<p>In this paper, we prove the existence of traveling train and impulse in FitzHugh-Nagumo system <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1371_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="225" /> </InlineMediaObject> <EquationSource Format="TEX">\(u_t=u_{xx}-u(u-1)(u-a)-v\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">xx</mi> </mrow> </msub> <mo>-</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>-</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>v</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1371_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">\(v_t=\varepsilon (u-\gamma v)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>=</mo> <mi>ε</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>-</mo> <mi>γ</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by studying a saddle-node bifurcation and a Bogdanov-Takens bifurcation of the corresponding three-dimensional system <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1371_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\({\dot{x}}=z\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1371_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">\({\dot{y}}=b(x-dy)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>d</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1371_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="210" /> </InlineMediaObject> <EquationSource Format="TEX">\({\dot{z}}=x(x-1)(x-a)+y+cz\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>z</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation>. The bifurcation analysis of the three-dimensional system indicates that the number of steady-state solutions in FitzHugh-Nagumo system can change through saddle-node bifurcation of the three-dimensional system, and there are different parameter values for which the three-dimensional system can have a limit cycle or a homoclinic loop, which implies that FitzHugh-Nagumo system can have a traveling train or an impulse for some specific parameters.</p>