✼✮ ♠♦str❛ ❛ t❡✐❛ ❞❡ ❛r❛♥❤❛ r❡❢❡r❡♥t❡ à ❡q✉❛çã♦ ✭✸✳✷✮
q✉❛♥❞♦ µ= 2,5 ❡x(0) = 0,1✳
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❋✐❣✉r❛ ✸✳✼✿ ❚❡✐❛ ❞❡ ❛r❛♥❤❛ ❞❡ x(n+ 1) =µx(n)(1−x(n))♣❛r❛ µ= 2,5✳
◆❡st❡ ❝❛s♦✱ ❛ ✜❣✉r❛ s✉❣❡r❡ q✉❡ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ x∗ = 0 é ✐♥stá✈❡❧ ❡ x∗ = 0,6
✸✹ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❛✉tô♥♦♠❛s✿ ❝❛s♦ r❡❛❧
❊①❡♠♣❧♦ ✶✳ ❖ ❢❡♥ô♠❡♥♦ ❞❛ t❡✐❛ ❞❡ ❛r❛♥❤❛ ❡♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ❡❝♦♥ô♠✐❝❛ ◆♦ ♠❡r❝❛❞♦ ✜♥❛♥❝❡✐r♦✱ ❛s ❞❡❝✐sõ❡s ❞♦s ♣r♦❞✉t♦r❡s q✉❛♥t♦ às q✉❛♥t✐❞❛❞❡s ❛ ♣r♦❞✉③✐r sã♦ t♦♠❛❞❛s ♥✉♠ ♣❡rí♦❞♦ ❛♥t❡s ❞❛ s✉❛ ✈❡♥❞❛✱ ♦✉ s❡❥❛✱ ❛ ♦❢❡rt❛ ❝♦rr❡♥t❡ ❞❡♣❡♥❞❡ ❞♦ ♣r❡ç♦ ❞♦ ❛♥♦ ❛♥t❡r✐♦r✳ ❉❡st❡ ♠♦❞♦✱ ❡st❛♠♦s ❞✐❛♥t❡ ❞❡ ✉♠ ♠♦❞❡❧♦ ✉s❛♥❞♦ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s✳ ❊st✉❞❛♥❞♦ ♦s ♣r❡ç♦s ❞❡ ✉♠❛ ❝❡rt❛ ♠❡r❝❛❞♦r✐❛✱ s❡❥❛ S(n) ♦ ♥ú♠❡r♦
❞❡ ✉♥✐❞❛❞❡s ♦❢❡r❡❝✐❞❛s✱ D(n) ♦ ♥ú♠❡r♦ ❞❡ ✉♥✐❞❛❞❡s ♣r♦❝✉r❛❞❛s ❡ p(n) ♦ ♣r❡ç♦ ♣♦r
✉♥✐❞❛❞❡ ❡♠ ✉♠ ♣❡rí♦❞♦ n✳ P❛rt✐♠♦s ❞♦ ♣r✐♥❝í♣✐♦ q✉❡ ❛ ♣r♦❝✉r❛ ✭❉✮ r❡❛❣❡ ❛♦ ♣r❡ç♦
✭♣✮ ✐♥st❛♥t❛♥❡❛♠❡♥t❡✱ ✐st♦ é✱ ❛ ♣r♦❝✉r❛ ❡♠ ✉♠ ❞❛❞♦ ♣❡rí♦❞♦ é ✉♠❛ ❢✉♥çã♦ ❞♦ ♣r❡ç♦ ♥❡ss❡ ♠❡s♠♦ ♣❡rí♦❞♦✱ D(n+ 1) =g(p(n+ 1))✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ♦❢❡rt❛ ✭❙✮ r❡❛❣❡ ❝♦♠ ♦
❞❡s❢❛s❛♠❡♥t♦ ❞❡ ✉♠ ♣❡rí♦❞♦✱ ✐st♦ é✱ ❛ ♦❢❡rt❛ ♥✉♠ ❞❛❞♦ ♣❡rí♦❞♦ é ✉♠❛ ❢✉♥çã♦ ❞♦ ♣r❡ç♦ ♥♦ ♣❡rí♦❞♦ ❛♥t❡r✐♦r✱ S(n+ 1) =h(p(n))✳
P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ❛❞♦t❛r❡♠♦s q✉❡ D(n)❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞❡p(n)❡ é ❞❡♥♦t❛❞♦ ♣♦r✿ D(n) =−mdp(n) +bd, md>0, bd >0.
❊st❛ ❡q✉❛çã♦ r❡❢❡r❡✲s❡ à r❡❧❛çã♦ ♣r❡ç♦✲♣r♦❝✉r❛ ❡ ✐♥❞✐❝❛ q✉❡ ✉♠ ❛✉♠❡♥t♦ ❞❡ ✉♠❛ ✉♥✐✲ ❞❛❞❡ ♥♦ ♣r❡ç♦ ♣r♦❞✉③ ✉♠❛ ❞✐♠✐♥✉✐çã♦ ❞❡ md ✉♥✐❞❛❞❡s ♥❛ ♣r♦❝✉r❛✱ ❝r✐❛♥❞♦ ✉♠❛ ❝✉r✈❛
❞❡ ✐♥❝❧✐♥❛çã♦ ♥❡❣❛t✐✈❛✳ ❚❛♠❜é♠ ❛ss✉♠✐♠♦s q✉❡ ❛ r❡❧❛çã♦ ♣r❡ç♦✲♦❢❡rt❛ r❡❧❛t❛ ♦ ❢♦r♥❡✲ ❝✐♠❡♥t♦ ❡♠ ❛❧❣✉♠ ♣❡rí♦❞♦ ♣❛r❛ ♦ ♣r❡ç♦ ❡♠ ✉♠ ♣❡rí♦❞♦ ❛♥t❡r✐♦r✱ ✐st♦ é✱
S(n+ 1) =msp(n) +bs.
❆ ❝♦♥st❛♥t❡ ms é ♣♦s✐t✐✈❛✱ ♣♦✐s ✉♠ ❛✉♠❡♥t♦ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ♥♦ ♣r❡ç♦ ❝❛✉s❛ ✉♠
❛✉♠❡♥t♦ ❞❡ ms ✉♥✐❞❛❞❡s ♥♦ ❢♦r♥❡❝✐♠❡♥t♦✱ ❝r✐❛♥❞♦ ✉♠❛ ❝✉r✈❛ ❞❡ ✐♥❝❧✐♥❛çã♦ ♣♦s✐t✐✈❛✳
P♦rt❛♥t♦✱ ❝♦♠♦ ❛ ♦❢❡rt❛ ♥♦ ♣❡rí♦❞♦ n+ 1❞❡♣❡♥❞❡ ❞♦ ♣r❡ç♦ ♥♦ ♣❡rí♦❞♦ ♣r❡❝❡❞❡♥t❡ n✱
✉♠ ❢❛❜r✐❝❛♥t❡ s❡rá t❡♥t❛❞♦ ❛ ♣r♦❞✉③✐r ♠❛✐s s❡ ♦ ♣r❡ç♦ ♥❛ é♣♦❝❛ ❛♥t❡r✐♦r s❡ ❡st❛❜❡❧❡❝❡✉ ❛ ✉♠ ♥í✈❡❧ ❡❧❡✈❛❞♦✳ ❆ ♣r♦❝✉r❛ ♥♦ ♣❡rí♦❞♦n+1❞❡♣❡♥❞❡ ❞♦ ♣r❡ç♦ ♥❡st❡ ♠❡s♠♦ ♣❡rí♦❞♦✳
❆❞♠✐t✐♥❞♦ q✉❡✱ ❡♠ ❝❛❞❛ ♣❡rí♦❞♦✱ ♦ ♠❡r❝❛❞♦ ❞❡t❡r♠✐♥❛ ♦ ♣r❡ç♦ ❞❡ t❛❧ ♠♦❞♦ q✉❡ ❡st❡ t♦r♥❛ ❛ ♣r♦❝✉r❛ ✐❣✉❛❧ ❛ ♦❢❡rt❛✱ ♦✉ s❡❥❛✱ ❛ ♣r♦❝✉r❛ ❛❜s♦r✈❡ ❡①❛t❛♠❡♥t❡ ❛s q✉❛♥t✐❞❛❞❡s ♦❢❡r❡❝✐❞❛s✱ t❡♠♦s✿
D(n+ 1) =S(n+ 1)
−mdp(n+ 1) +bd=msp(n) +bs,
♦✉
p(n+ 1) =Ap(n) +B =f(p(n))✱
✭✸✳✸✮ ♦♥❞❡ A=−ms
md, B =
bd−bs
md . ❡ f(x) =Ax+B✳
❊st❛ ❡q✉❛çã♦ é ✉♠❛ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s ❧✐♥❡❛r ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❖ ♣r❡ç♦ ❞❡ ❡q✉✐❧í❜r✐♦ p∗ é ❞❡✜♥✐❞♦ ♥❛ ❡❝♦♥♦♠✐❛ ❝♦♠♦ ♦ ♣r❡ç♦ q✉❡ r❡s✉❧t❛ ❡♠ ✉♠❛ ✐♥t❡rs❡❝çã♦ ❞♦
❢♦r♥❡❝✐♠❡♥t♦ S(n+ 1)❝♦♠ ❛ ❞❡♠❛♥❞❛ D(n)✳ ❚❛♠❜é♠✱p∗ é ♦ ú♥✐❝♦ ♣♦♥t♦ ✜①♦ ❞❡ f(p)
