]add DifferentialEquations
 Updating registry at `C:\Users\pauli\.julia\registries\General`  Updating git-repo `https://github.com/JuliaRegistries/General.git`  Resolving package versions...  Installed CloseOpenIntervals ─────────────── v0.1.4  Installed PreallocationTools ─────────────── v0.1.1  Installed NLsolve ────────────────────────── v4.5.1  Installed ArnoldiMethod ──────────────────── v0.1.0  Installed DifferentialEquations ──────────── v6.19.0  Installed Symbolics ──────────────────────── v3.5.0  Installed ModelingToolkit ────────────────── v6.7.1  Installed LabelledArrays ─────────────────── v1.6.5  Installed FastBroadcast ──────────────────── v0.1.10  Installed SymbolicUtils ──────────────────── v0.17.0  Installed DiffEqCallbacks ────────────────── v2.17.0  Installed SimpleTraits ───────────────────── v0.9.4  Installed NLSolversBase ──────────────────── v7.8.1  Installed ExponentialUtilities ───────────── v1.10.1  Installed DiffEqBase ─────────────────────── v6.75.0  Installed PolyesterWeave ─────────────────── v0.1.2  Installed DimensionalPlotRecipes ─────────── v1.2.0  Installed SafeTestsets ───────────────────── v0.0.1  Installed DomainSets ─────────────────────── v0.5.9  Installed DiffEqNoiseProcess ─────────────── v5.9.0  Installed StrideArraysCore ───────────────── v0.2.7  Installed FastClosures ───────────────────── v0.3.2  Installed DiffEqFinancial ────────────────── v2.4.0  Installed Tokenize ───────────────────────── v0.5.21  Installed BoundaryValueDiffEq ────────────── v2.7.1  Installed Optim ──────────────────────────── v1.4.1  Installed CSTParser ──────────────────────── v3.2.4  Installed SparseDiffTools ────────────────── v1.17.1  Installed TimerOutputs ───────────────────── v0.5.13  Installed RecursiveArrayTools ────────────── v2.20.0  Installed Inflate ────────────────────────── v0.1.2  Installed Random123 ──────────────────────── v1.4.2  Installed ResettableStacks ───────────────── v1.1.1  Installed StochasticDiffEq ───────────────── v6.40.0  Installed ParameterizedFunctions ─────────── v5.12.1  Installed SLEEFPirates ───────────────────── v0.6.28  Installed VectorizationBase ──────────────── v0.21.18  Installed MuladdMacro ────────────────────── v0.2.2  Installed RandomNumbers ──────────────────── v1.5.3  Installed MultivariatePolynomials ────────── v0.3.18  Installed FunctionWrappers ───────────────── v1.1.2  Installed Sundials ───────────────────────── v4.6.0  Installed RuntimeGeneratedFunctions ──────── v0.5.3  Installed LayoutPointers ─────────────────── v0.1.4  Installed LineSearches ───────────────────── v7.1.1  Installed VertexSafeGraphs ───────────────── v0.1.2  Installed DEDataArrays ───────────────────── v0.2.0  Installed CPUSummary ─────────────────────── v0.1.6  Installed Bijections ─────────────────────── v0.1.3  Installed PoissonRandom ──────────────────── v0.4.0  Installed TermInterface ──────────────────── v0.1.8  Installed BitTwiddlingConvenienceFunctions ─ v0.1.1  Installed Sundials_jll ───────────────────── v5.2.0+1  Installed Unitful ────────────────────────── v1.9.1  Installed LightGraphs ────────────────────── v1.3.5  Installed PositiveFactorizations ─────────── v0.2.4  Installed DiffEqPhysics ──────────────────── v3.9.0  Installed ArrayLayouts ───────────────────── v0.7.7  Installed MutableArithmetics ─────────────── v0.2.22  Installed DiffEqJump ─────────────────────── v7.3.1  Installed TriangularSolve ────────────────── v0.1.7  Installed JuliaFormatter ─────────────────── v0.18.1  Installed BandedMatrices ─────────────────── v0.16.11  Installed CommonMark ─────────────────────── v0.8.3  Installed Polyester ──────────────────────── v0.5.4  Installed SciMLBase ──────────────────────── v1.19.2  Installed HostCPUFeatures ────────────────── v0.1.5  Installed CompositeTypes ─────────────────── v0.1.2  Installed SteadyStateDiffEq ──────────────── v1.6.6  Installed OrdinaryDiffEq ─────────────────── v5.65.0  Installed MultiScaleArrays ───────────────── v1.8.1  Installed DynamicPolynomials ─────────────── v0.3.21  Installed CEnum ──────────────────────────── v0.4.1  Installed DelayDiffEq ────────────────────── v5.31.1  Installed LoopVectorization ──────────────── v0.12.89  Updating `C:\Users\pauli\.julia\environments\v1.6\Project.toml`  [0c46a032] + DifferentialEquations v6.19.0  Updating `C:\Users\pauli\.julia\environments\v1.6\Manifest.toml`  [ec485272] + ArnoldiMethod v0.1.0  [4c555306] + ArrayLayouts v0.7.7  [aae01518] + BandedMatrices v0.16.11  [e2ed5e7c] + Bijections v0.1.3  [62783981] + BitTwiddlingConvenienceFunctions v0.1.1  [764a87c0] + BoundaryValueDiffEq v2.7.1  [fa961155] + CEnum v0.4.1  [2a0fbf3d] + CPUSummary v0.1.6  [00ebfdb7] + CSTParser v3.2.4  [fb6a15b2] + CloseOpenIntervals v0.1.4  [a80b9123] + CommonMark v0.8.3  [b152e2b5] + CompositeTypes v0.1.2  [754358af] + DEDataArrays v0.2.0  [bcd4f6db] + DelayDiffEq v5.31.1  [2b5f629d] + DiffEqBase v6.75.0  [459566f4] + DiffEqCallbacks v2.17.0  [5a0ffddc] + DiffEqFinancial v2.4.0  [c894b116] + DiffEqJump v7.3.1  [77a26b50] + DiffEqNoiseProcess v5.9.0  [055956cb] + DiffEqPhysics v3.9.0  [0c46a032] + DifferentialEquations v6.19.0  [c619ae07] + DimensionalPlotRecipes v1.2.0  [5b8099bc] + DomainSets v0.5.9  [7c1d4256] + DynamicPolynomials v0.3.21  [d4d017d3] + ExponentialUtilities v1.10.1  [7034ab61] + FastBroadcast v0.1.10  [9aa1b823] + FastClosures v0.3.2  [6a86dc24] + FiniteDiff v2.8.1  [069b7b12] + FunctionWrappers v1.1.2  [3e5b6fbb] + HostCPUFeatures v0.1.5  [0e44f5e4] + Hwloc v2.0.0  [d25df0c9] + Inflate v0.1.2  [42fd0dbc] + IterativeSolvers v0.9.1  [98e50ef6] + JuliaFormatter v0.18.1  [2ee39098] + LabelledArrays v1.6.5  [10f19ff3] + LayoutPointers v0.1.4  [093fc24a] + LightGraphs v1.3.5  [d3d80556] + LineSearches v7.1.1  [bdcacae8] + LoopVectorization v0.12.89  [d125e4d3] + ManualMemory v0.1.6  [961ee093] + ModelingToolkit v6.7.1  [46d2c3a1] + MuladdMacro v0.2.2  [f9640e96] + MultiScaleArrays v1.8.1  [102ac46a] + MultivariatePolynomials v0.3.18  [d8a4904e] + MutableArithmetics v0.2.22  [d41bc354] + NLSolversBase v7.8.1  [2774e3e8] + NLsolve v4.5.1  [8913a72c] + NonlinearSolve v0.3.11  [429524aa] + Optim v1.4.1  [1dea7af3] + OrdinaryDiffEq v5.65.0  [65888b18] + ParameterizedFunctions v5.12.1  [e409e4f3] + PoissonRandom v0.4.0  [f517fe37] + Polyester v0.5.4  [1d0040c9] + PolyesterWeave v0.1.2  [85a6dd25] + PositiveFactorizations v0.2.4  [d236fae5] + PreallocationTools v0.1.1  [74087812] + Random123 v1.4.2  [e6cf234a] + RandomNumbers v1.5.3  [731186ca] + RecursiveArrayTools v2.20.0  [f2c3362d] + RecursiveFactorization v0.2.4  [ae5879a3] + ResettableStacks v1.1.1  [7e49a35a] + RuntimeGeneratedFunctions v0.5.3  [3cdde19b] + SIMDDualNumbers v0.1.0  [94e857df] + SIMDTypes v0.1.0  [476501e8] + SLEEFPirates v0.6.28  [1bc83da4] + SafeTestsets v0.0.1  [0bca4576] + SciMLBase v1.19.2  [699a6c99] + SimpleTraits v0.9.4  [47a9eef4] + SparseDiffTools v1.17.1  [9672c7b4] + SteadyStateDiffEq v1.6.6  [789caeaf] + StochasticDiffEq v6.40.0  [7792a7ef] + StrideArraysCore v0.2.7  [c3572dad] + Sundials v4.6.0  [d1185830] + SymbolicUtils v0.17.0  [0c5d862f] + Symbolics v3.5.0  [8ea1fca8] + TermInterface v0.1.8  [8290d209] + ThreadingUtilities v0.4.6  [a759f4b9] + TimerOutputs v0.5.13  [0796e94c] + Tokenize v0.5.21  [a2a6695c] + TreeViews v0.3.0  [d5829a12] + TriangularSolve v0.1.7  [1986cc42] + Unitful v1.9.1  [3d5dd08c] + VectorizationBase v0.21.18  [19fa3120] + VertexSafeGraphs v0.1.2  [e33a78d0] + Hwloc_jll v2.5.0+0  [fb77eaff] + Sundials_jll v5.2.0+1  [bea87d4a] + SuiteSparse_jll  Building Random123 → `C:\Users\pauli\.julia\scratchspaces\44cfe95a-1eb2-52ea-b672-e2afdf69b78f\0e8b146557ad1c6deb1367655e052276690e71a3\build.log` Precompiling project...  ✓ FunctionWrappers  ✓ SafeTestsets  ✓ TermInterface  ✓ ArnoldiMethod  ✓ Bijections  ✓ CompositeTypes  ✓ PositiveFactorizations  ✓ Inflate  ✓ DimensionalPlotRecipes  ✓ TimerOutputs  ✓ FastClosures  ✓ MuladdMacro  ✓ PoissonRandom  ✓ RandomNumbers  ✓ CEnum  ✓ RuntimeGeneratedFunctions  ✓ MutableArithmetics  ✓ Tokenize  ✓ ResettableStacks  ✓ SimpleTraits  ✓ BitTwiddlingConvenienceFunctions  ✓ CloseOpenIntervals  ✓ SuiteSparse_jll  ✓ CommonMark  ✓ ArrayLayouts  ✓ ExponentialUtilities  ✓ LayoutPointers  ✓ CPUSummary  ✓ LabelledArrays  ✓ RecursiveArrayTools  ✓ Random123  ✓ MultivariatePolynomials  ✓ DomainSets  ✓ NLSolversBase  ✓ HostCPUFeatures  ✓ CSTParser  ✓ Sundials_jll  ✓ StrideArraysCore  ✓ PolyesterWeave  ✓ PreallocationTools  ✓ Unitful  ✓ LightGraphs  ✓ BandedMatrices  ✓ DynamicPolynomials  ✓ Polyester  ✓ LineSearches  ✓ SciMLBase  ✓ JuliaFormatter  ✓ VertexSafeGraphs  ✓ FastBroadcast  ✓ NLsolve  ✓ DEDataArrays  ✓ Optim  ✓ SparseDiffTools  ✓ VectorizationBase  ✓ SymbolicUtils  ✓ SLEEFPirates  ✓ SIMDDualNumbers  ✓ Symbolics  ✓ LoopVectorization  ✓ TriangularSolve  ✓ RecursiveFactorization  ✓ NonlinearSolve  ✓ DiffEqBase  ✓ DiffEqNoiseProcess  ✓ DiffEqJump  ✓ BoundaryValueDiffEq  ✓ Sundials  ✓ DiffEqFinancial  ✓ OrdinaryDiffEq  ✓ DiffEqCallbacks  ✓ DelayDiffEq  ✓ StochasticDiffEq  ✓ DiffEqPhysics  ✓ SteadyStateDiffEq  ✓ MultiScaleArrays  ✓ ModelingToolkit  ✓ ParameterizedFunctions  ✓ DifferentialEquations 79 dependencies successfully precompiled in 301 seconds (264 already precompiled, 1 skipped during auto due to previous errors) 
using DifferentialEquations, OrdinaryDiffEq, RecursiveArrayTools
using LinearAlgebra, Plots

La ecuación de Van der Pol representa un modelo ampliamente usado para procesos oscilatorios en física, electrónica, biología, neurología, sociología y economía. Esta ecuación fue introducida en 1927 para describir oscilaciones de triodo en circuitos eléctricos por el ingeniero eléctrico holandés Balthazar van der Pol.

Algunas aplicaciones de la ecuación de Van der Pol consisten en

  • El propio Van der Pol construyó varios modelos de circuitos electrónicos del corazón humano para estudiar el rango de estabilidad de la dinámica cardíaca.

  • Para modelar neuronas acopladas en el circuito del molino gástrico del ganglio estomatogástrico.

  • La ecuación de Fitzhugh-Nagumo es un campo vectorial plano que extiende la ecuación de van der Pol como modelo para los potenciales de acción de las neuronas.

  • En sismología, la ecuación de van der Pol se ha utilizado en el desarrollo de un modelo de interacción de dos placas en una falla geológica.

La ecuación de Van der Pol se define como

formula not implemented
# Parameters
μ = 5.0
# Initial conditions
x0 = [0.5]
Dx0 = [1.0]
t = (0.0, 100.0)
function VdP(Dxx, Dx, x, μ, t)
    @. Dxx = -μ*(x^2-1)*Dx - x
end
VdP_ODE = SecondOrderODEProblem(VdP, Dx0, x0, t, μ)
VdP_solution = solve(VdP_ODE)
u = VdP_solution.u
plot(VdP_solution, vars = [1,2], xaxis = "t", yaxis = "x")
#plot(VdP_solution, vars = (1,2))
#plot(u_t, u_x, xaxis = "t", yaxis = "x")
anim = @animate for i in 1:n
    
function VdP_2D(du,u,μ,t)
    du[1] = μ*(u[1] - (1/3)*(u[1])^3 - u[2])
    du[2] = (1/μ)*u[1]
end
# Initial conditions and Parameters
μ = 50.0
u0 = [2.0,1.0]
tspan = (0.0,100.0)
ODE_VdP = ODEProblem(VdP_2D, u0, tspan, μ)
sol_VdP = solve(ODE_VdP)
plot()
u_t = [u[i][1] for i in 1:length(u)]
u_x = [u[i][2] for i in 1:length(u)]
#@show VdP_solution.t;
#@show VdP_solution.u;
Runtimes (1)