Ω s Q: How to find a ball such that x ( r ) B ¯ • ˆ x x ( r ) is a contraction ? T : B ¯ x ( r ) → B ¯ ν ¯ • x x ( r ) B ¯ Ball of radius r x ( r ) = ¯ x + B ( r ) B ¯ centered at 0 in the space Ω s
Ω s Q: How to find a ball such that x ( r ) B ¯ • ˆ x x ( r ) is a contraction ? T : B ¯ x ( r ) → B ¯ ν ¯ • x x ( r ) B ¯ Ball of radius r x ( r ) = ¯ x + B ( r ) B ¯ centered at 0 in the space Ω s A: : upper bounds satisfying Radii polynomials { p k ( r ) } k � − r � � � � � � � � T ν (¯ x ) − ¯ � + sup D x T ν (¯ x + b ) c ≤ p k ( r ) x � � � � ω s k k � � b,c ∈ B ( r ) k
Ω s Q: How to find a ball such that x ( r ) B ¯ • ˆ x x ( r ) is a contraction ? T : B ¯ x ( r ) → B ¯ ν ¯ • x x ( r ) B ¯ Ball of radius r x ( r ) = ¯ x + B ( r ) B ¯ centered at 0 in the space Ω s A: : upper bounds satisfying Radii polynomials { p k ( r ) } k � − r � � � � � � � � T ν (¯ x ) − ¯ � + sup D x T ν (¯ x + b ) c ≤ p k ( r ) x � � � � ω s k k � � b,c ∈ B ( r ) k Lemma: If there exists such that for all , p k ( r ) < 0 k r > 0 then there is a unique s.t. . f (ˆ x, ν ) = 0 x ( r ) x ∈ B ¯ ˆ proof. Banach fixed point theorem.
Analytic estimates to construct the polynomials Suppose there exist A 1 , A 2 , . . . , A n such that for every j ∈ { 1 , . . . , n } and every k ∈ Z d , we have that � ≤ A j � � � c ( j ) , � � k = | k 1 | s 1 · · · | k d | s d ω s k ω s k Then, for any k ∈ Z d , we get that ⌅ ⇧ n ⌥ α ( n ) � � ⇥ c (1) ∗ · · · ∗ c ( n ) ⇤ k A j . � ≤ � � ⌃ ω s � k k j =1 � � � � � � A 1 · · · A n � ⇥ c (1) ∗ · · · ∗ c ( n ) ⇤ � ⌦ c (1) k 1 · · · c ( n ) ⌦ � � = Proof. � � ≤ � k n � ω s ω s � � k � � k n k 1 k 1+ ··· + k n = k k 1+ ··· + k n = k � � k 1 ,..., k n ∈ Z d k 1 ,..., k n ∈ Z d � � ⌅ ⇧ ⌅ ⇧ n 1 ↵ ⌦ � = A j ⌃ ⌥ � ω s k 1 · · · ω s ⌃ ⌥ k n j =1 k 1+ ··· + k n = k k 1 ,..., k n ∈ Z d ⌅ ⇧ ⌅ ⇧ n d 1 ↵ ⌦ ↵ � A j = ⌃ ⌥ � ω s j j · · · ω s j ⌃ ⌥ k n k 1 j =1 j =1 k 1+ ··· + k n = k j k 1 ,..., k n ∈ Z d ⌅ ⇧ ⌅ ⇧ n d � 1 ↵ ↵ ⌦ � A j = � ω s j j · · · ω s j ⌃ ⌥ � k n k 1 j =1 j =1 k 1 j + ··· + kn ⌃ j = kj j ⌥ k 1 j ,...,k n j ∈ Z ⌅ ⇧ ⌅ ⇧ α ( n ) n d n ⌥ α ( n ) ↵ ↵ k j ↵ k A j = A j . ≤ ω s j ⌃ ⌥ ⌃ ω s k k j j =1 j =1 j =1 M. Gameiro & J.-P. L. Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs. Journal of Differential Equations , 2010.
Verifying the uniform ⇥ Radii polynomials { p k ( r, ∆ ν ) } contraction principle. : uniform contraction on [ ν 0 , ν 0 + ∆ ν ] ∃ r > 0 s.t. p k ( r, ∆ ν ) < 0 , ∀ k = ⇒ T The rigorous computational method x + ∆ ν ˙ x ν = ¯ x B x ν ( r ) f ( x, ν ) = 0 x ( r ) B ¯ • || x || s ¯ x • • ν 0 + ∆ ν ν 0 ν
Gluing the B + smooth pieces 1 B 0 B 1 • ¯ x 1 B − 1 • ¯ x 0 ν 0 ν 1 ν 2
Gluing the B + smooth pieces 1 B 0 B 1 • ¯ x 1 B − 1 • ¯ x 0 ν 0 ν 1 ν 2 { ( x, ν ) | f ( x, ν ) = 0 , ν ∈ [ ν 0 , ν 1 ] }
Gluing the B + smooth pieces 1 B 0 B 1 • ¯ x 1 B − 1 • ¯ x 0 ν 0 ν 1 ν 2 { ( x, ν ) | f ( x, ν ) = 0 , ν ∈ [ ν 0 , ν 2 ] }
Gluing the B + smooth pieces 1 B 0 B 1 • ¯ x 1 B − 1 • ¯ x 0 ν 0 ν 1 ν 2 { ( x, ν ) | f ( x, ν ) = 0 , ν ∈ [ ν 0 , ν 2 ] } • Global smooth curves of solutions. • Local uniqueness by the Banach fixed point theorem. • Proof of non existence of secondary bifurcations along the curves.
Applications • Initial value problems of ODEs (Chebyshev in time) • Boundary value problems of ODEs (Chebyshev in time) • Periodic solutions of ODEs (Fourier in time) • Connecting orbits of ODEs (Chebyshev in time + parameterization of invariant manifolds using power series) • Equilibria of PDEs (Fourier in space) • Periodic solutions of delay differential equations (Fourier in time) • Minimizers of action functionals (Chebyshev in time) • Periodic solutions of PDEs (Fourier in space and in time)
Applications • Initial value problems of ODEs (Chebyshev in time) • Boundary value problems of ODEs (Chebyshev in time) • Periodic solutions of ODEs (Fourier in time) • Connecting orbits of ODEs (Chebyshev in time + parameterization of invariant manifolds using power series) • Equilibria of PDEs (Fourier in space) • Periodic solutions of delay differential equations (Fourier in time) • Minimizers of action functionals (Chebyshev in time) • Periodic solutions of PDEs (Fourier in space and in time)
1. Homoclinic and heteroclinic orbits of ODEs (traveling waves) dx ODEs dt = f ( x ) t → ± ∞ x ( t ) = x ± ∈ R n lim x + = x − x + 6 = x − homoclinic orbit heteroclinic orbit
Rigorous Computations Connecting Orbits Compute a set of equilibria.
Rigorous Computations Connecting Orbits Compute a set of equilibria. Local representation of the invariant manifolds. Parameterization method
Rigorous Computations Connecting Orbits Compute a set of equilibria. Local representation of the invariant manifolds. Parameterization method Connecting orbits between the equilibria? Boundary value problem Chebyshev series Radii polynomials
2. Equilibria of PDEs Cahn-Hilliard 3D u t = − ∆ ( 1 ν ∆ u + u − u 3 ) , in Ω π π Ω = [0 , π ] × [0 , 1 . 001] × [0 , 1 . 002] ∂ n = ∂ ∆ u ∂ u = 0 , on ∂ Ω ∂ n 0.6 (1) (2) 0.5 (3) (4) (5) 0.4 (6) � u � (7) 0.3 0.2 0.1 0 0.5 1 1.5 2 2.5 3 3.5 ν 2 1
Systems of reaction-diffusion PDEs ∂ t x = d ∆ x + ( r 1 − a 1 ( x + y ) − b 1 z ) x + 1 1 − z − x z � � ⇥ ⇥ ⇤ y , ⌃ ⌃ ε N N ⌃ ⌃ ⌃ ⇧ ∂ t y = ( d + β N ) ∆ y + ( r 1 − a 1 ( x + y ) − b 1 z ) y − 1 1 − z − x z � � ⇥ ⇥ y , 0.4 ε N N ⌃ ⌃ ⌃ ⌃ ⌃ ∂ t z = d ∆ z + ( r 2 − b 2 ( x + y ) − a 2 z ) z. ⌅ 0.35 0.3 0.25 z (0) 0.2 0.15 0.1 0.05 d 0 0 0.005 0.01 0.015 0.02 0.025 0.03
Systems of reaction-diffusion PDEs ∂ t x = d ∆ x + ( r 1 − a 1 ( x + y ) − b 1 z ) x + 1 1 − z − x z � � ⇥ ⇥ ⇤ y , ⌃ ⌃ ε N N ⌃ ⌃ ⌃ ⇧ ∂ t y = ( d + β N ) ∆ y + ( r 1 − a 1 ( x + y ) − b 1 z ) y − 1 1 − z − x z � � ⇥ ⇥ y , 0.4 ε N N ⌃ ⌃ ⌃ ⌃ ⌃ • ∂ t z = d ∆ z + ( r 2 − b 2 ( x + y ) − a 2 z ) z. ⌅ 0.35 • 0.3 • 11 co-existing steady • states at d = 0 . 006 0.25 • z (0) 0.2 • 0.15 • • • 0.1 • • 0.05 d 0 0 0.005 0.01 0.015 0.02 0.025 0.03
3. Periodic solutions of delay equations y 0 ( t ) = F ( y ( t ) , y ( t − τ 1 ) , . . . , y ( t − τ d )) , 2 7 1.8 6 x 3 5 4 3 1.6 2 1 6 0 5 x 2 − 1 0 0.5 1 1.5 2 2.5 3 3.5 4 1.4 3 k x k ` 2 2 1 4 3.5 0 1.2 3 x 1 − 1 2.5 0 0.5 1 1.5 2 2.5 3 3.5 2 1.5 1 0.5 1 0 − 0.5 − 1 0 0.5 1 1.5 2 2.5 3 3.5 f ( x, ν ) = 0 0.8 2.5 3 3.5 4 ν y 0 ( t ) = − ν [ y ( t − τ 1 ) + y ( t − τ 2 )] [1 + y ( t )] ,
1.1 5 4 1.05 3 2 1 1 0 − 1 0 0.5 1 1.5 2 2.5 3 3.5 4 0.95 0.9 2.5 2 k x k ` 2 1.5 0.85 1 0.5 0 0.8 − 0.5 − 1 0 0.5 1 1.5 2 2.5 3 3.5 4 3 2.5 0.75 2 1.5 1 0.5 0.7 0 − 0.5 − 1 0 0.5 1 1.5 2 2.5 3 3.5 4 0.65 ν 0 0.05 0.1 0.15 0.2 0.25 y 0 ( t ) = − [2 . 425 y ( t − τ 1 ) + 2 . 425 y ( t − τ 2 ) + ν y ( t − τ 3 )] [1 + y ( t )] ,
������������������������������������������� �������������������������������� ����������� � ���������������������������������������������������������� 4. Minimizers of action functionals �������������� ������������������������������������������� �������������� Ginzburg–Landau energy: a model of superconductivity ����������������������������������������������������������������������������������� Z d φ 2 ( φ 2 − 2) + 2( φ 0 ) 2 G = G ( φ , a ) = 1 + 2 φ 2 a 2 + 2( a 0 − h e ) 2 � � ���������������������������������� dt. 2 d κ 2 � d φ > 0 � ������������������������������������������������� a � ����������������������� ���������������������������������� � ������������������������������������ � ���������������������� � �������������������������� ���������������������������������� ������������������� � �������� ��� ������������������������������ ���������������������������������� ����������������� ����������������������������������������������������������� �
������������������������������������������� ������������������������������������������� �������������������������������� �������������������������������� ����������� � ���������������������������������������������������������� ����������� � ���������������������������������������������������������� 4. Minimizers of action functionals �������������� ������������������������������������������� �������������� �������������� ������������������������������������������� �������������� Ginzburg–Landau energy: a model of superconductivity ����������������������������������������������������������������������������������� ����������������������������������������������������������������������������������� Z d φ 2 ( φ 2 − 2) + 2( φ 0 ) 2 G = G ( φ , a ) = 1 + 2 φ 2 a 2 + 2( a 0 − h e ) 2 � � ���������������������������������� ���������������������������������� dt. 2 d κ 2 � d � ������������������������������������������������� φ > 0 � ������������������������������������������������� � ����������������������� a � ����������������������� ���������������������������������� ���������������������������������� Parameters d � ������������������������������������ � ������������������������������������ h e � ���������������������� � ���������������������� κ � �������������������������� � �������������������������� ���������������������������������� ���������������������������������� ������������������� � �������� ��� ������������������� � �������� ��� ������������������������������ ������������������������������ ���������������������������������� ���������������������������������� ����������������� ����������������������������������������������������������� ����������������� ����������������������������������������������������������� � �
������������������������������������������� ������������������������������������������� �������������������������������� �������������������������������� ����������� � ���������������������������������������������������������� ����������� � ���������������������������������������������������������� 4. Minimizers of action functionals �������������� ������������������������������������������� �������������� �������������� ������������������������������������������� �������������� Ginzburg–Landau energy: a model of superconductivity ����������������������������������������������������������������������������������� ����������������������������������������������������������������������������������� Z d φ 2 ( φ 2 − 2) + 2( φ 0 ) 2 G = G ( φ , a ) = 1 + 2 φ 2 a 2 + 2( a 0 − h e ) 2 � � ���������������������������������� ���������������������������������� dt. 2 d κ 2 � d � ������������������������������������������������� φ > 0 � ������������������������������������������������� � ����������������������� a � ����������������������� κ = 0 . 3 , d = 4 Bifurcation diagram for kappa = 0.3, d = 4 ���������������������������������� ���������������������������������� Bifurcation Asym Parameters Sym 1 d � ������������������������������������ � ������������������������������������ h e � ���������������������� � ���������������������� 0.8 κ � �������������������������� � �������������������������� 0.6 φ ( d ) ���������������������������������� ���������������������������������� 0.4 Co-existence of ������������������� � �������� ��� ������������������� � �������� ��� nontrivial solutions 0.2 ������������������������������ ������������������������������ 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ���������������������������������� h h e ���������������������������������� ����������������� ����������������������������������������������������������� ����������������� ����������������������������������������������������������� � �
5. Periodic orbits of PDEs Kuramoto- Sivashinski equation ( u t = − ν u yyyy − u yy + 2 uu y (KS) u ( t, y ) = u ( t, y + 2 π ) , u ( t, − y ) = − u ( t, y )
5. Periodic orbits of PDEs Kuramoto- Sivashinski equation ( u t = − ν u yyyy − u yy + 2 uu y (KS) u ( t, y ) = u ( t, y + 2 π ) , u ( t, − y ) = − u ( t, y ) Popular model to analyze weak turbulence or spatiotemporal chaos
5. Periodic orbits of PDEs Kuramoto- Sivashinski equation ( u t = − ν u yyyy − u yy + 2 uu y (KS) u ( t, y ) = u ( t, y + 2 π ) , u ( t, − y ) = − u ( t, y ) Popular model to analyze weak turbulence or spatiotemporal chaos A common approach to study time-periodic solutions of (KS) is to construct a Poincaré map via numerical integration of the flow, and to look for fixed points of this map on a prescribed Poincaré section.
5. Periodic orbits of PDEs Kuramoto- Sivashinski equation ( u t = − ν u yyyy − u yy + 2 uu y (KS) u ( t, y ) = u ( t, y + 2 π ) , u ( t, − y ) = − u ( t, y ) Popular model to analyze weak turbulence or spatiotemporal chaos A common approach to study time-periodic solutions of (KS) is to construct a Poincaré map via numerical integration of the flow, and to look for fixed points of this map on a prescribed Poincaré section. Christiansen, Cvitanovic, Lan, Johnson, Jolly, Kevrekidis, Putkaradze, ...
5. Periodic orbits of PDEs Kuramoto- Sivashinski equation ( u t = − ν u yyyy − u yy + 2 uu y (KS) u ( t, y ) = u ( t, y + 2 π ) , u ( t, − y ) = − u ( t, y ) Popular model to analyze weak turbulence or spatiotemporal chaos A common approach to study time-periodic solutions of (KS) is to construct a Poincaré map via numerical integration of the flow, and to look for fixed points of this map on a prescribed Poincaré section. Christiansen, Cvitanovic, Lan, Johnson, Jolly, Kevrekidis, Putkaradze, ... Goal: propose an method (based on spectral methods and fixed point theory) to rigorously compute time periodic solutions of PDEs.
Letting L = 2 π p , the time-periodic solutions of period p of (KS) can be expanded using the Fourier expansion X where for k = ( k 1 , k 2 ) ∈ Z 2 , ψ k = e iLk 1 t e ik 2 y . u ( t, y ) = c k ψ k , k ∈ Z 2
Letting L = 2 π p , the time-periodic solutions of period p of (KS) can be expanded using the Fourier expansion X where for k = ( k 1 , k 2 ) ∈ Z 2 , ψ k = e iLk 1 t e ik 2 y . u ( t, y ) = c k ψ k , k ∈ Z 2 8 k = (0 , 0) L, > > k = (0 , k 2 ) , k 2 6 = 0 b k , < x k = ✓ ◆ a k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > b k : def def = Re ( c k ) and b k = Im ( c k ) . a k
Letting L = 2 π p , the time-periodic solutions of period p of (KS) can be expanded using the Fourier expansion X where for k = ( k 1 , k 2 ) ∈ Z 2 , ψ k = e iLk 1 t e ik 2 y . u ( t, y ) = c k ψ k , k ∈ Z 2 8 k = (0 , 0) L, Unknowns > > k = (0 , k 2 ) , k 2 6 = 0 b k , < x k = ✓ ◆ a k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > b k : def def = Re ( c k ) and b k = Im ( c k ) . a k
Letting L = 2 π p , the time-periodic solutions of period p of (KS) can be expanded using the Fourier expansion X where for k = ( k 1 , k 2 ) ∈ Z 2 , ψ k = e iLk 1 t e ik 2 y . u ( t, y ) = c k ψ k , k ∈ Z 2 8 k = (0 , 0) L, Unknowns > > k = (0 , k 2 ) , k 2 6 = 0 b k , < x k = ✓ ◆ a k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > b k : F def def = Re ( c k ) and b k = Im ( c k ) . a k Plugging the space-time Fourier expansion into (KS) results in solving, for all k 2 Z 2 X X def i k 1 2 c k 1 c k 2 = µ k c k � k 2 i c k 1 c k 2 = 0 , = µ k c k � 2 h k k 1 + k 2 = k k 1 + k 2 = k def = ik 1 L + ν k 4 2 � k 2 where µ k = µ k 1 ,k 2 2 .
Letting L = 2 π p , the time-periodic solutions of period p of (KS) can be expanded using the Fourier expansion X where for k = ( k 1 , k 2 ) ∈ Z 2 , ψ k = e iLk 1 t e ik 2 y . u ( t, y ) = c k ψ k , k ∈ Z 2 8 k = (0 , 0) L, Unknowns > > k = (0 , k 2 ) , k 2 6 = 0 b k , < x k = ✓ ◆ a k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > b k : F def def = Re ( c k ) and b k = Im ( c k ) . a k Plugging the space-time Fourier expansion into (KS) results in solving, for all k 2 Z 2 X X def i k 1 2 c k 1 c k 2 = µ k c k � k 2 i c k 1 c k 2 = 0 , = µ k c k � 2 h k k 1 + k 2 = k k 1 + k 2 = k def = ik 1 L + ν k 4 2 � k 2 where µ k = µ k 1 ,k 2 2 . X def ν k 4 2 � k 2 � � = Re ( h k ) = a k � ( k 1 L ) b k + 2 k 2 f k a k 1 b k 2 , 2 k 1 + k 2 = k X def ν k 4 2 � k 2 � � = Im ( h k ) = ( k 1 L ) a k + ( a k 1 a k 2 � b k 1 b k 2 ) . g k b k � k 2 2 k 1 + k 2 = k
Letting L = 2 π p , the time-periodic solutions of period p of (KS) can be expanded using the Fourier expansion X where for k = ( k 1 , k 2 ) ∈ Z 2 , ψ k = e iLk 1 t e ik 2 y . u ( t, y ) = c k ψ k , k ∈ Z 2 8 k = (0 , 0) L, Unknowns > > k = (0 , k 2 ) , k 2 6 = 0 b k , < x k = ✓ ◆ a k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > b k : F def def = Re ( c k ) and b k = Im ( c k ) . a k Plugging the space-time Fourier expansion into (KS) results in solving, for all k 2 Z 2 X X def i k 1 2 c k 1 c k 2 = µ k c k � k 2 i c k 1 c k 2 = 0 , = µ k c k � 2 h k k 1 + k 2 = k k 1 + k 2 = k def = ik 1 L + ν k 4 2 � k 2 where µ k = µ k 1 ,k 2 2 . Functions X def ν k 4 2 � k 2 � � = Re ( h k ) = a k � ( k 1 L ) b k + 2 k 2 f k a k 1 b k 2 , 2 k 1 + k 2 = k X def ν k 4 2 � k 2 � � = Im ( h k ) = ( k 1 L ) a k + ( a k 1 a k 2 � b k 1 b k 2 ) . g k b k � k 2 2 k 1 + k 2 = k
8 k = (0 , 0) L, Unknowns > > k = (0 , k 2 ) , k 2 6 = 0 b k , < x k = ✓ ◆ a k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > b k : Defining I = { (0 , 0) } [ { k = (0 , k 2 ) | k 2 6 = 0 } [ { k = ( k 1 , k 2 ) | k 1 6 = 0 and k 2 6 = 0 } , one can identify x = { x k } k ∈ I .
8 k = (0 , 0) L, Unknowns > > k = (0 , k 2 ) , k 2 6 = 0 b k , < x k = ✓ ◆ a k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > b k : Defining I = { (0 , 0) } [ { k = (0 , k 2 ) | k 2 6 = 0 } [ { k = ( k 1 , k 2 ) | k 1 6 = 0 and k 2 6 = 0 } , X � � one can identify x = { x k } k ∈ I . Finally, let us define F = {F k } k ∈ I component-wise by 8 k = (0 , 0) η , > > k = (0 , k 2 ) , k 2 6 = 0 g k , < F k = ✓ ◆ f k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > g k :
8 k = (0 , 0) L, Unknowns > > k = (0 , k 2 ) , k 2 6 = 0 b k , < x k = ✓ ◆ a k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > b k : Defining I = { (0 , 0) } [ { k = (0 , k 2 ) | k 2 6 = 0 } [ { k = ( k 1 , k 2 ) | k 1 6 = 0 and k 2 6 = 0 } , X � � one can identify x = { x k } k ∈ I . Finally, let us define F = {F k } k ∈ I component-wise by 8 k = (0 , 0) η , > > k = (0 , k 2 ) , k 2 6 = 0 g k , < F k = ✓ ◆ f k : k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > g k : Hence, Lemma. Finding time-periodic solutions u ( t, y ) of (KS) such that η = 0 is equivalent to find x such that F ( x ) = 0.
8 k = (0 , 0) L, Unknowns > > k = (0 , k 2 ) , k 2 6 = 0 b k , < x k = ✓ ◆ a k k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > b k : Defining I = { (0 , 0) } [ { k = (0 , k 2 ) | k 2 6 = 0 } [ { k = ( k 1 , k 2 ) | k 1 6 = 0 and k 2 6 = 0 } , X � � one can identify x = { x k } k ∈ I . Finally, let us define F = {F k } k ∈ I component-wise by 8 k = (0 , 0) η , > > k = (0 , k 2 ) , k 2 6 = 0 g k , < F k = ✓ ◆ f k : k = ( k 1 , k 2 ) , k 1 6 = 0 and k 2 6 = 0 . , > > g k : Hence, Lemma. Finding time-periodic solutions u ( t, y ) of (KS) such that η = 0 is equivalent to find x such that F ( x ) = 0. To solve rigorously in a Banach space
The Banach space j � Define the one-dimensional weights ω s k by ( 1 , if k = 0 ω s def = k | k | s , if k 6 = 0 . Using the 1-d weights, define the 2-dimensional weights, given k = ( k 1 , k 2 ) 2 Z 2 , = ω s 1 k 1 ω s 2 ω s def k 2 . k They are used to define the norm ω s k | x k | ∞ , k x k s = sup k ∈ I where | x k | ∞ is the sup norm of the vector x k , which is one or two dimensional, depending on k . Define the Banach space X s = { x | k x k s < 1 } , consisting of sequences with algebraically decaying tails according to the rate s .
The Banach space j � Define the one-dimensional weights ω s k by ( 1 , if k = 0 ω s def = k | k | s , if k 6 = 0 . Using the 1-d weights, define the 2-dimensional weights, given k = ( k 1 , k 2 ) 2 Z 2 , = ω s 1 k 1 ω s 2 ω s def k 2 . k They are used to define the norm ω s k | x k | ∞ , k x k s = sup k ∈ I where | x k | ∞ is the sup norm of the vector x k , which is one or two dimensional, depending on k . Define the Banach space Banach algebra under discrete X s = { x | k x k s < 1 } , convolution consisting of sequences with algebraically decaying tails according to the rate s .
For sake of simplicity of the presentation, for k = ( k 1 , k 2 ) with k 1 6 = 0 or k 2 6 = 0, let ✓ ◆ ν k 4 2 � k 2 � k 1 L def def = ν k 4 2 � k 2 2 R k ( ν , L ) = and R 0 ,k 2 ( ν , L ) 2 , ν k 4 2 � k 2 k 1 L 2 ✓ ◆ 2 a k 1 b k 2 X def N k ( x ) = � a k 1 a k 2 + b k 1 b k 2 k 1 + k 2 = k so that one has that F k ( x, ν ) = R k ( ν , L ) x k + k 2 N k ( x ) . Z
For sake of simplicity of the presentation, for k = ( k 1 , k 2 ) with k 1 6 = 0 or k 2 6 = 0, let ✓ ◆ ν k 4 2 � k 2 � k 1 L def def = ν k 4 2 � k 2 2 R k ( ν , L ) = and R 0 ,k 2 ( ν , L ) 2 , ν k 4 2 � k 2 k 1 L 2 ✓ ◆ 2 a k 1 b k 2 X def N k ( x ) = � a k 1 a k 2 + b k 1 b k 2 k 1 + k 2 = k so that one has that F k ( x, ν ) = R k ( ν , L ) x k + k 2 N k ( x ) . Z
For sake of simplicity of the presentation, for k = ( k 1 , k 2 ) with k 1 6 = 0 or k 2 6 = 0, let ✓ ◆ ν k 4 2 � k 2 � k 1 L def def = ν k 4 2 � k 2 2 R k ( ν , L ) = and R 0 ,k 2 ( ν , L ) 2 , ν k 4 2 � k 2 k 1 L 2 ✓ ◆ 2 a k 1 b k 2 X def N k ( x ) = � a k 1 a k 2 + b k 1 b k 2 k 1 + k 2 = k so that one has that F k ( x, ν ) = R k ( ν , L ) x k + k 2 N k ( x ) . Z Lemma. (Bootstrap) Consider a fixed decay rate s > (1 , 1) and assume the existence of M > (0 , 0) such that R k ( ν , L ) is invertible for all | k | > M . If there exists x ∈ X s such that F ( x ) = 0, then x ∈ X s 0 , for all s 0 > (1 , 1).
For sake of simplicity of the presentation, for k = ( k 1 , k 2 ) with k 1 6 = 0 or k 2 6 = 0, let ✓ ◆ ν k 4 2 � k 2 � k 1 L def def = ν k 4 2 � k 2 2 R k ( ν , L ) = and R 0 ,k 2 ( ν , L ) 2 , ν k 4 2 � k 2 k 1 L 2 ✓ ◆ 2 a k 1 b k 2 X def N k ( x ) = � a k 1 a k 2 + b k 1 b k 2 k 1 + k 2 = k so that one has that F k ( x, ν ) = R k ( ν , L ) x k + k 2 N k ( x ) . Z Lemma. (Bootstrap) Consider a fixed decay rate s > (1 , 1) and assume the existence of M > (0 , 0) such that R k ( ν , L ) is invertible for all | k | > M . If there exists x ∈ X s such that F ( x ) = 0, then x ∈ X s 0 , for all s 0 > (1 , 1). Hence, we focus our attention on looking for zeros of F within a Banach space with a fixed decay rate s>(1,1).
def = { k j 2 Z | | k j | < m j } . Given m = ( m 1 , m 2 ), define F m = F m 1 ⇥ F m 2 , where F m j def Consider a Galerkin projection of F of dimension n = n ( m ) = 2 m 1 m 2 � 2 m 1 � m 2 + 2 } k ∈ F m , where F ( m ) : R n ! R n , is given component-wise by = {F ( m ) def given by F ( m ) k F ( m ) def = F k ( x F m , 0 I m ) , ( x F m ) k 2 F m . k
def = { k j 2 Z | | k j | < m j } . Given m = ( m 1 , m 2 ), define F m = F m 1 ⇥ F m 2 , where F m j def Consider a Galerkin projection of F of dimension n = n ( m ) = 2 m 1 m 2 � 2 m 1 � m 2 + 2 } k ∈ F m , where F ( m ) : R n ! R n , is given component-wise by = {F ( m ) def given by F ( m ) k F ( m ) def = F k ( x F m , 0 I m ) , ( x F m ) k 2 F m . k def x F m such that F ( m ) (ˆ x F m , 0 I m ) 2 X s . Assume that the Consider ˆ x F m ) ⇡ 0. Let ˆ = (ˆ x Jacobian matrix D F ( m ) (ˆ x F m ) is non-singular and let A m an approximation for its inverse.
def = { k j 2 Z | | k j | < m j } . Given m = ( m 1 , m 2 ), define F m = F m 1 ⇥ F m 2 , where F m j def Consider a Galerkin projection of F of dimension n = n ( m ) = 2 m 1 m 2 � 2 m 1 � m 2 + 2 } k ∈ F m , where F ( m ) : R n ! R n , is given component-wise by = {F ( m ) def given by F ( m ) k F ( m ) def = F k ( x F m , 0 I m ) , ( x F m ) k 2 F m . k def x F m such that F ( m ) (ˆ x F m , 0 I m ) 2 X s . Assume that the Consider ˆ x F m ) ⇡ 0. Let ˆ = (ˆ x Jacobian matrix D F ( m ) (ˆ x F m ) is non-singular and let A m an approximation for its inverse. Define the action of the linear operator A on x = { x k } k ∈ I component-wise by 8 h i A m ( x F m ) if k 2 F m k , < h i def A ( x ) = R k ( ν , ˆ k L ) − 1 x k , if k 62 F m . : def T ( x ) = x � A F ( x ) .
def = { k j 2 Z | | k j | < m j } . Given m = ( m 1 , m 2 ), define F m = F m 1 ⇥ F m 2 , where F m j def Consider a Galerkin projection of F of dimension n = n ( m ) = 2 m 1 m 2 � 2 m 1 � m 2 + 2 } k ∈ F m , where F ( m ) : R n ! R n , is given component-wise by = {F ( m ) def given by F ( m ) k F ( m ) def = F k ( x F m , 0 I m ) , ( x F m ) k 2 F m . k def x F m such that F ( m ) (ˆ x F m , 0 I m ) 2 X s . Assume that the Consider ˆ x F m ) ⇡ 0. Let ˆ = (ˆ x Jacobian matrix D F ( m ) (ˆ x F m ) is non-singular and let A m an approximation for its inverse. Define the action of the linear operator A on x = { x k } k ∈ I component-wise by 8 h i A m ( x F m ) if k 2 F m k , < h i def A ( x ) = R k ( ν , ˆ k L ) − 1 x k , if k 62 F m . : def T ( x ) = x � A F ( x ) . (Newton-like operator)
def = { k j 2 Z | | k j | < m j } . Given m = ( m 1 , m 2 ), define F m = F m 1 ⇥ F m 2 , where F m j def Consider a Galerkin projection of F of dimension n = n ( m ) = 2 m 1 m 2 � 2 m 1 � m 2 + 2 } k ∈ F m , where F ( m ) : R n ! R n , is given component-wise by = {F ( m ) def given by F ( m ) k F ( m ) def = F k ( x F m , 0 I m ) , ( x F m ) k 2 F m . k def x F m such that F ( m ) (ˆ x F m , 0 I m ) 2 X s . Assume that the Consider ˆ x F m ) ⇡ 0. Let ˆ = (ˆ x Jacobian matrix D F ( m ) (ˆ x F m ) is non-singular and let A m an approximation for its inverse. Define the action of the linear operator A on x = { x k } k ∈ I component-wise by 8 h i A m ( x F m ) if k 2 F m k , < h i def A ( x ) = R k ( ν , ˆ k L ) − 1 x k , if k 62 F m . : def T ( x ) = x � A F ( x ) . (Newton-like operator) F � Lemma. Consider a Galerkin projection dimension m = ( m 1 , m 2 ) and let s = ( s 1 , s 2 ) > (1 , 1) a decay rate. The solutions of F = 0 are in one to one correspondence with the fixed points of T . Also, one has that T : X s ! X s .
The rigorous continuation method is based on the notion of the radii polynomials, which provide a numerically e ffi cient way to verify that the operator T is a contraction on a small x in X s . closed ball B (ˆ x, r ) centered at the numerical approximation ˆ
The rigorous continuation method is based on the notion of the radii polynomials, which provide a numerically e ffi cient way to verify that the operator T is a contraction on a small x in X s . closed ball B (ˆ x, r ) centered at the numerical approximation ˆ Ingredients to construct the radii polynomials • Convolution estimates • Interval arithmetic • Fast Fourier transform
The rigorous continuation method is based on the notion of the radii polynomials, which provide a numerically e ffi cient way to verify that the operator T is a contraction on a small x in X s . closed ball B (ˆ x, r ) centered at the numerical approximation ˆ Ingredients to construct the radii polynomials • Convolution estimates • Interval arithmetic • Fast Fourier transform The closed ball of radius r in X s , centered at the origin, is given by � d ( k ) � r , r Y def B ( r ) = , ω s ω s k k k ∈ I where d ( k ) = 1 if k = (0 , k 2 ) and d ( k ) = 2 otherwise. The closed ball of radius r centered at ˆ x is then def B (ˆ x, r ) = ˆ x + B ( r ) .
Consider now bounds Y k and Z k for all k 2 I , such that � � ⇥ ⇤ T (ˆ x ) � ˆ x � Y k , � � k � and � � ⇥ ⇤ sup DT (ˆ x + x 1 ) x 2 � Z k ( r ) . � � k � x 1 ,x 2 ∈ B ( r ) def If there exists an r > 0 such that k Y + Z k s < r , with Y = { Y k } k ∈ I and Lemma. def Z = { Z k } k ∈ I , then T is a contraction mapping on B (ˆ x, r ) with contraction constant at most k Y + Z k s /r < 1. Furthermore, there is a unique ˜ x 2 B (ˆ x, r ) such that F (˜ x ) = 0.
Consider now bounds Y k and Z k for all k 2 I , such that � � ⇥ ⇤ T (ˆ x ) � ˆ x � Y k , � � k � and � � ⇥ ⇤ sup DT (ˆ x + x 1 ) x 2 � Z k ( r ) . � � k � x 1 ,x 2 ∈ B ( r ) def If there exists an r > 0 such that k Y + Z k s < r , with Y = { Y k } k ∈ I and Lemma. def Z = { Z k } k ∈ I , then T is a contraction mapping on B (ˆ x, r ) with contraction constant at most k Y + Z k s /r < 1. Furthermore, there is a unique ˜ x 2 B (ˆ x, r ) such that F (˜ x ) = 0. Define the finite radii polynomials { p k ( r ) } k ∈ F M by = Y k + Z k ( r ) � r def I d ( k ) , p k ( r ) ! s k and the tail radii polynomial by = ˜ def p M ( r ) ˜ Z M ( r ) � 1 .
Consider now bounds Y k and Z k for all k 2 I , such that � � ⇥ ⇤ T (ˆ x ) � ˆ x � Y k , � � k � and � � ⇥ ⇤ sup DT (ˆ x + x 1 ) x 2 � Z k ( r ) . � � k � x 1 ,x 2 ∈ B ( r ) def If there exists an r > 0 such that k Y + Z k s < r , with Y = { Y k } k ∈ I and Lemma. def Z = { Z k } k ∈ I , then T is a contraction mapping on B (ˆ x, r ) with contraction constant at most k Y + Z k s /r < 1. Furthermore, there is a unique ˜ x 2 B (ˆ x, r ) such that F (˜ x ) = 0. Define the finite radii polynomials { p k ( r ) } k ∈ F M by = Y k + Z k ( r ) � r def I d ( k ) , p k ( r ) ! s k and the tail radii polynomial by asymptotic bound s for Z in X k = ˜ def p M ( r ) ˜ Z M ( r ) � 1 .
Consider now bounds Y k and Z k for all k 2 I , such that � � ⇥ ⇤ T (ˆ x ) � ˆ x � Y k , � � k � and � � ⇥ ⇤ sup DT (ˆ x + x 1 ) x 2 � Z k ( r ) . � � k � x 1 ,x 2 ∈ B ( r ) def If there exists an r > 0 such that k Y + Z k s < r , with Y = { Y k } k ∈ I and Lemma. def Z = { Z k } k ∈ I , then T is a contraction mapping on B (ˆ x, r ) with contraction constant at most k Y + Z k s /r < 1. Furthermore, there is a unique ˜ x 2 B (ˆ x, r ) such that F (˜ x ) = 0. Define the finite radii polynomials { p k ( r ) } k ∈ F M by = Y k + Z k ( r ) � r def I d ( k ) , p k ( r ) ! s k and the tail radii polynomial by asymptotic bound s for Z in X k = ˜ def p M ( r ) ˜ Z M ( r ) � 1 . Lemma. If there exists r > 0 such that p k ( r ) < 0 for all k 2 F M and ˜ p M ( r ) < 0, then there is a unique ˜ x 2 B (ˆ x, r ) such that F (˜ x ) = 0.
Results Kuramoto- Sivashinski equation ( u t = − ν u yyyy − u yy + 2 uu y (KS) u ( t, y ) = u ( t, y + 2 π ) , u ( t, − y ) = − u ( t, y )
Results Kuramoto- Sivashinski equation ( u t = − ν u yyyy − u yy + 2 uu y (KS) u ( t, y ) = u ( t, y + 2 π ) , u ( t, − y ) = − u ( t, y ) m = (77 , 15), M = (229 , 43), s = ( 3 2 , 3 2 ) # of time Fourier modes decay rates # of space Fourier modes
Results Kuramoto- Sivashinski equation ( u t = − ν u yyyy − u yy + 2 uu y (KS) u ( t, y ) = u ( t, y + 2 π ) , u ( t, − y ) = − u ( t, y ) m = (77 , 15), M = (229 , 43), s = ( 3 2 , 3 2 ) # of time Fourier modes decay rates # of space Fourier modes ν ∈ { . 127 , . 12707 , . 12715 , . 12725 , . 12739 , . 12756 , . 12777 }
Results Kuramoto- Sivashinski equation ( u t = − ν u yyyy − u yy + 2 uu y (KS) u ( t, y ) = u ( t, y + 2 π ) , u ( t, − y ) = − u ( t, y ) m = (77 , 15), M = (229 , 43), s = ( 3 2 , 3 2 ) # of time Fourier modes decay rates # of space Fourier modes Y ν ∈ { . 127 , . 12707 , . 12715 , . 12725 , . 12739 , . 12756 , . 12777 } ∈ I # d ( k ) " − 3 × 10 − 4 , 3 × 10 − 4 Y ⊂ X ( 3 2 , 3 2 ) x ∈ B (ˆ ˜ x, r ) = ˆ x + k 3 / 2 k 3 / 2 k 3 / 2 k 3 / 2 1 2 1 2 k ∈ I
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