From f21d3e209c7e0e94627df02ce66482f268bf55f1 Mon Sep 17 00:00:00 2001
From: Qingyu Qu <52615090+ErikQQY@users.noreply.github.com>
Date: Wed, 26 Apr 2023 10:01:33 +0800
Subject: [PATCH] Add noncommensurate lyapunov exponents (#66)

* Add noncommensurate lyapunov exponents

Signed-off-by: ErikQQY <2283984853@qq.com>

* Fix comparing function

Signed-off-by: ErikQQY <2283984853@qq.com>

* Delete test file

Signed-off-by: ErikQQY <2283984853@qq.com>

* Fix CI failing

Signed-off-by: ErikQQY <2283984853@qq.com>

---------

Signed-off-by: ErikQQY <2283984853@qq.com>
---
 examples/RF.jl                                |   2 -
 .../lyapunov/rabinovich_fabrikant_lyapunov.jl |  30 +++++
 src/lyapunov.jl                               | 118 +++++++++++++++++-
 test/FOLEtest.jl                              |  17 ++-
 4 files changed, 162 insertions(+), 5 deletions(-)
 create mode 100644 examples/lyapunov/rabinovich_fabrikant_lyapunov.jl

diff --git a/examples/RF.jl b/examples/RF.jl
index 73fe610c..74f741c1 100644
--- a/examples/RF.jl
+++ b/examples/RF.jl
@@ -1,7 +1,5 @@
 using FractionalDiffEq, Plots
 
-plotlyjs()
-
 function RF(du, u, t, p)
     du[1] = u[2]*(u[3]-1+u[1]*u[1])+0.1*u[1];
     du[2] = u[1]*(3*u[3]+1-u[1]*u[1])+0.1*u[2];
diff --git a/examples/lyapunov/rabinovich_fabrikant_lyapunov.jl b/examples/lyapunov/rabinovich_fabrikant_lyapunov.jl
new file mode 100644
index 00000000..cceb1f7e
--- /dev/null
+++ b/examples/lyapunov/rabinovich_fabrikant_lyapunov.jl
@@ -0,0 +1,30 @@
+# Commensurate case
+
+using FractionalDiffEq, Plots
+
+function RF(du, u, t)
+    du[1] = u[2]*(u[3]-1+u[1]*u[1])+0.1*u[1]
+    du[2] = u[1]*(3*u[3]+1-u[1]*u[1])+0.1*u[2]
+    du[3] = -2*u[3]*(0.98+u[1]*u[2])
+end
+
+(LE, tspan)=FOLyapunov(RF, 0.999, 0, 0.02, 300, [0.1; 0.1; 0.1], 0.005, 1000)
+plot(tspan, LE[1, :])
+plot!(tspan, LE[2, :])
+plot!(tspan, LE[3, :])
+
+
+
+# Noncommensurate case
+
+using FractionalDiffEq, Plots
+
+function LE_RF_TEST(du, u, p, t)
+    du[1] = u[2]*(u[3]-1+u[1]^2) + 0.1*u[1]
+    du[2] = u[1]*(3*u[3]+1-u[1]^2) + 0.1*u[2]
+    du[3] = -2*u[3]*(0.98+u[1]*u[2])
+end
+(LE, tspan) = FOLyapunov(LE_RF_TEST, [0.995, 0.992, 0.996], 0, 0.1, 1000, [1,1,1], 0.01, 1000)
+plot(tspan, LE[1, :])
+plot!(tspan, LE[2, :])
+plot!(tspan, LE[3, :])
\ No newline at end of file
diff --git a/src/lyapunov.jl b/src/lyapunov.jl
index 4e3e1c69..8b5c02c3 100644
--- a/src/lyapunov.jl
+++ b/src/lyapunov.jl
@@ -14,8 +14,124 @@ Computing fractional order Lyapunov exponent of a fractionl order system.
 }
 ```
 """
-function FOLyapunov(fun, order, t_start, h_norm, t_end, u0, h, out)# TODO: Generate the Lyapunov exponent plot
+function FOLyapunov(fun, order, t_start, h_norm, t_end, u0, h, out)
+    if is_all_equal(order)
+        LE, tspan = commensurate_lyapunov(fun, order, t_start, h_norm, t_end, u0, h, out)
+    else
+        LE, tspan = noncommensurate_lyapunov(fun, order, t_start, h_norm, t_end, u0, h, out)
+    end
+    return LE, tspan
+end
+
+@inline function is_all_equal(order::AbstractArray)
+    length(order) < 2 && return true
+    element1 = order[1]
+    i = 2
+    @inbounds for i=2:length(order)
+        order[i] == element1 || return false
+    end
+    return true
+end
+
+function noncommensurate_lyapunov(fun, order, t_start, h_norm, t_end, u0, h, out)# TODO: Generate the Lyapunov exponent plot
+    ne::Int = length(u0) # System dimension
+
+    tspan = Float64[]
+    LE = Float64[]
+
+    # Generate extend system with jacobian
+    function extend_fun(du, u, p, t)
+        current_du = zeros(ne)
+        fun(current_du, u[1:ne], p, t)
+        extend_du = jacobian_of_fdefun(fun, t, u[1:ne], p)
+        extend_du_mul_u = extend_du*reshape(u[ne+1:end], ne, ne)
+        du[:] = [current_du[:]; extend_du_mul_u[:]]
+    end
+
+    x = zeros(Float64, ne*(ne+1))
+    x0 = zeros(Float64, ne*(ne+1))
+    c = zeros(Float64, ne)
+    gsc = zeros(Float64, ne)
+    zn = zeros(Float64, ne)
+    n_it = round(Int, (t_end-t_start)/h_norm)
+    x[1:ne] = u0
+    q = repeat(order, ne+1, 1) # fractional order of the extend system
+    for i=1:ne
+        x[(ne+1)*i]=1.0
+    end
+    t = t_start
+    LExp = zeros(ne)
+    for it=1:n_it
+        prob = FODESystem(extend_fun, q[:], x[:], (t, t+h_norm))
+        sol = solve(prob, h, PECE())
+        Y = sol.u
+        t = t+h_norm
+        Y = Y'
+
+        x = Y[size(Y, 1), :]
+        for i=1:ne
+            for j=1:ne
+                x0[ne*i+j]=x[ne*j+i]
+            end
+        end
+        zn[1] = 0.0
+        for j=1:ne
+            zn[1] = zn[1]+x0[ne*j+1]^2
+        end
+        zn[1] = sqrt(zn[1])
+        for j=1:ne
+            x0[ne*j+1] = x0[ne*j+1]/zn[1]
+        end
+        for j=2:ne
+            for k=1:(j-1)
+                gsc[k] = 0.0
+                for l=1:ne
+                    gsc[k] = gsc[k]+x0[ne*l+j]*x0[ne*l+k]
+                end
+            end
+            for k=1:ne
+                for l=1:j-1
+                    x0[ne*k+j]=x0[ne*k+j]-gsc[l]*x0[ne*k+l]
+                end
+            end
+            zn[j]=0.0
+            for k=1:ne
+                zn[j]=zn[j]+x0[ne*k+j]^2
+            end
+            zn[j]=sqrt(zn[j])
+            for k=1:ne
+                x0[ne*k+j]=x0[ne*k+j]/zn[j]
+            end
+        end
+        for k=1:ne
+            c[k]=c[k]+log(zn[k])
+            LExp[k]=c[k]/(t-t_start)
+        end
+        for i=1:ne
+            for j=1:ne
+                x[ne*j+i]=x0[ne*i+j]
+            end
+        end
+        x=transpose(x)
+        mod(it, out)==0 ? println(LExp) : nothing
+        LE = [LE; LExp]
+        tspan = [tspan; t]
+    end
+    LE = reshape(LE, ne, :)
+    return LE, tspan
+end
+
+function jacobian_of_fdefun(f, t, y, p)
+    ForwardDiff.jacobian(y) do y
+    du = similar(y)
+    f(du, y, p, t)
+    du
+    end
+end
+
+function commensurate_lyapunov(fun, order, t_start, h_norm, t_end, u0, h, out)# TODO: Generate the Lyapunov exponent plot
     ne::Int = length(u0) # System dimension
+    order = order[1] # Since this is the commensurate case, we only need one element in order array
     
     Jfdefun(t, u) = jacobian_of_fdefun(fun, t, u)
 
diff --git a/test/FOLEtest.jl b/test/FOLEtest.jl
index f73994f4..c54f5ac4 100644
--- a/test/FOLEtest.jl
+++ b/test/FOLEtest.jl
@@ -1,17 +1,30 @@
 using FractionalDiffEq
 using Test
 
-@testset "Test Lyapunov exponents" begin
+@testset "Test Commensurate Lyapunov exponents" begin
     function RF(du, u, t)
         du[1] = u[2]*(u[3]-1+u[1]*u[1])+0.1*u[1]
         du[2] = u[1]*(3*u[3]+1-u[1]*u[1])+0.1*u[2]
         du[3] = -2*u[3]*(0.98+u[1]*u[2])
     end
     
-    (LE, tspan)=FOLyapunov(RF, 0.999, 0, 0.02, 300, [0.1; 0.1; 0.1], 0.005, 1000)
+    (LE, tspan)=FOLyapunov(RF, [0.999, 0.999, 0.999], 0, 0.02, 300, [0.1; 0.1; 0.1], 0.005, 1000)
 
     # Test the latest computed(final) Lyapunov exponents
     @test isapprox(LE[end-2:end], [0.06111650166568285
     0.0038981396237095034
    -1.8324646820425692]; atol=1e-3)
 end
+
+@testset "Test Noncommensurate Lyapunov exponents" begin
+    function LE_RF_TEST(du, u, p, t)
+        du[1] = u[2]*(u[3]-1+u[1]^2) + 0.1*u[1]
+        du[2] = u[1]*(3*u[3]+1-u[1]^2) + 0.1*u[2]
+        du[3] = -2*u[3]*(0.98+u[1]*u[2])
+    end
+    (LE, tspan) = FOLyapunov(LE_RF_TEST, [0.995, 0.992, 0.996], 0, 0.1, 1000, [1,1,1], 0.01, 1000)
+    
+    @test isapprox(LE[end-2:end], [-0.0033493403496077058
+    -0.0050780674255360226
+    -1.774132087623718]; atol=1e-3)
+end
\ No newline at end of file